lib/math/sincos_helpers.cl

/*
 * Copyright (c) 2014 Advanced Micro Devices, Inc.
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in
 * all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
 * THE SOFTWARE.
 */

#include <clc/clc.h>

#include "math.h"
#include "tables.h"
#include "sincos_helpers.h"

#define bitalign(hi, lo, shift) \
  ((hi) << (32 - (shift))) | ((lo) >> (shift));

#define bytealign(src0, src1, src2) \
  ((uint) (((((long)(src0)) << 32) | (long)(src1)) >> (((src2) & 3)*8)))

_CLC_DEF float __clc_sinf_piby4(float x, float y) {
    // Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
    // = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
    // = x * f(w)
    // where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
    // We use a minimax approximation of (f(w) - 1) / w
    // because this produces an expansion in even powers of x.

    const float c1 = -0.1666666666e0f;
    const float c2 = 0.8333331876e-2f;
    const float c3 = -0.198400874e-3f;
    const float c4 = 0.272500015e-5f;
    const float c5 = -2.5050759689e-08f; // 0xb2d72f34
    const float c6 = 1.5896910177e-10f;	 // 0x2f2ec9d3

    float z = x * x;
    float v = z * x;
    float r = mad(z, mad(z, mad(z, mad(z, c6, c5), c4), c3), c2);
    float ret = x - mad(v, -c1, mad(z, mad(y, 0.5f, -v*r), -y));

    return ret;
}

_CLC_DEF float __clc_cosf_piby4(float x, float y) {
    // Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
    // = f(w)
    // where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
    // We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
    // because this produces an expansion in even powers of x.

    const float c1 = 0.416666666e-1f;
    const float c2 = -0.138888876e-2f;
    const float c3 = 0.248006008e-4f;
    const float c4 = -0.2730101334e-6f;
    const float c5 = 2.0875723372e-09f;	 // 0x310f74f6
    const float c6 = -1.1359647598e-11f; // 0xad47d74e

    float z = x * x;
    float r = z * mad(z, mad(z, mad(z, mad(z, mad(z, c6,  c5), c4), c3), c2), c1);

    // if |x| < 0.3
    float qx = 0.0f;

    int ix = as_int(x) & EXSIGNBIT_SP32;

    //  0.78125 > |x| >= 0.3
    float xby4 = as_float(ix - 0x01000000);
    qx = (ix >= 0x3e99999a) & (ix <= 0x3f480000) ? xby4 : qx;

    // x > 0.78125
    qx = ix > 0x3f480000 ? 0.28125f : qx;

    float hz = mad(z, 0.5f, -qx);
    float a = 1.0f - qx;
    float ret = a - (hz - mad(z, r, -x*y));
    return ret;
}

_CLC_DEF float __clc_tanf_piby4(float x, int regn)
{
    // Core Remez [1,2] approximation to tan(x) on the interval [0,pi/4].
    float r = x * x;

    float a = mad(r, -0.0172032480471481694693109f, 0.385296071263995406715129f);

    float b = mad(r,
	          mad(r, 0.01844239256901656082986661f, -0.51396505478854532132342f),
	          1.15588821434688393452299f);

    float t = mad(x*r, native_divide(a, b), x);
    float tr = -MATH_RECIP(t);

    return regn & 1 ? tr : t;
}

_CLC_DEF void __clc_fullMulS(float *hi, float *lo, float a, float b, float bh, float bt)
{
    if (HAVE_HW_FMA32()) {
        float ph = a * b;
        *hi = ph;
        *lo = fma(a, b, -ph);
    } else {
        float ah = as_float(as_uint(a) & 0xfffff000U);
        float at = a - ah;
        float ph = a * b;
        float pt = mad(at, bt, mad(at, bh, mad(ah, bt, mad(ah, bh, -ph))));
        *hi = ph;
        *lo = pt;
    }
}

_CLC_DEF float __clc_removePi2S(float *hi, float *lo, float x)
{
    // 72 bits of pi/2
    const float fpiby2_1 = (float) 0xC90FDA / 0x1.0p+23f;
    const float fpiby2_1_h = (float) 0xC90 / 0x1.0p+11f;
    const float fpiby2_1_t = (float) 0xFDA / 0x1.0p+23f;

    const float fpiby2_2 = (float) 0xA22168 / 0x1.0p+47f;
    const float fpiby2_2_h = (float) 0xA22 / 0x1.0p+35f;
    const float fpiby2_2_t = (float) 0x168 / 0x1.0p+47f;

    const float fpiby2_3 = (float) 0xC234C4 / 0x1.0p+71f;
    const float fpiby2_3_h = (float) 0xC23 / 0x1.0p+59f;
    const float fpiby2_3_t = (float) 0x4C4 / 0x1.0p+71f;

    const float twobypi = 0x1.45f306p-1f;

    float fnpi2 = trunc(mad(x, twobypi, 0.5f));

