xref: /dports/graphics/py-openimageio/oiio-Release-2.2.16.0/src/include/OpenImageIO/fmath.h

// Copyright 2008-present Contributors to the OpenImageIO project.
// SPDX-License-Identifier: BSD-3-Clause
// https://github.com/OpenImageIO/oiio/blob/master/LICENSE.md

/*
  A few bits here are based upon code from NVIDIA that was also released
  under the same 3-Clause BSD license, and marked as:
     Copyright 2004 NVIDIA Corporation. All Rights Reserved.

  Some parts of this file were first open-sourced in Open Shading Language,
  also 3-Clause BSD license, then later moved here. The original copyright
  notice was:
     Copyright (c) 2009-2014 Sony Pictures Imageworks Inc., et al.

  Many of the math functions were copied from or inspired by other
  public domain sources or open source packages with compatible licenses.
  The individual functions give references were applicable.
*/


// clang-format off

/// \file
///
/// A variety of floating-point math helper routines (and, slight
/// misnomer, some int stuff as well).
///


#pragma once
#define OIIO_FMATH_H 1

#include <algorithm>
#include <cmath>
#include <cstring>
#include <limits>
#include <typeinfo>
#include <type_traits>

#include <OpenImageIO/Imath.h>
#include <OpenImageIO/span.h>
#include <OpenImageIO/dassert.h>
#include <OpenImageIO/oiioversion.h>
#include <OpenImageIO/platform.h>
#include <OpenImageIO/simd.h>


OIIO_NAMESPACE_BEGIN

/// Occasionally there may be a tradeoff where the best/fastest
/// implementation of a math function in an ordinary scalar context does
/// things that prevent autovectorization (or OMP pragma vectorization) of a
/// loop containing that operation, and the only way to make it SIMD
/// friendly slows it down as a scalar op.
///
/// By default, we always prefer doing it as correctly and quickly as
/// possible in scalar mode, but if OIIO_FMATH_SIMD_FRIENDLY is defined to
/// be nonzero, we prefer ensuring that the SIMD loops are as efficient as
/// possible, even at the expense of scalar performance.
///
/// Because everything here consists of inline functions, and it doesn't
/// really matter for OIIO internals themselves, this is primarily intended
/// for the sake of 3rd party apps that happen to have OIIO as a dependency
/// and use the fmath.h functions. Such a downstream dependency just needs
/// to ensure that OIIO_FMATH_SIMD_FRIENDLY is defined to 1 prior to
/// including fmath.h.
///
#ifndef OIIO_FMATH_SIMD_FRIENDLY
#    define OIIO_FMATH_SIMD_FRIENDLY 0
#endif


// Helper template to let us tell if two types are the same.
// C++11 defines this, keep in OIIO namespace for back compat.
// DEPRECATED(2.0) -- clients should switch OIIO::is_same -> std::is_same.
using std::is_same;


// For back compatibility: expose these in the OIIO namespace.
// DEPRECATED(2.0) -- clients should switch OIIO:: -> std:: for these.
using std::isfinite;
using std::isinf;
using std::isnan;


// Define math constants just in case they aren't included (Windows is a
// little finicky about this, only defining these if _USE_MATH_DEFINES is
// defined before <cmath> is included, which is hard to control).
#ifndef M_PI
#    define M_PI 3.14159265358979323846264338327950288
#endif
#ifndef M_PI_2
#    define M_PI_2 1.57079632679489661923132169163975144
#endif
#ifndef M_PI_4
#    define M_PI_4 0.785398163397448309615660845819875721
#endif
#ifndef M_TWO_PI
#    define M_TWO_PI (M_PI * 2.0)
#endif
#ifndef M_1_PI
#    define M_1_PI 0.318309886183790671537767526745028724
#endif
#ifndef M_2_PI
#    define M_2_PI 0.636619772367581343075535053490057448
#endif
#ifndef M_SQRT2
#    define M_SQRT2 1.41421356237309504880168872420969808
#endif
#ifndef M_SQRT1_2
#    define M_SQRT1_2 0.707106781186547524400844362104849039
#endif
#ifndef M_LN2
#    define M_LN2 0.69314718055994530941723212145817656
#endif
#ifndef M_LN10
#    define M_LN10 2.30258509299404568401799145468436421
#endif
#ifndef M_E
#    define M_E 2.71828182845904523536028747135266250
#endif
#ifndef M_LOG2E
#    define M_LOG2E 1.44269504088896340735992468100189214
#endif



////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// INTEGER HELPER FUNCTIONS
//
// A variety of handy functions that operate on integers.
//

/// Quick test for whether an integer is a power of 2.
///
template<typename T>
inline OIIO_HOSTDEVICE OIIO_CONSTEXPR14 bool
ispow2(T x) noexcept
{
    // Numerous references for this bit trick are on the web.  The
    // principle is that x is a power of 2 <=> x == 1<<b <=> x-1 is
    // all 1 bits for bits < b.
    return (x & (x - 1)) == 0 && (x >= 0);
}



/// Round up to next higher power of 2 (return x if it's already a power
/// of 2).
inline OIIO_HOSTDEVICE OIIO_CONSTEXPR14 int
ceil2(int x) noexcept
{
    // Here's a version with no loops.
    if (x < 0)
        return 0;
    // Subtract 1, then round up to a power of 2, that way if we are
    // already a power of 2, we end up with the same number.
    --x;
    // Make all bits past the first 1 also be 1, i.e. 0001xxxx -> 00011111
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    // Now we have 2^n-1, by adding 1, we make it a power of 2 again
    return x + 1;
}



/// Round down to next lower power of 2 (return x if it's already a power
/// of 2).
inline OIIO_HOSTDEVICE OIIO_CONSTEXPR14 int
floor2(int x) noexcept
{
    // Make all bits past the first 1 also be 1, i.e. 0001xxxx -> 00011111
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    // Strip off all but the high bit, i.e. 00011111 -> 00010000
    // That's the power of two <= the original x
    return x & ~(x >> 1);
}


// Old names -- DEPRECATED(2.1)
inline OIIO_HOSTDEVICE int pow2roundup(int x) { return ceil2(x); }
inline OIIO_HOSTDEVICE int pow2rounddown(int x) { return floor2(x); }



/// Round value up to the next whole multiple.
/// For example, round_to_multiple(7,10) returns 10.
template <typename V, typename M>
inline OIIO_HOSTDEVICE V round_to_multiple (V value, M multiple)
{
    return V (((value + V(multiple) - 1) / V(multiple)) * V(multiple));
}



/// Round up to the next whole multiple of m, for the special case where
/// m is definitely a power of 2 (somewhat simpler than the more general
/// round_to_multiple). This is a template that should work for any
/// integer type.
template<typename T>
inline OIIO_HOSTDEVICE T
round_to_multiple_of_pow2(T x, T m)
{
    OIIO_DASSERT(ispow2(m));
    return (x + m - 1) & (~(m - 1));
}



/// Multiply two unsigned 32-bit ints safely, carefully checking for
/// overflow, and clamping to uint32_t's maximum value.
inline OIIO_HOSTDEVICE uint32_t
clamped_mult32(uint32_t a, uint32_t b)
{
    const uint32_t Err = std::numeric_limits<uint32_t>::max();
    uint64_t r = (uint64_t)a * (uint64_t)b;  // Multiply into a bigger int
    return r < Err ? (uint32_t)r : Err;
}



/// Multiply two unsigned 64-bit ints safely, carefully checking for
/// overflow, and clamping to uint64_t's maximum value.
inline OIIO_HOSTDEVICE uint64_t
clamped_mult64(uint64_t a, uint64_t b)
{
    uint64_t ab = a * b;
    if (b && ab / b != a)
        return std::numeric_limits<uint64_t>::max();
    else
        return ab;
}



/// Bitwise circular rotation left by `s` bits (for any unsigned integer
/// type).  For info on the C++20 std::rotl(), see
/// http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2019/p0553r4.html
// FIXME: this should be constexpr, but we're leaving that out for now
// because the Cuda specialization uses an intrinsic that isn't constexpr.
// Come back to this later when more of the Cuda library is properly
// constexpr.
template<class T>
OIIO_NODISCARD OIIO_FORCEINLINE OIIO_HOSTDEVICE
// constexpr
T rotl(T x, int s) noexcept
{
    static_assert(std::is_unsigned<T>::value && std::is_integral<T>::value,
                  "rotl only works for unsigned integer types");
    return (x << s) | (x >> ((sizeof(T) * 8) - s));
}


#if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 320
// Cuda has an intrinsic for 32 bit unsigned int rotation
// FIXME: This should be constexpr, but __funnelshift_lc seems not to be
// marked as such.
template<>
OIIO_NODISCARD OIIO_FORCEINLINE OIIO_HOSTDEVICE
// constexpr
uint32_t rotl(uint32_t x, int s) noexcept
{
    return __funnelshift_lc(x, x, s);
}
#endif



// Old names -- DEPRECATED(2.1)
OIIO_FORCEINLINE OIIO_HOSTDEVICE uint32_t
rotl32(uint32_t x, int k)
{
#if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 320
    return __funnelshift_lc(x, x, k);
#else
    return (x << k) | (x >> (32 - k));
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE uint64_t
rotl64(uint64_t x, int k)
{
    return (x << k) | (x >> (64 - k));
}



// safe_mod is like integer a%b, but safely returns 0 when b==0.
template <class T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
safe_mod(T a, T b)
{
    return b ? (a % b) : T(0);
}

// (end of integer helper functions)
////////////////////////////////////////////////////////////////////////////



////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// FLOAT UTILITY FUNCTIONS
//
// Helper/utility functions: clamps, blends, interpolations...
//


/// clamp a to bounds [low,high].
template <class T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
clamp (const T& a, const T& low, const T& high)
{
#if 1
    // This looks clunky, but it generates minimal code. For float, it
    // should result in just a max and min instruction, thats it.
    // This implementation is courtesy of Alex Wells, Intel, via OSL.
    T val = a;
    if (!(low <= val))  // Forces clamp(NaN,low,high) to return low
        val = low;
    if (val > high)
        val = high;
    return val;
#else
    // The naive implementation we originally had, below, only does the 2nd
    // comparison in the else block, which will generate extra code in a
    // SIMD masking scenario. I (LG) can confirm that benchmarks show that
    // the above implementation is indeed faster than the one below, even
    // for non-SIMD use.
    return (a >= low) ? ((a <= high) ? a : high) : low;
#endif
}


