src/math/exp2.rs

// origin: FreeBSD /usr/src/lib/msun/src/s_exp2.c */
//-
// Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions
// are met:
// 1. Redistributions of source code must retain the above copyright
//    notice, this list of conditions and the following disclaimer.
// 2. Redistributions in binary form must reproduce the above copyright
//    notice, this list of conditions and the following disclaimer in the
//    documentation and/or other materials provided with the distribution.
//
// THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
// ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
// FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
// DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
// OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
// HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
// LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
// OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
// SUCH DAMAGE.

use super::scalbn;

const TBLSIZE: usize = 256;

#[cfg_attr(rustfmt, rustfmt_skip)]
static TBL: [u64; TBLSIZE * 2] = [
    //  exp2(z + eps)          eps
    0x3fe6a09e667f3d5d, 0x3d39880000000000,
    0x3fe6b052fa751744, 0x3cd8000000000000,
    0x3fe6c012750bd9fe, 0xbd28780000000000,
    0x3fe6cfdcddd476bf, 0x3d1ec00000000000,
    0x3fe6dfb23c651a29, 0xbcd8000000000000,
    0x3fe6ef9298593ae3, 0xbcbc000000000000,
    0x3fe6ff7df9519386, 0xbd2fd80000000000,
    0x3fe70f7466f42da3, 0xbd2c880000000000,
    0x3fe71f75e8ec5fc3, 0x3d13c00000000000,
    0x3fe72f8286eacf05, 0xbd38300000000000,
    0x3fe73f9a48a58152, 0xbd00c00000000000,
    0x3fe74fbd35d7ccfc, 0x3d2f880000000000,
    0x3fe75feb564267f1, 0x3d03e00000000000,
    0x3fe77024b1ab6d48, 0xbd27d00000000000,
    0x3fe780694fde5d38, 0xbcdd000000000000,
    0x3fe790b938ac1d00, 0x3ce3000000000000,
    0x3fe7a11473eb0178, 0xbced000000000000,
    0x3fe7b17b0976d060, 0x3d20400000000000,
    0x3fe7c1ed0130c133, 0x3ca0000000000000,
    0x3fe7d26a62ff8636, 0xbd26900000000000,
    0x3fe7e2f336cf4e3b, 0xbd02e00000000000,
    0x3fe7f3878491c3e8, 0xbd24580000000000,
    0x3fe80427543e1b4e, 0x3d33000000000000,
    0x3fe814d2add1071a, 0x3d0f000000000000,
    0x3fe82589994ccd7e, 0xbd21c00000000000,
    0x3fe8364c1eb942d0, 0x3d29d00000000000,
    0x3fe8471a4623cab5, 0x3d47100000000000,
    0x3fe857f4179f5bbc, 0x3d22600000000000,
    0x3fe868d99b4491af, 0xbd32c40000000000,
    0x3fe879cad931a395, 0xbd23000000000000,
    0x3fe88ac7d98a65b8, 0xbd2a800000000000,
    0x3fe89bd0a4785800, 0xbced000000000000,
    0x3fe8ace5422aa223, 0x3d33280000000000,
    0x3fe8be05bad619fa, 0x3d42b40000000000,
    0x3fe8cf3216b54383, 0xbd2ed00000000000,
    0x3fe8e06a5e08664c, 0xbd20500000000000,
    0x3fe8f1ae99157807, 0x3d28280000000000,
    0x3fe902fed0282c0e, 0xbd1cb00000000000,
    0x3fe9145b0b91ff96, 0xbd05e00000000000,