    // subtract n * pi/2 from x
    float rhead, rtail;
    __clc_fullMulS(&rhead, &rtail, fnpi2, fpiby2_1, fpiby2_1_h, fpiby2_1_t);
    float v = x - rhead;
    float rem = v + (((x - v) - rhead) - rtail);

    float rhead2, rtail2;
    __clc_fullMulS(&rhead2, &rtail2, fnpi2, fpiby2_2, fpiby2_2_h, fpiby2_2_t);
    v = rem - rhead2;
    rem = v + (((rem - v) - rhead2) - rtail2);

    float rhead3, rtail3;
    __clc_fullMulS(&rhead3, &rtail3, fnpi2, fpiby2_3, fpiby2_3_h, fpiby2_3_t);
    v = rem - rhead3;

    *hi = v + ((rem - v) - rhead3);
    *lo = -rtail3;
    return fnpi2;
}

_CLC_DEF int __clc_argReductionSmallS(float *r, float *rr, float x)
{
    float fnpi2 = __clc_removePi2S(r, rr, x);
    return (int)fnpi2 & 0x3;
}

#define FULL_MUL(A, B, HI, LO) \
    LO = A * B; \
    HI = mul_hi(A, B)

#define FULL_MAD(A, B, C, HI, LO) \
    LO = ((A) * (B) + (C)); \
    HI = mul_hi(A, B); \
    HI += LO < C

_CLC_DEF int __clc_argReductionLargeS(float *r, float *rr, float x)
{
    int xe = (int)(as_uint(x) >> 23) - 127;
    uint xm = 0x00800000U | (as_uint(x) & 0x7fffffU);

    // 224 bits of 2/PI: . A2F9836E 4E441529 FC2757D1 F534DDC0 DB629599 3C439041 FE5163AB
    const uint b6 = 0xA2F9836EU;
    const uint b5 = 0x4E441529U;
    const uint b4 = 0xFC2757D1U;
    const uint b3 = 0xF534DDC0U;
    const uint b2 = 0xDB629599U;
    const uint b1 = 0x3C439041U;
    const uint b0 = 0xFE5163ABU;

    uint p0, p1, p2, p3, p4, p5, p6, p7, c0, c1;

    FULL_MUL(xm, b0, c0, p0);
    FULL_MAD(xm, b1, c0, c1, p1);
    FULL_MAD(xm, b2, c1, c0, p2);
    FULL_MAD(xm, b3, c0, c1, p3);
    FULL_MAD(xm, b4, c1, c0, p4);
    FULL_MAD(xm, b5, c0, c1, p5);
    FULL_MAD(xm, b6, c1, p7, p6);

    uint fbits = 224 + 23 - xe;

    // shift amount to get 2 lsb of integer part at top 2 bits
    //   min: 25 (xe=18) max: 134 (xe=127)
    uint shift = 256U - 2 - fbits;

    // Shift by up to 134/32 = 4 words
    int c = shift > 31;
    p7 = c ? p6 : p7;
    p6 = c ? p5 : p6;
    p5 = c ? p4 : p5;
    p4 = c ? p3 : p4;
    p3 = c ? p2 : p3;
    p2 = c ? p1 : p2;
    p1 = c ? p0 : p1;
    shift -= (-c) & 32;

    c = shift > 31;
    p7 = c ? p6 : p7;
    p6 = c ? p5 : p6;
    p5 = c ? p4 : p5;
    p4 = c ? p3 : p4;
    p3 = c ? p2 : p3;
    p2 = c ? p1 : p2;
    shift -= (-c) & 32;

    c = shift > 31;
    p7 = c ? p6 : p7;
    p6 = c ? p5 : p6;
    p5 = c ? p4 : p5;
    p4 = c ? p3 : p4;
    p3 = c ? p2 : p3;
    shift -= (-c) & 32;

    c = shift > 31;
    p7 = c ? p6 : p7;
    p6 = c ? p5 : p6;
    p5 = c ? p4 : p5;
    p4 = c ? p3 : p4;
    shift -= (-c) & 32;