#ifndef __CUDA_ARCH__
// Specialization of clamp for vfloat4
template<> OIIO_FORCEINLINE simd::vfloat4
clamp (const simd::vfloat4& a, const simd::vfloat4& low, const simd::vfloat4& high)
{
    return simd::min (high, simd::max (low, a));
}

template<> OIIO_FORCEINLINE simd::vfloat8
clamp (const simd::vfloat8& a, const simd::vfloat8& low, const simd::vfloat8& high)
{
    return simd::min (high, simd::max (low, a));
}

template<> OIIO_FORCEINLINE simd::vfloat16
clamp (const simd::vfloat16& a, const simd::vfloat16& low, const simd::vfloat16& high)
{
    return simd::min (high, simd::max (low, a));
}

// Specialization of clamp for vint4
template<> OIIO_FORCEINLINE simd::vint4
clamp (const simd::vint4& a, const simd::vint4& low, const simd::vint4& high)
{
    return simd::min (high, simd::max (low, a));
}

template<> OIIO_FORCEINLINE simd::vint8
clamp (const simd::vint8& a, const simd::vint8& low, const simd::vint8& high)
{
    return simd::min (high, simd::max (low, a));
}

template<> OIIO_FORCEINLINE simd::vint16
clamp (const simd::vint16& a, const simd::vint16& low, const simd::vint16& high)
{
    return simd::min (high, simd::max (low, a));
}
#endif



// For the multply+add (or sub) operations below, note that the results may
// differ slightly on different hardware, depending on whether true fused
// multiply and add is available or if the code generated just does an old
// fashioned multiply followed by a separate add. So please interpret these
// as "do a multiply and add as fast as possible for this hardware, with at
// least as much precision as a multiply followed by a separate add."

/// Fused multiply and add: (a*b + c)
OIIO_FORCEINLINE OIIO_HOSTDEVICE float madd (float a, float b, float c) {
    // NOTE: GCC/ICC will turn this (for float) into a FMA unless
    // explicitly asked not to, clang will do so if -ffp-contract=fast.
    OIIO_CLANG_PRAGMA(clang fp contract(fast))
    return a * b + c;
}


/// Fused multiply and subtract: (a*b - c)
OIIO_FORCEINLINE OIIO_HOSTDEVICE float msub (float a, float b, float c) {
    OIIO_CLANG_PRAGMA(clang fp contract(fast))
    return a * b - c;
}



/// Fused negative multiply and add: -(a*b) + c
OIIO_FORCEINLINE OIIO_HOSTDEVICE float nmadd (float a, float b, float c) {
    OIIO_CLANG_PRAGMA(clang fp contract(fast))
    return c - (a * b);
}



/// Negative fused multiply and subtract: -(a*b) - c
OIIO_FORCEINLINE OIIO_HOSTDEVICE float nmsub (float a, float b, float c) {
    OIIO_CLANG_PRAGMA(clang fp contract(fast))
    return -(a * b) - c;
}



/// Linearly interpolate values v0-v1 at x: v0*(1-x) + v1*x.
/// This is a template, and so should work for any types.
template <class T, class Q>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
lerp (const T& v0, const T& v1, const Q& x)
{
    // NOTE: a*(1-x) + b*x is much more numerically stable than a+x*(b-a)
    return v0*(Q(1)-x) + v1*x;
}



/// Bilinearly interoplate values v0-v3 (v0 upper left, v1 upper right,
/// v2 lower left, v3 lower right) at coordinates (s,t) and return the
/// result.  This is a template, and so should work for any types.
template <class T, class Q>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
bilerp(const T& v0, const T& v1, const T& v2, const T& v3, const Q& s, const Q& t)
{
    // NOTE: a*(t-1) + b*t is much more numerically stable than a+t*(b-a)
    Q s1 = Q(1) - s;
    return T ((Q(1)-t)*(v0*s1 + v1*s) + t*(v2*s1 + v3*s));
}



/// Bilinearly interoplate arrays of values v0-v3 (v0 upper left, v1
/// upper right, v2 lower left, v3 lower right) at coordinates (s,t),
/// storing the results in 'result'.  These are all vectors, so do it
/// for each of 'n' contiguous values (using the same s,t interpolants).
template <class T, class Q>
inline OIIO_HOSTDEVICE void
bilerp (const T *v0, const T *v1,
        const T *v2, const T *v3,
        Q s, Q t, int n, T *result)
{
    Q s1 = Q(1) - s;
    Q t1 = Q(1) - t;
    for (int i = 0;  i < n;  ++i)
        result[i] = T (t1*(v0[i]*s1 + v1[i]*s) + t*(v2[i]*s1 + v3[i]*s));
}



/// Bilinearly interoplate arrays of values v0-v3 (v0 upper left, v1
/// upper right, v2 lower left, v3 lower right) at coordinates (s,t),
/// SCALING the interpolated value by 'scale' and then ADDING to
/// 'result'.  These are all vectors, so do it for each of 'n'
/// contiguous values (using the same s,t interpolants).
template <class T, class Q>
inline OIIO_HOSTDEVICE void
bilerp_mad (const T *v0, const T *v1,
            const T *v2, const T *v3,
            Q s, Q t, Q scale, int n, T *result)
{
    Q s1 = Q(1) - s;
    Q t1 = Q(1) - t;
    for (int i = 0;  i < n;  ++i)
        result[i] += T (scale * (t1*(v0[i]*s1 + v1[i]*s) +
                                  t*(v2[i]*s1 + v3[i]*s)));
}



/// Trilinearly interoplate arrays of values v0-v7 (v0 upper left top, v1
/// upper right top, ...) at coordinates (s,t,r), and return the
/// result.  This is a template, and so should work for any types.
template <class T, class Q>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
trilerp (T v0, T v1, T v2, T v3, T v4, T v5, T v6, T v7, Q s, Q t, Q r)
{
    // NOTE: a*(t-1) + b*t is much more numerically stable than a+t*(b-a)
    Q s1 = Q(1) - s;
    Q t1 = Q(1) - t;
    Q r1 = Q(1) - r;
    return T (r1*(t1*(v0*s1 + v1*s) + t*(v2*s1 + v3*s)) +
               r*(t1*(v4*s1 + v5*s) + t*(v6*s1 + v7*s)));
}



/// Trilinearly interoplate arrays of values v0-v7 (v0 upper left top, v1
/// upper right top, ...) at coordinates (s,t,r),
/// storing the results in 'result'.  These are all vectors, so do it
/// for each of 'n' contiguous values (using the same s,t,r interpolants).
template <class T, class Q>
inline OIIO_HOSTDEVICE void
trilerp (const T *v0, const T *v1, const T *v2, const T *v3,
         const T *v4, const T *v5, const T *v6, const T *v7,
         Q s, Q t, Q r, int n, T *result)
{
    Q s1 = Q(1) - s;
    Q t1 = Q(1) - t;
    Q r1 = Q(1) - r;
    for (int i = 0;  i < n;  ++i)
        result[i] = T (r1*(t1*(v0[i]*s1 + v1[i]*s) + t*(v2[i]*s1 + v3[i]*s)) +
                        r*(t1*(v4[i]*s1 + v5[i]*s) + t*(v6[i]*s1 + v7[i]*s)));
}



/// Trilinearly interoplate arrays of values v0-v7 (v0 upper left top, v1
/// upper right top, ...) at coordinates (s,t,r),
/// SCALING the interpolated value by 'scale' and then ADDING to
/// 'result'.  These are all vectors, so do it for each of 'n'
/// contiguous values (using the same s,t,r interpolants).
template <class T, class Q>
inline OIIO_HOSTDEVICE void
trilerp_mad (const T *v0, const T *v1, const T *v2, const T *v3,
             const T *v4, const T *v5, const T *v6, const T *v7,
             Q s, Q t, Q r, Q scale, int n, T *result)
{
    Q r1 = Q(1) - r;
    bilerp_mad (v0, v1, v2, v3, s, t, scale*r1, n, result);
    bilerp_mad (v4, v5, v6, v7, s, t, scale*r, n, result);
}



/// Evaluate B-spline weights in w[0..3] for the given fraction.  This
/// is an important component of performing a cubic interpolation.
template <typename T>
inline OIIO_HOSTDEVICE void evalBSplineWeights (T w[4], T fraction)
{
    T one_frac = 1 - fraction;
    w[0] = T(1.0 / 6.0) * one_frac * one_frac * one_frac;
    w[1] = T(2.0 / 3.0) - T(0.5) * fraction * fraction * (2 - fraction);
    w[2] = T(2.0 / 3.0) - T(0.5) * one_frac * one_frac * (2 - one_frac);
    w[3] = T(1.0 / 6.0) * fraction * fraction * fraction;
}


/// Evaluate B-spline derivative weights in w[0..3] for the given
/// fraction.  This is an important component of performing a cubic
/// interpolation with derivatives.
template <typename T>
inline OIIO_HOSTDEVICE void evalBSplineWeightDerivs (T dw[4], T fraction)
{
    T one_frac = 1 - fraction;
    dw[0] = -T(0.5) * one_frac * one_frac;
    dw[1] =  T(0.5) * fraction * (3 * fraction - 4);
    dw[2] = -T(0.5) * one_frac * (3 * one_frac - 4);
    dw[3] =  T(0.5) * fraction * fraction;
}



/// Bicubically interoplate arrays of pointers arranged in a 4x4 pattern
/// with val[0] pointing to the data in the upper left corner, val[15]
/// pointing to the lower right) at coordinates (s,t), storing the
/// results in 'result'.  These are all vectors, so do it for each of
/// 'n' contiguous values (using the same s,t interpolants).
template <class T>
inline OIIO_HOSTDEVICE void
bicubic_interp (const T **val, T s, T t, int n, T *result)
{
    for (int c = 0;  c < n;  ++c)
        result[c] = T(0);
    T wx[4]; evalBSplineWeights (wx, s);
    T wy[4]; evalBSplineWeights (wy, t);
    for (int j = 0;  j < 4;  ++j) {
        for (int i = 0;  i < 4;  ++i) {
            T w = wx[i] * wy[j];
            for (int c = 0;  c < n;  ++c)
                result[c] += w * val[j*4+i][c];
        }
    }
}