    0x3fe925c353aa2ff9, 0x3cf5400000000000,
    0x3fe93737b0cdc64a, 0x3d17200000000000,
    0x3fe948b82b5f98ae, 0xbd09000000000000,
    0x3fe95a44cbc852cb, 0x3d25680000000000,
    0x3fe96bdd9a766f21, 0xbd36d00000000000,
    0x3fe97d829fde4e2a, 0xbd01000000000000,
    0x3fe98f33e47a23a3, 0x3d2d000000000000,
    0x3fe9a0f170ca0604, 0xbd38a40000000000,
    0x3fe9b2bb4d53ff89, 0x3d355c0000000000,
    0x3fe9c49182a3f15b, 0x3d26b80000000000,
    0x3fe9d674194bb8c5, 0xbcec000000000000,
    0x3fe9e86319e3238e, 0x3d17d00000000000,
    0x3fe9fa5e8d07f302, 0x3d16400000000000,
    0x3fea0c667b5de54d, 0xbcf5000000000000,
    0x3fea1e7aed8eb8f6, 0x3d09e00000000000,
    0x3fea309bec4a2e27, 0x3d2ad80000000000,
    0x3fea42c980460a5d, 0xbd1af00000000000,
    0x3fea5503b23e259b, 0x3d0b600000000000,
    0x3fea674a8af46213, 0x3d38880000000000,
    0x3fea799e1330b3a7, 0x3d11200000000000,
    0x3fea8bfe53c12e8d, 0x3d06c00000000000,
    0x3fea9e6b5579fcd2, 0xbd29b80000000000,
    0x3feab0e521356fb8, 0x3d2b700000000000,
    0x3feac36bbfd3f381, 0x3cd9000000000000,
    0x3fead5ff3a3c2780, 0x3ce4000000000000,
    0x3feae89f995ad2a3, 0xbd2c900000000000,
    0x3feafb4ce622f367, 0x3d16500000000000,
    0x3feb0e07298db790, 0x3d2fd40000000000,
    0x3feb20ce6c9a89a9, 0x3d12700000000000,
    0x3feb33a2b84f1a4b, 0x3d4d470000000000,
    0x3feb468415b747e7, 0xbd38380000000000,
    0x3feb59728de5593a, 0x3c98000000000000,
    0x3feb6c6e29f1c56a, 0x3d0ad00000000000,
    0x3feb7f76f2fb5e50, 0x3cde800000000000,
    0x3feb928cf22749b2, 0xbd04c00000000000,
    0x3feba5b030a10603, 0xbd0d700000000000,
    0x3febb8e0b79a6f66, 0x3d0d900000000000,
    0x3febcc1e904bc1ff, 0x3d02a00000000000,
    0x3febdf69c3f3a16f, 0xbd1f780000000000,
    0x3febf2c25bd71db8, 0xbd10a00000000000,
    0x3fec06286141b2e9, 0xbd11400000000000,
    0x3fec199bdd8552e0, 0x3d0be00000000000,
    0x3fec2d1cd9fa64ee, 0xbd09400000000000,
    0x3fec40ab5fffd02f, 0xbd0ed00000000000,
    0x3fec544778fafd15, 0x3d39660000000000,
    0x3fec67f12e57d0cb, 0xbd1a100000000000,
    0x3fec7ba88988c1b6, 0xbd58458000000000,
    0x3fec8f6d9406e733, 0xbd1a480000000000,
    0x3feca3405751c4df, 0x3ccb000000000000,
    0x3fecb720dcef9094, 0x3d01400000000000,
    0x3feccb0f2e6d1689, 0x3cf0200000000000,
    0x3fecdf0b555dc412, 0x3cf3600000000000,
    0x3fecf3155b5bab3b, 0xbd06900000000000,
    0x3fed072d4a0789bc, 0x3d09a00000000000,
    0x3fed1b532b08c8fa, 0xbd15e00000000000,
    0x3fed2f87080d8a85, 0x3d1d280000000000,
    0x3fed43c8eacaa203, 0x3d01a00000000000,
    0x3fed5818dcfba491, 0x3cdf000000000000,
    0x3fed6c76e862e6a1, 0xbd03a00000000000,
    0x3fed80e316c9834e, 0xbd0cd80000000000,
    0x3fed955d71ff6090, 0x3cf4c00000000000,
    0x3feda9e603db32ae, 0x3cff900000000000,
    0x3fedbe7cd63a8325, 0x3ce9800000000000,
    0x3fedd321f301b445, 0xbcf5200000000000,
    0x3fede7d5641c05bf, 0xbd1d700000000000,