    // bitalign cannot handle a shift of 32
    c = shift > 0;
    shift = 32 - shift;
    uint t7 = bitalign(p7, p6, shift);
    uint t6 = bitalign(p6, p5, shift);
    uint t5 = bitalign(p5, p4, shift);
    p7 = c ? t7 : p7;
    p6 = c ? t6 : p6;
    p5 = c ? t5 : p5;

    // Get 2 lsb of int part and msb of fraction
    int i = p7 >> 29;

    // Scoot up 2 more bits so only fraction remains
    p7 = bitalign(p7, p6, 30);
    p6 = bitalign(p6, p5, 30);
    p5 = bitalign(p5, p4, 30);

    // Subtract 1 if msb of fraction is 1, i.e. fraction >= 0.5
    uint flip = i & 1 ? 0xffffffffU : 0U;
    uint sign = i & 1 ? 0x80000000U : 0U;
    p7 = p7 ^ flip;
    p6 = p6 ^ flip;
    p5 = p5 ^ flip;

    // Find exponent and shift away leading zeroes and hidden bit
    xe = clz(p7) + 1;
    shift = 32 - xe;
    p7 = bitalign(p7, p6, shift);
    p6 = bitalign(p6, p5, shift);

    // Most significant part of fraction
    float q1 = as_float(sign | ((127 - xe) << 23) | (p7 >> 9));

    // Shift out bits we captured on q1
    p7 = bitalign(p7, p6, 32-23);

    // Get 24 more bits of fraction in another float, there are not long strings of zeroes here
    int xxe = clz(p7) + 1;
    p7 = bitalign(p7, p6, 32-xxe);
    float q0 = as_float(sign | ((127 - (xe + 23 + xxe)) << 23) | (p7 >> 9));

    // At this point, the fraction q1 + q0 is correct to at least 48 bits
    // Now we need to multiply the fraction by pi/2
    // This loses us about 4 bits
    // pi/2 = C90 FDA A22 168 C23 4C4

    const float pio2h = (float)0xc90fda / 0x1.0p+23f;
    const float pio2hh = (float)0xc90 / 0x1.0p+11f;
    const float pio2ht = (float)0xfda / 0x1.0p+23f;
    const float pio2t = (float)0xa22168 / 0x1.0p+47f;

    float rh, rt;

    if (HAVE_HW_FMA32()) {
        rh = q1 * pio2h;
        rt = fma(q0, pio2h, fma(q1, pio2t, fma(q1, pio2h, -rh)));
    } else {
        float q1h = as_float(as_uint(q1) & 0xfffff000);
        float q1t = q1 - q1h;
        rh = q1 * pio2h;
        rt = mad(q1t, pio2ht, mad(q1t, pio2hh, mad(q1h, pio2ht, mad(q1h, pio2hh, -rh))));
        rt = mad(q0, pio2h, mad(q1, pio2t, rt));
    }

    float t = rh + rt;
    rt = rt - (t - rh);

    *r = t;
    *rr = rt;
    return ((i >> 1) + (i & 1)) & 0x3;
}

_CLC_DEF int __clc_argReductionS(float *r, float *rr, float x)
{
    if (x < 0x1.0p+23f)
        return __clc_argReductionSmallS(r, rr, x);
    else
        return __clc_argReductionLargeS(r, rr, x);
}

#ifdef cl_khr_fp64

#pragma OPENCL EXTENSION cl_khr_fp64 : enable

// Reduction for medium sized arguments
_CLC_DEF void __clc_remainder_piby2_medium(double x, double *r, double *rr, int *regn) {
    // How many pi/2 is x a multiple of?
    const double two_by_pi = 0x1.45f306dc9c883p-1;
    double dnpi2 = trunc(fma(x, two_by_pi, 0.5));

    const double piby2_h = -7074237752028440.0 / 0x1.0p+52;
    const double piby2_m = -2483878800010755.0 / 0x1.0p+105;
    const double piby2_t = -3956492004828932.0 / 0x1.0p+158;

    // Compute product of npi2 with 159 bits of 2/pi
    double p_hh = piby2_h * dnpi2;
    double p_ht = fma(piby2_h, dnpi2, -p_hh);
    double p_mh = piby2_m * dnpi2;
    double p_mt = fma(piby2_m, dnpi2, -p_mh);
    double p_th = piby2_t * dnpi2;
    double p_tt = fma(piby2_t, dnpi2, -p_th);