/// Return floor(x) cast to an int.
OIIO_FORCEINLINE OIIO_HOSTDEVICE int
ifloor (float x)
{
    return (int)floorf(x);
}



/// Return (x-floor(x)) and put (int)floor(x) in *xint.  This is similar
/// to the built-in modf, but returns a true int, always rounds down
/// (compared to modf which rounds toward 0), and always returns
/// frac >= 0 (comapred to modf which can return <0 if x<0).
inline OIIO_HOSTDEVICE float
floorfrac (float x, int *xint)
{
#if 1
    float f = std::floor(x);
    *xint = int(f);
    return x - f;
#else /* vvv This idiom is slower */
    int i = ifloor(x);
    *xint = i;
    return x - static_cast<float>(i);   // Return the fraction left over
#endif
}


#ifndef __CUDA_ARCH__
inline simd::vfloat4 floorfrac (const simd::vfloat4& x, simd::vint4 *xint) {
    simd::vfloat4 f = simd::floor(x);
    *xint = simd::vint4(f);
    return x - f;
}

inline simd::vfloat8 floorfrac (const simd::vfloat8& x, simd::vint8 *xint) {
    simd::vfloat8 f = simd::floor(x);
    *xint = simd::vint8(f);
    return x - f;
}

inline simd::vfloat16 floorfrac (const simd::vfloat16& x, simd::vint16 *xint) {
    simd::vfloat16 f = simd::floor(x);
    *xint = simd::vint16(f);
    return x - f;
}
#endif




/// Convert degrees to radians.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T radians (T deg) { return deg * T(M_PI / 180.0); }

/// Convert radians to degrees
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T degrees (T rad) { return rad * T(180.0 / M_PI); }


/// Faster floating point negation, in cases where you aren't too picky
/// about the difference between +0 and -0. (Courtesy of Alex Wells, Intel,
/// via code in OSL.)
///
/// Beware: fast_neg(0.0f) returns 0.0f, NOT -0.0f. All other values,
/// including -0.0 and +/- Inf, work identically to `-x`. For many use
/// cases, that's fine. (When was the last time you wanted a -0.0f anyway?)
///
/// The reason it's faster is that `-x` (for float x) is implemented by
/// compilers by xor'ing x against a bitmask that flips the sign bit, and
/// that bitmask had to come from somewhere -- memory! -- which can be
/// expensive, depending on whether/where it is in cache. But  `0 - x`
/// generates a zero in a register (with xor, no memory access needed) and
/// then subtracts, so that's sometimes faster because there is no memory
/// read. This works for SIMD types as well!
///
/// It's also safe (though pointless) to use fast_neg for integer types,
/// where both `-x` and `0-x` are implemented as a `neg` instruction.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T
fast_neg(const T &x) {
    return T(0) - x;
}



inline OIIO_HOSTDEVICE void
sincos (float x, float* sine, float* cosine)
{
#if defined(__GNUC__) && defined(__linux__) && !defined(__clang__)
    __builtin_sincosf(x, sine, cosine);
#elif defined(__CUDA_ARCH__)
    // Explicitly select the single-precision CUDA library function
    sincosf(x, sine, cosine);
#else
    *sine = std::sin(x);
    *cosine = std::cos(x);
#endif
}

inline OIIO_HOSTDEVICE void
sincos (double x, double* sine, double* cosine)
{
#if defined(__GNUC__) && defined(__linux__) && !defined(__clang__)
    __builtin_sincos(x, sine, cosine);
#elif defined(__CUDA_ARCH__)
    // Use of anonymous namespace resolves to the CUDA library function and
    // avoids infinite recursion
    ::sincos(x, sine, cosine);
#else
    *sine = std::sin(x);
    *cosine = std::cos(x);
#endif
}


inline OIIO_HOSTDEVICE float sign (float x)
{
    return x < 0.0f ? -1.0f : (x==0.0f ? 0.0f : 1.0f);
}




// (end of float helper functions)
////////////////////////////////////////////////////////////////////////////



////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// CONVERSION
//
// Type and range conversion helper functions and classes.


template <typename IN_TYPE, typename OUT_TYPE>
OIIO_FORCEINLINE OIIO_HOSTDEVICE OUT_TYPE bit_cast (const IN_TYPE& in) {
    // NOTE: this is the only standards compliant way of doing this type of casting,
    // luckily the compilers we care about know how to optimize away this idiom.
    static_assert(sizeof(IN_TYPE) == sizeof(OUT_TYPE),
                  "bit_cast must be between objects of the same size");
    OUT_TYPE out;
    memcpy ((void *)&out, &in, sizeof(IN_TYPE));
    return out;
}

#if defined(__INTEL_COMPILER)
    // On x86/x86_64 for certain compilers we can use Intel CPU intrinsics
    // for some common bit_cast cases that might be even more understandable
    // to the compiler and generate better code without its getting confused
    // about the memcpy in the general case.
    // FIXME: The intrinsics are not in clang <= 9 nor gcc <= 9.1. Check
    // future releases.
    template<> OIIO_FORCEINLINE uint32_t bit_cast<float, uint32_t> (const float val) {
          return static_cast<uint32_t>(_castf32_u32(val));
    }
    template<> OIIO_FORCEINLINE int32_t bit_cast<float, int32_t> (const float val) {
          return static_cast<int32_t>(_castf32_u32(val));
    }
    template<> OIIO_FORCEINLINE float bit_cast<uint32_t, float> (const uint32_t val) {
          return _castu32_f32(val);
    }
    template<> OIIO_FORCEINLINE float bit_cast<int32_t, float> (const int32_t val) {
          return _castu32_f32(val);
    }
    template<> OIIO_FORCEINLINE uint64_t bit_cast<double, uint64_t> (const double val) {
          return static_cast<uint64_t>(_castf64_u64(val));
    }
    template<> OIIO_FORCEINLINE int64_t bit_cast<double, int64_t> (const double val) {
          return static_cast<int64_t>(_castf64_u64(val));
    }
    template<> OIIO_FORCEINLINE double bit_cast<uint64_t, double> (const uint64_t val) {
          return _castu64_f64(val);
    }
    template<> OIIO_FORCEINLINE double bit_cast<int64_t, double> (const int64_t val) {
          return _castu64_f64(val);
    }
#endif


OIIO_FORCEINLINE OIIO_HOSTDEVICE int bitcast_to_int (float x) {
    return bit_cast<float,int>(x);
}
OIIO_FORCEINLINE OIIO_HOSTDEVICE float bitcast_to_float (int x) {
    return bit_cast<int,float>(x);
}



/// Change endian-ness of one or more data items that are each 2, 4,
/// or 8 bytes.  This should work for any of short, unsigned short, int,
/// unsigned int, float, long long, pointers.
template<class T>
inline OIIO_HOSTDEVICE void
swap_endian (T *f, int len=1)
{
    for (char *c = (char *) f;  len--;  c += sizeof(T)) {
        if (sizeof(T) == 2) {
            std::swap (c[0], c[1]);
        } else if (sizeof(T) == 4) {
            std::swap (c[0], c[3]);
            std::swap (c[1], c[2]);
        } else if (sizeof(T) == 8) {
            std::swap (c[0], c[7]);
            std::swap (c[1], c[6]);
            std::swap (c[2], c[5]);
            std::swap (c[3], c[4]);
        }
    }
}



// big_enough_float<T>::float_t is a floating-point type big enough to
// handle the range and precision of a <T>. It's a float, unless T is big.
template <typename T> struct big_enough_float    { typedef float float_t; };
template<> struct big_enough_float<int>          { typedef double float_t; };
template<> struct big_enough_float<unsigned int> { typedef double float_t; };
template<> struct big_enough_float<int64_t>      { typedef double float_t; };
template<> struct big_enough_float<uint64_t>     { typedef double float_t; };
template<> struct big_enough_float<double>       { typedef double float_t; };


/// Multiply src by scale, clamp to [min,max], and round to the nearest D
/// (presumed to be integer).  This is just a helper for the convert_type
/// templates, it probably has no other use.
template<typename S, typename D, typename F>
inline OIIO_HOSTDEVICE D
scaled_conversion(const S& src, F scale, F min, F max)
{
    if (std::numeric_limits<S>::is_signed) {
        F s = src * scale;
        s += (s < 0 ? (F)-0.5 : (F)0.5);
        return (D)clamp(s, min, max);
    } else {
        return (D)clamp((F)src * scale + (F)0.5, min, max);
    }
}



/// Convert n consecutive values from the type of S to the type of D.
/// The conversion is not a simple cast, but correctly remaps the
/// 0.0->1.0 range from and to the full positive range of integral
/// types.  Take a memcpy shortcut if both types are the same and no
/// conversion is necessary.  Optional arguments can give nonstandard
/// quantizations.
//
// FIXME: make table-based specializations for common types with only a
// few possible src values (like unsigned char -> float).
template<typename S, typename D>
void convert_type (const S *src, D *dst, size_t n, D _min, D _max)
{
    if (std::is_same<S,D>::value) {
        // They must be the same type.  Just memcpy.
        memcpy (dst, src, n*sizeof(D));
        return;
    }
    typedef typename big_enough_float<D>::float_t F;
    F scale = std::numeric_limits<S>::is_integer ?
        (F(1)) / F(std::numeric_limits<S>::max()) : F(1);
    if (std::numeric_limits<D>::is_integer) {
        // Converting to an integer-like type.
        F min = (F)_min;  // std::numeric_limits<D>::min();
        F max = (F)_max;  // std::numeric_limits<D>::max();
        scale *= _max;
        // Unroll loop for speed
        for ( ; n >= 16; n -= 16) {
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
        }
        while (n--)
            *dst++ = scaled_conversion<S,D,F> (*src++, scale, min, max);
    } else {
        // Converting to a float-like type, so we don't need to remap
        // the range
        // Unroll loop for speed
        for ( ; n >= 16; n -= 16) {
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
            *dst++ = (D)((*src++) * scale);
        }
        while (n--)
            *dst++ = (D)((*src++) * scale);
    }
}


#ifndef __CUDA_ARCH__
template<>
inline void convert_type<uint8_t,float> (const uint8_t *src,
                                         float *dst, size_t n,
                                         float /*_min*/, float /*_max*/)
{
    float scale (1.0f/std::numeric_limits<uint8_t>::max());
    simd::vfloat4 scale_simd (scale);
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 s_simd (src);
        simd::vfloat4 d_simd = s_simd * scale_simd;
        d_simd.store (dst);
    }
    while (n--)
        *dst++ = (*src++) * scale;
}



template<>
inline void convert_type<uint16_t,float> (const uint16_t *src,
                                          float *dst, size_t n,
                                          float /*_min*/, float /*_max*/)
{
    float scale (1.0f/std::numeric_limits<uint16_t>::max());
    simd::vfloat4 scale_simd (scale);
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 s_simd (src);
        simd::vfloat4 d_simd = s_simd * scale_simd;
        d_simd.store (dst);
    }
    while (n--)
        *dst++ = (*src++) * scale;
}


#if defined(_HALF_H_) || defined(IMATH_HALF_H_)
template<>
inline void convert_type<half,float> (const half *src,
                                      float *dst, size_t n,
                                      float /*_min*/, float /*_max*/)
{
#if OIIO_SIMD >= 8 && OIIO_F16C_ENABLED
    // If f16c ops are enabled, it's worth doing this by 8's
    for ( ; n >= 8; n -= 8, src += 8, dst += 8) {
        simd::vfloat8 s_simd (src);
        s_simd.store (dst);
    }
#endif
#if OIIO_SIMD >= 4
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 s_simd (src);
        s_simd.store (dst);
    }
#endif
    while (n--)
        *dst++ = (*src++);
}
#endif



template<>
inline void
convert_type<float,uint16_t> (const float *src, uint16_t *dst, size_t n,
                              uint16_t /*_min*/, uint16_t /*_max*/)
{
    float min = std::numeric_limits<uint16_t>::min();
    float max = std::numeric_limits<uint16_t>::max();
    float scale = max;
    simd::vfloat4 max_simd (max);
    simd::vfloat4 zero_simd (0.0f);
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 scaled = simd::round (simd::vfloat4(src) * max_simd);
        simd::vfloat4 clamped = clamp (scaled, zero_simd, max_simd);
        simd::vint4 i (clamped);
        i.store (dst);
    }
    while (n--)
        *dst++ = scaled_conversion<float,uint16_t,float> (*src++, scale, min, max);
}


template<>
inline void
convert_type<float,uint8_t> (const float *src, uint8_t *dst, size_t n,
                             uint8_t /*_min*/, uint8_t /*_max*/)
{
    float min = std::numeric_limits<uint8_t>::min();
    float max = std::numeric_limits<uint8_t>::max();
    float scale = max;
    simd::vfloat4 max_simd (max);
    simd::vfloat4 zero_simd (0.0f);
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 scaled = simd::round (simd::vfloat4(src) * max_simd);
        simd::vfloat4 clamped = clamp (scaled, zero_simd, max_simd);
        simd::vint4 i (clamped);
        i.store (dst);
    }
    while (n--)
        *dst++ = scaled_conversion<float,uint8_t,float> (*src++, scale, min, max);
}


#if defined(_HALF_H_) || defined(IMATH_HALF_H_)
template<>
inline void
convert_type<float,half> (const float *src, half *dst, size_t n,
                          half /*_min*/, half /*_max*/)
{
#if OIIO_SIMD >= 8 && OIIO_F16C_ENABLED
    // If f16c ops are enabled, it's worth doing this by 8's
    for ( ; n >= 8; n -= 8, src += 8, dst += 8) {
        simd::vfloat8 s (src);
        s.store (dst);
    }
#endif
#if OIIO_SIMD >= 4
    for ( ; n >= 4; n -= 4, src += 4, dst += 4) {
        simd::vfloat4 s (src);
        s.store (dst);
    }
#endif
    while (n--)
        *dst++ = *src++;
}
#endif
#endif



template<typename S, typename D>
inline void convert_type (const S *src, D *dst, size_t n)
{
    convert_type<S,D> (src, dst, n,
                       std::numeric_limits<D>::min(),
                       std::numeric_limits<D>::max());
}




/// Convert a single value from the type of S to the type of D.
/// The conversion is not a simple cast, but correctly remaps the
/// 0.0->1.0 range from and to the full positive range of integral
/// types.  Take a copy shortcut if both types are the same and no
/// conversion is necessary.
template<typename S, typename D>
inline D
convert_type (const S &src)
{
    if (std::is_same<S,D>::value) {
        // They must be the same type.  Just return it.
        return (D)src;
    }
    typedef typename big_enough_float<D>::float_t F;
    F scale = std::numeric_limits<S>::is_integer ?
        F(1) / F(std::numeric_limits<S>::max()) : F(1);
    if (std::numeric_limits<D>::is_integer) {
        // Converting to an integer-like type.
        F min = (F) std::numeric_limits<D>::min();
        F max = (F) std::numeric_limits<D>::max();
        scale *= max;
        return scaled_conversion<S,D,F> (src, scale, min, max);
    } else {
        // Converting to a float-like type, so we don't need to remap
        // the range
        return (D)((F)src * scale);
    }
}



/// Helper function to convert channel values between different bit depths.
/// Roughly equivalent to:
///
/// out = round (in * (pow (2, TO_BITS) - 1) / (pow (2, FROM_BITS) - 1));
///
/// but utilizes an integer math trick for speed. It can be proven that the
/// absolute error of this method is less or equal to 1, with an average error
/// (with respect to the entire domain) below 0.2.
///
/// It is assumed that the original value is a valid FROM_BITS integer, i.e.
/// shifted fully to the right.
template<unsigned int FROM_BITS, unsigned int TO_BITS>
inline OIIO_HOSTDEVICE unsigned int bit_range_convert(unsigned int in) {
    unsigned int out = 0;
    int shift = TO_BITS - FROM_BITS;
    for (; shift > 0; shift -= FROM_BITS)
        out |= in << shift;
    out |= in >> -shift;
    return out;
}



// non-templated version.  Slow but general
inline OIIO_HOSTDEVICE unsigned int
bit_range_convert(unsigned int in, unsigned int FROM_BITS, unsigned int TO_BITS)
{
    unsigned int out = 0;
    int shift = TO_BITS - FROM_BITS;
    for (; shift > 0; shift -= FROM_BITS)
        out |= in << shift;
    out |= in >> -shift;
    return out;
}



/// Append the `n` LSB bits of `val` into a bit sting `T out[]`, where the
/// `filled` MSB bits of `*out` are already filled in. Increment `out` and
/// adjust `filled` as required. Type `T` should be uint8_t, uint16_t, or
/// uint32_t.
template<typename T>
inline void
bitstring_add_n_bits (T* &out, int& filled, uint32_t val, int n)
{
    static_assert(std::is_same<T,uint8_t>::value ||
                  std::is_same<T,uint16_t>::value ||
                  std::is_same<T,uint32_t>::value,
                  "bitstring_add_n_bits must be unsigned int 8/16/32");
    const int Tbits = sizeof(T) * 8;
    // val:         | don't care     | copy me    |
    //                                <- n bits ->
    //
    // *out:        | don't touch |   fill in here       |
    //               <- filled  -> <- (Tbits - filled) ->
    while (n > 0) {
        // Make sure val doesn't have any cruft in bits > n
        val &= ~(0xffffffff << n);
        // Initialize any new byte we're filling in
        if (filled == 0)
            *out = 0;
        // How many bits can we fill in `*out` without having to increment
        // to the next byte?
        int bits_left_in_out = Tbits - filled;
        int b = 0;   // bit pattern to 'or' with *out
        int nb = 0;  // number of bits to add
        if (n <= bits_left_in_out) { // can fit completely in this byte
            b = val << (bits_left_in_out - n);
            nb = n;
        } else { // n > bits_left_in_out, will spill to next byte
            b = val >> (n - bits_left_in_out);
            nb = bits_left_in_out;
        }
        *out |= b;
        filled += nb;
        OIIO_DASSERT (filled <= Tbits);
        n -= nb;
        if (filled == Tbits) {
            ++out;
            filled = 0;
        }
    }
}



/// Pack values from `T in[0..n-1]` (where `T` is expected to be a uint8,
/// uint16, or uint32, into successive raw outbits-bit pieces of `out[]`,
/// where outbits is expected to be less than the number of bits in a `T`.
template<typename T>
inline void
bit_pack(cspan<T> data, void* out, int outbits)
{
    static_assert(std::is_same<T,uint8_t>::value ||
                  std::is_same<T,uint16_t>::value ||
                  std::is_same<T,uint32_t>::value,
                  "bit_pack must be unsigned int 8/16/32");
    unsigned char* outbuffer = (unsigned char*)out;
    int filled = 0;
    for (size_t i = 0, e = data.size(); i < e; ++i)
        bitstring_add_n_bits (outbuffer, filled, data[i], outbits);
}



/// Decode n packed inbits-bits values from in[...] into normal uint8,
/// uint16, or uint32 representation of `T out[0..n-1]`. In other words,
/// each successive `inbits` of `in` (allowing spanning of byte boundaries)
/// will be stored in a successive out[i].
template<typename T>
inline void
bit_unpack(int n, const unsigned char* in, int inbits, T* out)
{
    static_assert(std::is_same<T,uint8_t>::value ||
                  std::is_same<T,uint16_t>::value ||
                  std::is_same<T,uint32_t>::value,
                  "bit_unpack must be unsigned int 8/16/32");
    OIIO_DASSERT(inbits >= 1 && inbits < 32);  // surely bugs if not
    // int highest = (1 << inbits) - 1;
    int B = 0, b = 0;
    // Invariant:
    // So far, we have used in[0..B-1] and the high b bits of in[B].
    for (int i = 0; i < n; ++i) {
        long long val = 0;
        int valbits   = 0;  // bits so far we've accumulated in val
        while (valbits < inbits) {
            // Invariant: we have already accumulated valbits of the next
            // needed value (of a total of inbits), living in the valbits
            // low bits of val.
            int out_left = inbits - valbits;  // How much more we still need
            int in_left  = 8 - b;             // Bits still available in in[B].
            if (in_left <= out_left) {
                // Eat the rest of this byte:
                //   |---------|--------|
                //        b      in_left
                val <<= in_left;
                val |= in[B] & ~(0xffffffff << in_left);
                ++B;
                b = 0;
                valbits += in_left;
            } else {
                // Eat just the bits we need:
                //   |--|---------|-----|
                //    b  out_left  extra
                val <<= out_left;
                int extra = 8 - b - out_left;
                val |= (in[B] >> extra) & ~(0xffffffff << out_left);
                b += out_left;
                valbits = inbits;
            }
        }
        out[i] = val; //T((val * 0xff) / highest);
    }
}