    // Reduce to 159 bits
    double ph = p_hh;
    double pm = p_ht + p_mh;
    double t = p_mh - (pm - p_ht);
    double pt = p_th + t + p_mt + p_tt;
    t = ph + pm; pm = pm - (t - ph); ph = t;
    t = pm + pt; pt = pt - (t - pm); pm = t;

    // Subtract from x
    t = x + ph;
    double qh = t + pm;
    double qt = pm - (qh - t) + pt;

    *r = qh;
    *rr = qt;
    *regn = (int)(long)dnpi2 & 0x3;
}

// Given positive argument x, reduce it to the range [-pi/4,pi/4] using
// extra precision, and return the result in r, rr.
// Return value "regn" tells how many lots of pi/2 were subtracted
// from x to put it in the range [-pi/4,pi/4], mod 4.

_CLC_DEF void __clc_remainder_piby2_large(double x, double *r, double *rr, int *regn) {

    long ux = as_long(x);
    int e = (int)(ux >> 52) -  1023;
    int i = max(23, (e >> 3) + 17);
    int j = 150 - i;
    int j16 = j & ~0xf;
    double fract_temp;

    // The following extracts 192 consecutive bits of 2/pi aligned on an arbitrary byte boundary
    uint4 q0 = USE_TABLE(pibits_tbl, j16);
    uint4 q1 = USE_TABLE(pibits_tbl, (j16 + 16));
    uint4 q2 = USE_TABLE(pibits_tbl, (j16 + 32));

    int k = (j >> 2) & 0x3;
    int4 c = (int4)k == (int4)(0, 1, 2, 3);

    uint u0, u1, u2, u3, u4, u5, u6;

    u0 = c.s1 ? q0.s1 : q0.s0;
    u0 = c.s2 ? q0.s2 : u0;
    u0 = c.s3 ? q0.s3 : u0;

    u1 = c.s1 ? q0.s2 : q0.s1;
    u1 = c.s2 ? q0.s3 : u1;
    u1 = c.s3 ? q1.s0 : u1;

    u2 = c.s1 ? q0.s3 : q0.s2;
    u2 = c.s2 ? q1.s0 : u2;
    u2 = c.s3 ? q1.s1 : u2;

    u3 = c.s1 ? q1.s0 : q0.s3;
    u3 = c.s2 ? q1.s1 : u3;
    u3 = c.s3 ? q1.s2 : u3;

    u4 = c.s1 ? q1.s1 : q1.s0;
    u4 = c.s2 ? q1.s2 : u4;
    u4 = c.s3 ? q1.s3 : u4;

    u5 = c.s1 ? q1.s2 : q1.s1;
    u5 = c.s2 ? q1.s3 : u5;
    u5 = c.s3 ? q2.s0 : u5;

    u6 = c.s1 ? q1.s3 : q1.s2;
    u6 = c.s2 ? q2.s0 : u6;
    u6 = c.s3 ? q2.s1 : u6;

    uint v0 = bytealign(u1, u0, j);
    uint v1 = bytealign(u2, u1, j);
    uint v2 = bytealign(u3, u2, j);
    uint v3 = bytealign(u4, u3, j);
    uint v4 = bytealign(u5, u4, j);
    uint v5 = bytealign(u6, u5, j);

    // Place those 192 bits in 4 48-bit doubles along with correct exponent
    // If i > 1018 we would get subnormals so we scale p up and x down to get the same product
    i = 2 + 8*i;
    x *= i > 1018 ? 0x1.0p-136 : 1.0;
    i -= i > 1018 ? 136 : 0;

    uint ua = (uint)(1023 + 52 - i) << 20;
    double a = as_double((uint2)(0, ua));
    double p0 = as_double((uint2)(v0, ua | (v1 & 0xffffU))) - a;
    ua += 0x03000000U;
    a = as_double((uint2)(0, ua));
    double p1 = as_double((uint2)((v2 << 16) | (v1 >> 16), ua | (v2 >> 16))) - a;
    ua += 0x03000000U;
    a = as_double((uint2)(0, ua));
    double p2 = as_double((uint2)(v3, ua | (v4 & 0xffffU))) - a;
    ua += 0x03000000U;
    a = as_double((uint2)(0, ua));
    double p3 = as_double((uint2)((v5 << 16) | (v4 >> 16), ua | (v5 >> 16))) - a;