/// A DataProxy<I,E> looks like an (E &), but it really holds an (I &)
/// and does conversions (via convert_type) as it reads in and out.
/// (I and E are for INTERNAL and EXTERNAL data types, respectively).
template<typename I, typename E>
struct DataProxy {
    DataProxy (I &data) : m_data(data) { }
    E operator= (E newval) { m_data = convert_type<E,I>(newval); return newval; }
    operator E () const { return convert_type<I,E>(m_data); }
private:
    DataProxy& operator = (const DataProxy&); // Do not implement
    I &m_data;
};



/// A ConstDataProxy<I,E> looks like a (const E &), but it really holds
/// a (const I &) and does conversions (via convert_type) as it reads.
/// (I and E are for INTERNAL and EXTERNAL data types, respectively).
template<typename I, typename E>
struct ConstDataProxy {
    ConstDataProxy (const I &data) : m_data(data) { }
    operator E () const { return convert_type<E,I>(*m_data); }
private:
    const I &m_data;
};



/// A DataArrayProxy<I,E> looks like an (E *), but it really holds an (I *)
/// and does conversions (via convert_type) as it reads in and out.
/// (I and E are for INTERNAL and EXTERNAL data types, respectively).
template<typename I, typename E>
struct DataArrayProxy {
    DataArrayProxy (I *data=NULL) : m_data(data) { }
    E operator* () const { return convert_type<I,E>(*m_data); }
    E operator[] (int i) const { return convert_type<I,E>(m_data[i]); }
    DataProxy<I,E> operator[] (int i) { return DataProxy<I,E> (m_data[i]); }
    void set (I *data) { m_data = data; }
    I * get () const { return m_data; }
    const DataArrayProxy<I,E> & operator+= (int i) {
        m_data += i;  return *this;
    }
private:
    I *m_data;
};



/// A ConstDataArrayProxy<I,E> looks like an (E *), but it really holds an
/// (I *) and does conversions (via convert_type) as it reads in and out.
/// (I and E are for INTERNAL and EXTERNAL data types, respectively).
template<typename I, typename E>
struct ConstDataArrayProxy {
    ConstDataArrayProxy (const I *data=NULL) : m_data(data) { }
    E operator* () const { return convert_type<I,E>(*m_data); }
    E operator[] (int i) const { return convert_type<I,E>(m_data[i]); }
    void set (const I *data) { m_data = data; }
    const I * get () const { return m_data; }
    const ConstDataArrayProxy<I,E> & operator+= (int i) {
        m_data += i;  return *this;
    }
private:
    const I *m_data;
};



/// Fast table-based conversion of 8-bit to other types.  Declare this
/// as static to avoid the expensive ctr being called all the time.
template <class T=float>
class EightBitConverter {
public:
    EightBitConverter () noexcept { init(); }
    T operator() (unsigned char c) const noexcept { return val[c]; }
private:
    T val[256];
    void init () noexcept {
        float scale = 1.0f / 255.0f;
        if (std::numeric_limits<T>::is_integer)
            scale *= (float)std::numeric_limits<T>::max();
        for (int i = 0;  i < 256;  ++i)
            val[i] = (T)(i * scale);
    }
};



/// Simple conversion of a (presumably non-negative) float into a
/// rational.  This does not attempt to find the simplest fraction
/// that approximates the float, for example 52.83 will simply
/// return 5283/100.  This does not attempt to gracefully handle
/// floats that are out of range that could be easily int/int.
inline OIIO_HOSTDEVICE void
float_to_rational (float f, unsigned int &num, unsigned int &den)
{
    if (f <= 0) {   // Trivial case of zero, and handle all negative values
        num = 0;
        den = 1;
    } else if ((int)(1.0/f) == (1.0/f)) { // Exact results for perfect inverses
        num = 1;
        den = (int)f;
    } else {
        num = (int)f;
        den = 1;
        while (fabsf(f-static_cast<float>(num)) > 0.00001f && den < 1000000) {
            den *= 10;
            f *= 10;
            num = (int)f;
        }
    }
}



/// Simple conversion of a float into a rational.  This does not attempt
/// to find the simplest fraction that approximates the float, for
/// example 52.83 will simply return 5283/100.  This does not attempt to
/// gracefully handle floats that are out of range that could be easily
/// int/int.
inline OIIO_HOSTDEVICE void
float_to_rational (float f, int &num, int &den)
{
    unsigned int n, d;
    float_to_rational (fabsf(f), n, d);
    num = (f >= 0) ? (int)n : -(int)n;
    den = (int) d;
}


// (end of conversion helpers)
////////////////////////////////////////////////////////////////////////////




////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// SAFE MATH
//
// The functions named "safe_*" are versions with various internal clamps
// or other deviations from IEEE standards with the specific intent of
// never producing NaN or Inf values or throwing exceptions. But within the
// valid range, they should be full precision and match IEEE standards.
//


/// Safe (clamping) sqrt: safe_sqrt(x<0) returns 0, not NaN.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_sqrt (T x) {
    return x >= T(0) ? std::sqrt(x) : T(0);
}

/// Safe (clamping) inverse sqrt: safe_inversesqrt(x<=0) returns 0.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_inversesqrt (T x) {
    return x > T(0) ? T(1) / std::sqrt(x) : T(0);
}


/// Safe (clamping) arcsine: clamp to the valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_asin (T x) {
    if (x <= T(-1)) return T(-M_PI_2);
    if (x >= T(+1)) return T(+M_PI_2);
    return std::asin(x);
}

/// Safe (clamping) arccosine: clamp to the valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_acos (T x) {
    if (x <= T(-1)) return T(M_PI);
    if (x >= T(+1)) return T(0);
    return std::acos(x);
}


/// Safe log2: clamp to valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_log2 (T x) {
    // match clamping from fast version
    if (x < std::numeric_limits<T>::min()) x = std::numeric_limits<T>::min();
    if (x > std::numeric_limits<T>::max()) x = std::numeric_limits<T>::max();
    return std::log2(x);
}

/// Safe log: clamp to valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_log (T x) {
    // slightly different than fast version since clamping happens before scaling
    if (x < std::numeric_limits<T>::min()) x = std::numeric_limits<T>::min();
    if (x > std::numeric_limits<T>::max()) x = std::numeric_limits<T>::max();
    return std::log(x);
}

/// Safe log10: clamp to valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_log10 (T x) {
    // slightly different than fast version since clamping happens before scaling
    if (x < std::numeric_limits<T>::min()) x = std::numeric_limits<T>::min();
    if (x > std::numeric_limits<T>::max()) x = std::numeric_limits<T>::max();
    return log10f(x);
}

/// Safe logb: clamp to valid domain.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_logb (T x) {
    return (x != T(0)) ? std::logb(x) : -std::numeric_limits<T>::max();
}

/// Safe pow: clamp the domain so it never returns Inf or NaN or has divide
/// by zero error.
template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T safe_pow (T x, T y) {
    if (y == T(0)) return T(1);
    if (x == T(0)) return T(0);
    // if x is negative, only deal with integer powers
    if ((x < T(0)) && (y != floor(y))) return T(0);
    // FIXME: this does not match "fast" variant because clamping limits are different
    T r = std::pow(x, y);
    // Clamp to avoid returning Inf.
    const T big = std::numeric_limits<T>::max();
    return clamp (r, -big, big);
}


/// Safe fmod: guard against b==0.0 (in which case, return 0.0). Also, it
/// seems that this is much faster than std::fmod!
OIIO_FORCEINLINE OIIO_HOSTDEVICE float safe_fmod (float a, float b)
{
    if (OIIO_LIKELY(b != 0.0f)) {
#if 0
        return std::fmod (a,b);
        // std::fmod was getting called serially instead of vectorizing, so
        // we will just do the calculation ourselves.
#else
        // This formulation is equivalent but much faster in our benchmarks,
        // also vectorizes better.
        // The floating-point remainder of the division operation
        // a/b is a - N*b, where N = a/b with its fractional part truncated.
        int N = static_cast<int>(a/b);
        return a - N*b;
#endif
    }
    return 0.0f;
}

#define OIIO_FMATH_HAS_SAFE_FMOD 1

// (end of safe_* functions)
////////////////////////////////////////////////////////////////////////////



////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// FAST & APPROXIMATE MATH
//
// The functions named "fast_*" provide a set of replacements to libm that
// are much faster at the expense of some accuracy and robust handling of
// extreme values. One design goal for these approximation was to avoid
// branches as much as possible and operate on single precision values only
// so that SIMD versions should be straightforward ports. We also try to
// implement "safe" semantics (ie: clamp to valid range where possible)
// natively since wrapping these inline calls in another layer would be
// wasteful.
//
// The "fast_*" functions are not only possibly differing in value
// (slightly) from the std versions of these functions, but also we do not
// guarantee that the results of "fast" will exactly match from platform to
// platform. This is because if a particular platform (HW, library, or
// compiler) provides an intrinsic that is even faster than our usual "fast"
// implementation, we may substitute it.
//
// Some functions are fast_safe_*, which is both a faster approximation as
// well as clamped input domain to ensure no NaN, Inf, or divide by zero.
//
// NB: When compiling for CUDA devices, selection of the 'fast' intrinsics
//     is influenced by compiler options (-ffast-math for clang/gcc,
//     --use-fast-math for NVCC).
//
// TODO: Quantify the performance and accuracy of the CUDA Math functions
//       relative to the definitions in this file. It may be better in some
//       cases to use the approximate versions defined below.


/// Round to nearest integer, returning as an int.
/// Note that this differs from std::rint, which returns a float; it's more
/// akin to std::lrint, which returns a long (int). Sorry for the naming
/// confusion.
OIIO_FORCEINLINE OIIO_HOSTDEVICE int fast_rint (float x) {
    // used by sin/cos/tan range reduction
#if OIIO_SIMD_SSE >= 4
    // single roundps instruction on SSE4.1+ (for gcc/clang at least)
    return static_cast<int>(std::rint(x));
#else
    // emulate rounding by adding/substracting 0.5
    return static_cast<int>(x + copysignf(0.5f, x));
#endif
}

#ifndef __CUDA_ARCH__
OIIO_FORCEINLINE simd::vint4 fast_rint (const simd::vfloat4& x) {
    return simd::rint (x);
}
#endif


OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_sin (float x) {
#ifndef __CUDA_ARCH__
    // very accurate argument reduction from SLEEF
    // starts failing around x=262000
    // Results on: [-2pi,2pi]
    // Examined 2173837240 values of sin: 0.00662760244 avg ulp diff, 2 max ulp, 1.19209e-07 max error
    int q = fast_rint (x * float(M_1_PI));
    float qf = float(q);
    x = madd(qf, -0.78515625f*4, x);
    x = madd(qf, -0.00024187564849853515625f*4, x);
    x = madd(qf, -3.7747668102383613586e-08f*4, x);
    x = madd(qf, -1.2816720341285448015e-12f*4, x);
    x = float(M_PI_2) - (float(M_PI_2) - x); // crush denormals
    float s = x * x;
    if ((q & 1) != 0) x = -x;
    // this polynomial approximation has very low error on [-pi/2,+pi/2]
    // 1.19209e-07 max error in total over [-2pi,+2pi]
    float u = 2.6083159809786593541503e-06f;
    u = madd(u, s, -0.0001981069071916863322258f);
    u = madd(u, s, +0.00833307858556509017944336f);
    u = madd(u, s, -0.166666597127914428710938f);
    u = madd(s, u * x, x);
    // For large x, the argument reduction can fail and the polynomial can
    // be evaluated with arguments outside the valid internal. Just clamp
    // the bad values away.
    return clamp(u, -1.0f, 1.0f);
#else
    return __sinf(x);
#endif
}


OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_cos (float x) {
#ifndef __CUDA_ARCH__
    // same argument reduction as fast_sin
    int q = fast_rint (x * float(M_1_PI));
    float qf = float(q);
    x = madd(qf, -0.78515625f*4, x);
    x = madd(qf, -0.00024187564849853515625f*4, x);
    x = madd(qf, -3.7747668102383613586e-08f*4, x);
    x = madd(qf, -1.2816720341285448015e-12f*4, x);
    x = float(M_PI_2) - (float(M_PI_2) - x); // crush denormals
    float s = x * x;
    // polynomial from SLEEF's sincosf, max error is
    // 4.33127e-07 over [-2pi,2pi] (98% of values are "exact")
    float u = -2.71811842367242206819355e-07f;
    u = madd(u, s, +2.47990446951007470488548e-05f);
    u = madd(u, s, -0.00138888787478208541870117f);
    u = madd(u, s, +0.0416666641831398010253906f);
    u = madd(u, s, -0.5f);
    u = madd(u, s, +1.0f);
    if ((q & 1) != 0) u = -u;
    return clamp(u, -1.0f, 1.0f);
#else
    return __cosf(x);
#endif
}


OIIO_FORCEINLINE OIIO_HOSTDEVICE void fast_sincos (float x, float* sine, float* cosine) {
#ifndef __CUDA_ARCH__
    // same argument reduction as fast_sin
    int q = fast_rint (x * float(M_1_PI));
    float qf = float(q);
    x = madd(qf, -0.78515625f*4, x);
    x = madd(qf, -0.00024187564849853515625f*4, x);
    x = madd(qf, -3.7747668102383613586e-08f*4, x);
    x = madd(qf, -1.2816720341285448015e-12f*4, x);
    x = float(M_PI_2) - (float(M_PI_2) - x); // crush denormals
    float s = x * x;
    // NOTE: same exact polynomials as fast_sin and fast_cos above
    if ((q & 1) != 0) x = -x;
    float su = 2.6083159809786593541503e-06f;
    su = madd(su, s, -0.0001981069071916863322258f);
    su = madd(su, s, +0.00833307858556509017944336f);
    su = madd(su, s, -0.166666597127914428710938f);
    su = madd(s, su * x, x);
    float cu = -2.71811842367242206819355e-07f;
    cu = madd(cu, s, +2.47990446951007470488548e-05f);
    cu = madd(cu, s, -0.00138888787478208541870117f);
    cu = madd(cu, s, +0.0416666641831398010253906f);
    cu = madd(cu, s, -0.5f);
    cu = madd(cu, s, +1.0f);
    if ((q & 1) != 0) cu = -cu;
    *sine   = clamp(su, -1.0f, 1.0f);;
    *cosine = clamp(cu, -1.0f, 1.0f);;
#else
    __sincosf(x, sine, cosine);
#endif
}

// NOTE: this approximation is only valid on [-8192.0,+8192.0], it starts becoming
// really poor outside of this range because the reciprocal amplifies errors
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_tan (float x) {
#ifndef __CUDA_ARCH__
    // derived from SLEEF implementation
    // note that we cannot apply the "denormal crush" trick everywhere because
    // we sometimes need to take the reciprocal of the polynomial
    int q = fast_rint (x * float(2 * M_1_PI));
    float qf = float(q);
    x = madd(qf, -0.78515625f*2, x);
    x = madd(qf, -0.00024187564849853515625f*2, x);
    x = madd(qf, -3.7747668102383613586e-08f*2, x);
    x = madd(qf, -1.2816720341285448015e-12f*2, x);
    if ((q & 1) == 0)
        x = float(M_PI_4) - (float(M_PI_4) - x); // crush denormals (only if we aren't inverting the result later)
    float s = x * x;
    float u = 0.00927245803177356719970703f;
    u = madd(u, s, 0.00331984995864331722259521f);
    u = madd(u, s, 0.0242998078465461730957031f);
    u = madd(u, s, 0.0534495301544666290283203f);
    u = madd(u, s, 0.133383005857467651367188f);
    u = madd(u, s, 0.333331853151321411132812f);
    u = madd(s, u * x, x);
    if ((q & 1) != 0)
        u = -1.0f / u;
    return u;
#else
    return __tanf(x);
#endif
}

/// Fast, approximate sin(x*M_PI) with maximum absolute error of 0.000918954611.
/// Adapted from http://devmaster.net/posts/9648/fast-and-accurate-sine-cosine#comment-76773
/// Note that this is MUCH faster, but much less accurate than fast_sin.
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_sinpi (float x)
{
#ifndef __CUDA_ARCH__
	// Fast trick to strip the integral part off, so our domain is [-1,1]
	const float z = x - ((x + 25165824.0f) - 25165824.0f);
    const float y = z - z * fabsf(z);
    const float Q = 3.10396624f;
    const float P = 3.584135056f; // P = 16-4*Q
    return y * (Q + P * fabsf(y));
    /* The original article used used inferior constants for Q and P and
     * so had max error 1.091e-3.
     *
     * The optimal value for Q was determined by exhaustive search, minimizing
     * the absolute numerical error relative to float(std::sin(double(phi*M_PI)))
     * over the interval [0,2] (which is where most of the invocations happen).
     *
     * The basic idea of this approximation starts with the coarse approximation:
     *      sin(pi*x) ~= f(x) =  4 * (x - x * abs(x))
     *
     * This approximation always _over_ estimates the target. On the otherhand, the
     * curve:
     *      sin(pi*x) ~= f(x) * abs(f(x)) / 4
     *
     * always lies _under_ the target. Thus we can simply numerically search for the
     * optimal constant to LERP these curves into a more precise approximation.
     * After folding the constants together and simplifying the resulting math, we
     * end up with the compact implementation below.
     *
     * NOTE: this function actually computes sin(x * pi) which avoids one or two
     * mults in many cases and guarantees exact values at integer periods.
     */
#else
    return sinpif(x);
#endif
}

/// Fast approximate cos(x*M_PI) with ~0.1% absolute error.
/// Note that this is MUCH faster, but much less accurate than fast_cos.
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_cospi (float x)
{
#ifndef __CUDA_ARCH__
    return fast_sinpi (x+0.5f);
#else
    return cospif(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_acos (float x) {
#ifndef __CUDA_ARCH__
    const float f = fabsf(x);
    const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f; // clamp and crush denormals
    // based on http://www.pouet.net/topic.php?which=9132&page=2
    // 85% accurate (ulp 0)
    // Examined 2130706434 values of acos: 15.2000597 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // without "denormal crush"
    // Examined 2130706434 values of acos: 15.2007108 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // with "denormal crush"
    const float a = sqrtf(1.0f - m) * (1.5707963267f + m * (-0.213300989f + m * (0.077980478f + m * -0.02164095f)));
    return x < 0 ? float(M_PI) - a : a;
#else
    return acosf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_asin (float x) {
#ifndef __CUDA_ARCH__
    // based on acosf approximation above
    // max error is 4.51133e-05 (ulps are higher because we are consistently off by a little amount)
    const float f = fabsf(x);
    const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f; // clamp and crush denormals
    const float a = float(M_PI_2) - sqrtf(1.0f - m) * (1.5707963267f + m * (-0.213300989f + m * (0.077980478f + m * -0.02164095f)));
    return copysignf(a, x);
#else
    return asinf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_atan (float x) {
#ifndef __CUDA_ARCH__
    const float a = fabsf(x);
    const float k = a > 1.0f ? 1 / a : a;
    const float s = 1.0f - (1.0f - k); // crush denormals
    const float t = s * s;
    // http://mathforum.org/library/drmath/view/62672.html
    // the coefficients were tuned in mathematica with the assumption that we want atan(1)=pi/4
    // (slightly higher error but no discontinuities)
    // Examined 4278190080 values of atan: 2.53989068 avg ulp diff, 315 max ulp, 9.17912e-06 max error      // (with  denormals)
    // Examined 4278190080 values of atan: 171160502 avg ulp diff, 855638016 max ulp, 9.17912e-06 max error // (crush denormals)
    float r = s * madd(0.430165678f, t, 1.0f) / madd(madd(0.0579354987f, t, 0.763007998f), t, 1.0f);
    if (a > 1.0f) r = 1.570796326794896557998982f - r;
    return copysignf(r, x);
#else
    return atanf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_atan2 (float y, float x) {
#ifndef __CUDA_ARCH__
    // based on atan approximation above
    // the special cases around 0 and infinity were tested explicitly
    // the only case not handled correctly is x=NaN,y=0 which returns 0 instead of nan
    const float a = fabsf(x);
    const float b = fabsf(y);
    bool b_is_greater_than_a = b > a;