    // Exact multiply
    double f0h = p0 * x;
    double f0l = fma(p0, x, -f0h);
    double f1h = p1 * x;
    double f1l = fma(p1, x, -f1h);
    double f2h = p2 * x;
    double f2l = fma(p2, x, -f2h);
    double f3h = p3 * x;
    double f3l = fma(p3, x, -f3h);

    // Accumulate product into 4 doubles
    double s, t;

    double f3 = f3h + f2h;
    t = f2h - (f3 - f3h);
    s = f3l + t;
    t = t - (s - f3l);

    double f2 = s + f1h;
    t = f1h - (f2 - s) + t;
    s = f2l + t;
    t = t - (s - f2l);

    double f1 = s + f0h;
    t = f0h - (f1 - s) + t;
    s = f1l + t;

    double f0 = s + f0l;

    // Strip off unwanted large integer bits
    f3 = 0x1.0p+10 * fract(f3 * 0x1.0p-10, &fract_temp);
    f3 += f3 + f2 < 0.0 ? 0x1.0p+10 : 0.0;

    // Compute least significant integer bits
    t = f3 + f2;
    double di = t - fract(t, &fract_temp);
    i = (float)di;

    // Shift out remaining integer part
    f3 -= di;
    s = f3 + f2; t = f2 - (s - f3); f3 = s; f2 = t;
    s = f2 + f1; t = f1 - (s - f2); f2 = s; f1 = t;
    f1 += f0;

    // Subtract 1 if fraction is >= 0.5, and update regn
    int g = f3 >= 0.5;
    i += g;
    f3 -= (float)g;

    // Shift up bits
    s = f3 + f2; t = f2 -(s - f3); f3 = s; f2 = t + f1;

    // Multiply precise fraction by pi/2 to get radians
    const double p2h = 7074237752028440.0 / 0x1.0p+52;
    const double p2t = 4967757600021510.0 / 0x1.0p+106;

    double rhi = f3 * p2h;
    double rlo = fma(f2, p2h, fma(f3, p2t, fma(f3, p2h, -rhi)));

    *r = rhi + rlo;
    *rr = rlo - (*r - rhi);
    *regn = i & 0x3;
}


_CLC_DEF double2 __clc_sincos_piby4(double x, double xx) {
    // Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
    //                      = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
    //                      = x * f(w)
    // where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
    // We use a minimax approximation of (f(w) - 1) / w
    // because this produces an expansion in even powers of x.
    // If xx (the tail of x) is non-zero, we add a correction
    // term g(x,xx) = (1-x*x/2)*xx to the result, where g(x,xx)
    // is an approximation to cos(x)*sin(xx) valid because
    // xx is tiny relative to x.

    // Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
    //                      = f(w)
    // where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
    // We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
    // because this produces an expansion in even powers of x.
    // If xx (the tail of x) is non-zero, we subtract a correction
    // term g(x,xx) = x*xx to the result, where g(x,xx)
    // is an approximation to sin(x)*sin(xx) valid because
    // xx is tiny relative to x.

    const double sc1 = -0.166666666666666646259241729;
    const double sc2 =  0.833333333333095043065222816e-2;
    const double sc3 = -0.19841269836761125688538679e-3;
    const double sc4 =  0.275573161037288022676895908448e-5;
    const double sc5 = -0.25051132068021699772257377197e-7;
    const double sc6 =  0.159181443044859136852668200e-9;

    const double cc1 =  0.41666666666666665390037e-1;
    const double cc2 = -0.13888888888887398280412e-2;
    const double cc3 =  0.248015872987670414957399e-4;
    const double cc4 = -0.275573172723441909470836e-6;
    const double cc5 =  0.208761463822329611076335e-8;
    const double cc6 = -0.113826398067944859590880e-10;

    double x2 = x * x;
    double x3 = x2 * x;
    double r = 0.5 * x2;
    double t = 1.0 - r;

    double sp = fma(fma(fma(fma(sc6, x2, sc5), x2, sc4), x2, sc3), x2, sc2);

    double cp = t + fma(fma(fma(fma(fma(fma(cc6, x2, cc5), x2, cc4), x2, cc3), x2, cc2), x2, cc1),
                        x2*x2, fma(x, xx, (1.0 - t) - r));

    double2 ret;
    ret.lo = x - fma(-x3, sc1, fma(fma(-x3, sp, 0.5*xx), x2, -xx));
    ret.hi = cp;

    return ret;
}

#endif