#if OIIO_FMATH_SIMD_FRIENDLY
    // When applying to all lanes in SIMD, we end up doing extra masking and
    // 2 divides. So lets just do 1 divide and swap the parameters instead.
    // And if we are going to do a doing a divide anyway, when a == b it
    // should be 1.0f anyway so lets not bother special casing it.
    float sa = b_is_greater_than_a ? b : a;
    float sb = b_is_greater_than_a ? a : b;
    const float k = (b == 0) ? 0.0f : sb/sa;
#else
    const float k = (b == 0) ? 0.0f : ((a == b) ? 1.0f : (b > a ? a / b : b / a));
#endif

    const float s = 1.0f - (1.0f - k); // crush denormals
    const float t = s * s;

    float r = s * madd(0.430165678f, t, 1.0f) / madd(madd(0.0579354987f, t, 0.763007998f), t, 1.0f);

    if (b_is_greater_than_a)
        r = 1.570796326794896557998982f - r; // account for arg reduction
    // TODO:  investigate if testing x < 0.0f is more efficient
    if (bit_cast<float, unsigned>(x) & 0x80000000u) // test sign bit of x
        r = float(M_PI) - r;
    return copysignf(r, y);
#else
    return atan2f(y, x);
#endif
}


template<typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_log2 (const T& xval) {
    using namespace simd;
    typedef typename T::int_t intN;
    // See float fast_log2 for explanations
    T x = clamp (xval, T(std::numeric_limits<float>::min()), T(std::numeric_limits<float>::max()));
    intN bits = bitcast_to_int(x);
    intN exponent = srl (bits, 23) - intN(127);
    T f = bitcast_to_float ((bits & intN(0x007FFFFF)) | intN(0x3f800000)) - T(1.0f);
    T f2 = f * f;
    T f4 = f2 * f2;
    T hi = madd(f, T(-0.00931049621349f), T( 0.05206469089414f));
    T lo = madd(f, T( 0.47868480909345f), T(-0.72116591947498f));
    hi = madd(f, hi, T(-0.13753123777116f));
    hi = madd(f, hi, T( 0.24187369696082f));
    hi = madd(f, hi, T(-0.34730547155299f));
    lo = madd(f, lo, T( 1.442689881667200f));
    return ((f4 * hi) + (f * lo)) + T(exponent);
}


template<>
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_log2 (const float& xval) {
#ifndef __CUDA_ARCH__
    // NOTE: clamp to avoid special cases and make result "safe" from large negative values/nans
    float x = clamp (xval, std::numeric_limits<float>::min(), std::numeric_limits<float>::max());
    // based on https://github.com/LiraNuna/glsl-sse2/blob/master/source/vec4.h
    unsigned bits = bit_cast<float, unsigned>(x);
    int exponent = int(bits >> 23) - 127;
    float f = bit_cast<unsigned, float>((bits & 0x007FFFFF) | 0x3f800000) - 1.0f;
    // Examined 2130706432 values of log2 on [1.17549435e-38,3.40282347e+38]: 0.0797524457 avg ulp diff, 3713596 max ulp, 7.62939e-06 max error
    // ulp histogram:
    //  0  = 97.46%
    //  1  =  2.29%
    //  2  =  0.11%
    float f2 = f * f;
    float f4 = f2 * f2;
    float hi = madd(f, -0.00931049621349f,  0.05206469089414f);
    float lo = madd(f,  0.47868480909345f, -0.72116591947498f);
    hi = madd(f, hi, -0.13753123777116f);
    hi = madd(f, hi,  0.24187369696082f);
    hi = madd(f, hi, -0.34730547155299f);
    lo = madd(f, lo,  1.442689881667200f);
    return ((f4 * hi) + (f * lo)) + exponent;
#else
    return __log2f(xval);
#endif
}



template<typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_log (const T& x) {
    // Examined 2130706432 values of logf on [1.17549435e-38,3.40282347e+38]: 0.313865375 avg ulp diff, 5148137 max ulp, 7.62939e-06 max error
    return fast_log2(x) * T(M_LN2);
}

#ifdef __CUDA_ARCH__
template<>
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_log(const float& x)
{
     return __logf(x);
}
#endif


template<typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_log10 (const T& x) {
    // Examined 2130706432 values of log10f on [1.17549435e-38,3.40282347e+38]: 0.631237033 avg ulp diff, 4471615 max ulp, 3.8147e-06 max error
    return fast_log2(x) * T(M_LN2 / M_LN10);
}

#ifdef __CUDA_ARCH__
template<>
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_log10(const float& x)
{
     return __log10f(x);
}
#endif

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_logb (float x) {
#ifndef __CUDA_ARCH__
    // don't bother with denormals
    x = fabsf(x);
    if (x < std::numeric_limits<float>::min()) x = std::numeric_limits<float>::min();
    if (x > std::numeric_limits<float>::max()) x = std::numeric_limits<float>::max();
    unsigned bits = bit_cast<float, unsigned>(x);
    return float (int(bits >> 23) - 127);
#else
    return logbf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_log1p (float x) {
#ifndef __CUDA_ARCH__
    if (fabsf(x) < 0.01f) {
        float y = 1.0f - (1.0f - x); // crush denormals
        return copysignf(madd(-0.5f, y * y, y), x);
    } else {
        return fast_log(x + 1);
    }
#else
    return log1pf(x);
#endif
}



template<typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_exp2 (const T& xval) {
    using namespace simd;
    typedef typename T::int_t intN;
#if OIIO_SIMD_SSE
    // See float specialization for explanations
    T x = clamp (xval, T(-126.0f), T(126.0f));
    intN m (x); x -= T(m);
    T one (1.0f);
    x = one - (one - x); // crush denormals (does not affect max ulps!)
    const T kA (1.33336498402e-3f);
    const T kB (9.810352697968e-3f);
    const T kC (5.551834031939e-2f);
    const T kD (0.2401793301105f);
    const T kE (0.693144857883f);
    T r (kA);
    r = madd(x, r, kB);
    r = madd(x, r, kC);
    r = madd(x, r, kD);
    r = madd(x, r, kE);
    r = madd(x, r, one);
    return bitcast_to_float (bitcast_to_int(r) + (m << 23));
#else
    T r;
    for (int i = 0; i < r.elements; ++i)
        r[i] = fast_exp2(xval[i]);
    for (int i = r.elements; i < r.paddedelements; ++i)
        r[i] = 0.0f;
    return r;
#endif
}


template<>
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_exp2 (const float& xval) {
#if OIIO_NON_INTEL_CLANG && OIIO_FMATH_SIMD_FRIENDLY
    // Clang was unhappy using the bitcast/memcpy/reinter_cast/union inside
    // an explicit SIMD loop, so revert to calling the standard version.
    return std::exp2(xval);
#elif !defined(__CUDA_ARCH__)
    // clamp to safe range for final addition
    float x = clamp (xval, -126.0f, 126.0f);
    // range reduction
    int m = int(x); x -= m;
    x = 1.0f - (1.0f - x); // crush denormals (does not affect max ulps!)
    // 5th degree polynomial generated with sollya
    // Examined 2247622658 values of exp2 on [-126,126]: 2.75764912 avg ulp diff, 232 max ulp
    // ulp histogram:
    //  0  = 87.81%
    //  1  =  4.18%
    float r = 1.33336498402e-3f;
    r = madd(x, r, 9.810352697968e-3f);
    r = madd(x, r, 5.551834031939e-2f);
    r = madd(x, r, 0.2401793301105f);
    r = madd(x, r, 0.693144857883f);
    r = madd(x, r, 1.0f);
    // multiply by 2 ^ m by adding in the exponent
    // NOTE: left-shift of negative number is undefined behavior
    return bit_cast<unsigned, float>(bit_cast<float, unsigned>(r) + (unsigned(m) << 23));
    // Clang: loop not vectorized: unsafe dependent memory operations in loop.
    // This is why we special case the OIIO_FMATH_SIMD_FRIENDLY above.
    // FIXME: as clang releases continue to improve, periodically check if
    // this is still the case.
#else
    return exp2f(xval);
#endif
}




template <typename T>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_exp (const T& x) {
    // Examined 2237485550 values of exp on [-87.3300018,87.3300018]: 2.6666452 avg ulp diff, 230 max ulp
    return fast_exp2(x * T(1 / M_LN2));
}

#ifdef __CUDA_ARCH__
template<>
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_exp (const float& x) {
    return __expf(x);
}
#endif



/// Faster float exp than is in libm, but still 100% accurate
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_correct_exp (float x)
{
#if defined(__x86_64__) && defined(__GNU_LIBRARY__) && defined(__GLIBC__ ) && defined(__GLIBC_MINOR__) && __GLIBC__ <= 2 && __GLIBC_MINOR__ < 16
    // On x86_64, versions of glibc < 2.16 have an issue where expf is
    // much slower than the double version.  This was fixed in glibc 2.16.
    return static_cast<float>(std::exp(static_cast<double>(x)));
#else
    return std::exp(x);
#endif
}


OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_exp10 (float x) {
#ifndef __CUDA_ARCH__
    // Examined 2217701018 values of exp10 on [-37.9290009,37.9290009]: 2.71732409 avg ulp diff, 232 max ulp
    return fast_exp2(x * float(M_LN10 / M_LN2));
#else
    return __exp10f(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_expm1 (float x) {
#ifndef __CUDA_ARCH__
    if (fabsf(x) < 0.03f) {
        float y = 1.0f - (1.0f - x); // crush denormals
        return copysignf(madd(0.5f, y * y, y), x);
    } else
        return fast_exp(x) - 1.0f;
#else
    return expm1f(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_sinh (float x) {
#ifndef __CUDA_ARCH__
    float a = fabsf(x);
    if (a > 1.0f) {
        // Examined 53389559 values of sinh on [1,87.3300018]: 33.6886442 avg ulp diff, 178 max ulp
        float e = fast_exp(a);
        return copysignf(0.5f * e - 0.5f / e, x);
    } else {
        a = 1.0f - (1.0f - a); // crush denorms
        float a2 = a * a;
        // degree 7 polynomial generated with sollya
        // Examined 2130706434 values of sinh on [-1,1]: 1.19209e-07 max error
        float r = 2.03945513931e-4f;
        r = madd(r, a2, 8.32990277558e-3f);
        r = madd(r, a2, 0.1666673421859f);
        r = madd(r * a, a2, a);
        return copysignf(r, x);
    }
#else
    return sinhf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_cosh (float x) {
#ifndef __CUDA_ARCH__
    // Examined 2237485550 values of cosh on [-87.3300018,87.3300018]: 1.78256726 avg ulp diff, 178 max ulp
    float e = fast_exp(fabsf(x));
    return 0.5f * e + 0.5f / e;
#else
    return coshf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_tanh (float x) {
#ifndef __CUDA_ARCH__
    // Examined 4278190080 values of tanh on [-3.40282347e+38,3.40282347e+38]: 3.12924e-06 max error
    // NOTE: ulp error is high because of sub-optimal handling around the origin
    float e = fast_exp(2.0f * fabsf(x));
    return copysignf(1 - 2 / (1 + e), x);
#else
    return tanhf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_safe_pow (float x, float y) {
    if (y == 0) return 1.0f; // x^0=1
    if (x == 0) return 0.0f; // 0^y=0
    // be cheap & exact for special case of squaring and identity
    if (y == 1.0f)
        return x;
    if (y == 2.0f) {
#ifndef __CUDA_ARCH__
        return std::min (x*x, std::numeric_limits<float>::max());
#else
        return fminf (x*x, std::numeric_limits<float>::max());
#endif
    }
    float sign = 1.0f;
    if (OIIO_UNLIKELY(x < 0.0f)) {
        // if x is negative, only deal with integer powers
        // powf returns NaN for non-integers, we will return 0 instead
        int ybits = bit_cast<float, int>(y) & 0x7fffffff;
        if (ybits >= 0x4b800000) {
            // always even int, keep positive
        } else if (ybits >= 0x3f800000) {
            // bigger than 1, check
            int k = (ybits >> 23) - 127;  // get exponent
            int j =  ybits >> (23 - k);   // shift out possible fractional bits
            if ((j << (23 - k)) == ybits) // rebuild number and check for a match
                sign = bit_cast<int, float>(0x3f800000 | (j << 31)); // +1 for even, -1 for odd
            else
                return 0.0f; // not integer
        } else {
            return 0.0f; // not integer
        }
    }
    return sign * fast_exp2(y * fast_log2(fabsf(x)));
}


// Fast simd pow that only needs to work for positive x
template<typename T, typename U>
OIIO_FORCEINLINE OIIO_HOSTDEVICE T fast_pow_pos (const T& x, const U& y) {
    return fast_exp2(y * fast_log2(x));
}


// Fast cube root (performs better that using fast_pow's above with y=1/3)
OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_cbrt (float x) {
#ifndef __CUDA_ARCH__
    float x0 = fabsf(x);
    // from hacker's delight
    float a = bit_cast<int, float>(0x2a5137a0 + bit_cast<float, int>(x0) / 3); // Initial guess.
    // Examined 14272478 values of cbrt on [-9.99999935e-39,9.99999935e-39]: 8.14687e-14 max error
    // Examined 2131958802 values of cbrt on [9.99999935e-39,3.40282347e+38]: 2.46930719 avg ulp diff, 12 max ulp
    a = 0.333333333f * (2.0f * a + x0 / (a * a));  // Newton step.
    a = 0.333333333f * (2.0f * a + x0 / (a * a));  // Newton step again.
    a = (x0 == 0) ? 0 : a; // Fix 0 case
    return copysignf(a, x);
#else
    return cbrtf(x);
#endif
}


OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_erf (float x)
{
#ifndef __CUDA_ARCH__
    // Examined 1082130433 values of erff on [0,4]: 1.93715e-06 max error
    // Abramowitz and Stegun, 7.1.28
    const float a1 = 0.0705230784f;
    const float a2 = 0.0422820123f;
    const float a3 = 0.0092705272f;
    const float a4 = 0.0001520143f;
    const float a5 = 0.0002765672f;
    const float a6 = 0.0000430638f;
    const float a = fabsf(x);
    const float b = 1.0f - (1.0f - a); // crush denormals
    const float r = madd(madd(madd(madd(madd(madd(a6, b, a5), b, a4), b, a3), b, a2), b, a1), b, 1.0f);
    const float s = r * r; // ^2
    const float t = s * s; // ^4
    const float u = t * t; // ^8
    const float v = u * u; // ^16
    return copysignf(1.0f - 1.0f / v, x);
#else
    return erff(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_erfc (float x)
{
#ifndef __CUDA_ARCH__
    // Examined 2164260866 values of erfcf on [-4,4]: 1.90735e-06 max error
    // ulp histogram:
    //   0  = 80.30%
    return 1.0f - fast_erf(x);
#else
    return erfcf(x);
#endif
}

OIIO_FORCEINLINE OIIO_HOSTDEVICE float fast_ierf (float x)
{
    // from: Approximating the erfinv function by Mike Giles
    // to avoid trouble at the limit, clamp input to 1-eps
    float a = fabsf(x); if (a > 0.99999994f) a = 0.99999994f;
    float w = -fast_log((1.0f - a) * (1.0f + a)), p;
    if (w < 5.0f) {
        w = w - 2.5f;
        p =  2.81022636e-08f;
        p = madd(p, w,  3.43273939e-07f);
        p = madd(p, w, -3.5233877e-06f );
        p = madd(p, w, -4.39150654e-06f);
        p = madd(p, w,  0.00021858087f );
        p = madd(p, w, -0.00125372503f );
        p = madd(p, w, -0.00417768164f );
        p = madd(p, w,  0.246640727f   );
        p = madd(p, w,  1.50140941f    );
    } else {
        w = sqrtf(w) - 3.0f;
        p = -0.000200214257f;
        p = madd(p, w,  0.000100950558f);
        p = madd(p, w,  0.00134934322f );
        p = madd(p, w, -0.00367342844f );
        p = madd(p, w,  0.00573950773f );
        p = madd(p, w, -0.0076224613f  );
        p = madd(p, w,  0.00943887047f );
        p = madd(p, w,  1.00167406f    );
        p = madd(p, w,  2.83297682f    );
    }
    return p * x;
}

// (end of fast* functions)
////////////////////////////////////////////////////////////////////////////




////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
//
// MISCELLANEOUS NUMERICAL METHODS
//


/// Solve for the x for which func(x) == y on the interval [xmin,xmax].
/// Use a maximum of maxiter iterations, and stop any time the remaining
/// search interval or the function evaluations <= eps.  If brack is
/// non-NULL, set it to true if y is in [f(xmin), f(xmax)], otherwise
/// false (in which case the caller should know that the results may be
/// unreliable.  Results are undefined if the function is not monotonic
/// on that interval or if there are multiple roots in the interval (it
/// may not converge, or may converge to any of the roots without
/// telling you that there are more than one).
template<class T, class Func> OIIO_HOSTDEVICE
T invert (Func &func, T y, T xmin=0.0, T xmax=1.0,
          int maxiters=32, T eps=1.0e-6, bool *brack=0)
{
    // Use the Regula Falsi method, falling back to bisection if it
    // hasn't converged after 3/4 of the maximum number of iterations.
    // See, e.g., Numerical Recipes for the basic ideas behind both
    // methods.
    T v0 = func(xmin), v1 = func(xmax);
    T x = xmin, v = v0;
    bool increasing = (v0 < v1);
    T vmin = increasing ? v0 : v1;
    T vmax = increasing ? v1 : v0;
    bool bracketed = (y >= vmin && y <= vmax);
    if (brack)
        *brack = bracketed;
    if (! bracketed) {
        // If our bounds don't bracket the zero, just give up, and
        // return the appropriate "edge" of the interval
        return ((y < vmin) == increasing) ? xmin : xmax;
    }
    if (fabs(v0-v1) < eps)   // already close enough
        return x;
    int rfiters = (3*maxiters)/4;   // how many times to try regula falsi
    for (int iters = 0;  iters < maxiters;  ++iters) {
        T t;  // interpolation factor
        if (iters < rfiters) {
            // Regula falsi
            t = (y-v0)/(v1-v0);
            if (t <= T(0) || t >= T(1))
                t = T(0.5);  // RF convergence failure -- bisect instead
        } else {
            t = T(0.5);            // bisection
        }
        x = lerp (xmin, xmax, t);
        v = func(x);
        if ((v < y) == increasing) {
            xmin = x; v0 = v;
        } else {
            xmax = x; v1 = v;
        }
        if (fabs(xmax-xmin) < eps || fabs(v-y) < eps)
            return x;   // converged
    }
    return x;
}



/// Linearly interpolate a list of evenly-spaced knots y[0..len-1] with
/// y[0] corresponding to the value at x==0.0 and y[len-1] corresponding to
/// x==1.0.
inline OIIO_HOSTDEVICE float
interpolate_linear (float x, span_strided<const float> y)
{
#ifndef __CUDA_ARCH__
    DASSERT_MSG (y.size() >= 2, "interpolate_linear needs at least 2 knot values (%zd)", y.size());
#endif
    x = clamp (x, float(0.0), float(1.0));
    int nsegs = int(y.size()) - 1;
    int segnum;
    x = floorfrac (x*nsegs, &segnum);
#ifndef __CUDA_ARCH__
    int nextseg = std::min (segnum+1, nsegs);
#else
    int nextseg = min (segnum+1, nsegs);
#endif
    return lerp (y[segnum], y[nextseg], x);
}

// (end miscellaneous numerical methods)
////////////////////////////////////////////////////////////////////////////



OIIO_NAMESPACE_END