    0x3fedfc97337b9aec, 0xbd16140000000000,
    0x3fee11676b197d5e, 0x3d0b480000000000,
    0x3fee264614f5a3e7, 0x3d40ce0000000000,
    0x3fee3b333b16ee5c, 0x3d0c680000000000,
    0x3fee502ee78b3fb4, 0xbd09300000000000,
    0x3fee653924676d68, 0xbce5000000000000,
    0x3fee7a51fbc74c44, 0xbd07f80000000000,
    0x3fee8f7977cdb726, 0xbcf3700000000000,
    0x3feea4afa2a490e8, 0x3ce5d00000000000,
    0x3feeb9f4867ccae4, 0x3d161a0000000000,
    0x3feecf482d8e680d, 0x3cf5500000000000,
    0x3feee4aaa2188514, 0x3cc6400000000000,
    0x3feefa1bee615a13, 0xbcee800000000000,
    0x3fef0f9c1cb64106, 0xbcfa880000000000,
    0x3fef252b376bb963, 0xbd2c900000000000,
    0x3fef3ac948dd7275, 0x3caa000000000000,
    0x3fef50765b6e4524, 0xbcf4f00000000000,
    0x3fef6632798844fd, 0x3cca800000000000,
    0x3fef7bfdad9cbe38, 0x3cfabc0000000000,
    0x3fef91d802243c82, 0xbcd4600000000000,
    0x3fefa7c1819e908e, 0xbd0b0c0000000000,
    0x3fefbdba3692d511, 0xbcc0e00000000000,
    0x3fefd3c22b8f7194, 0xbd10de8000000000,
    0x3fefe9d96b2a23ee, 0x3cee430000000000,
    0x3ff0000000000000, 0x0,
    0x3ff00b1afa5abcbe, 0xbcb3400000000000,
    0x3ff0163da9fb3303, 0xbd12170000000000,
    0x3ff02168143b0282, 0x3cba400000000000,
    0x3ff02c9a3e77806c, 0x3cef980000000000,
    0x3ff037d42e11bbca, 0xbcc7400000000000,
    0x3ff04315e86e7f89, 0x3cd8300000000000,
    0x3ff04e5f72f65467, 0xbd1a3f0000000000,
    0x3ff059b0d315855a, 0xbd02840000000000,
    0x3ff0650a0e3c1f95, 0x3cf1600000000000,
    0x3ff0706b29ddf71a, 0x3d15240000000000,
    0x3ff07bd42b72a82d, 0xbce9a00000000000,
    0x3ff0874518759bd0, 0x3ce6400000000000,
    0x3ff092bdf66607c8, 0xbd00780000000000,
    0x3ff09e3ecac6f383, 0xbc98000000000000,
    0x3ff0a9c79b1f3930, 0x3cffa00000000000,
    0x3ff0b5586cf988fc, 0xbcfac80000000000,
    0x3ff0c0f145e46c8a, 0x3cd9c00000000000,
    0x3ff0cc922b724816, 0x3d05200000000000,
    0x3ff0d83b23395dd8, 0xbcfad00000000000,
    0x3ff0e3ec32d3d1f3, 0x3d1bac0000000000,
    0x3ff0efa55fdfa9a6, 0xbd04e80000000000,
    0x3ff0fb66affed2f0, 0xbd0d300000000000,
    0x3ff1073028d7234b, 0x3cf1500000000000,
    0x3ff11301d0125b5b, 0x3cec000000000000,
    0x3ff11edbab5e2af9, 0x3d16bc0000000000,
    0x3ff12abdc06c31d5, 0x3ce8400000000000,
    0x3ff136a814f2047d, 0xbd0ed00000000000,
    0x3ff1429aaea92de9, 0x3ce8e00000000000,
    0x3ff14e95934f3138, 0x3ceb400000000000,
    0x3ff15a98c8a58e71, 0x3d05300000000000,
    0x3ff166a45471c3df, 0x3d03380000000000,
    0x3ff172b83c7d5211, 0x3d28d40000000000,
    0x3ff17ed48695bb9f, 0xbd05d00000000000,
    0x3ff18af9388c8d93, 0xbd1c880000000000,
    0x3ff1972658375d66, 0x3d11f00000000000,
    0x3ff1a35beb6fcba7, 0x3d10480000000000,
    0x3ff1af99f81387e3, 0xbd47390000000000,
    0x3ff1bbe084045d54, 0x3d24e40000000000,
    0x3ff1c82f95281c43, 0xbd0a200000000000,
    0x3ff1d4873168b9b2, 0x3ce3800000000000,
    0x3ff1e0e75eb44031, 0x3ceac00000000000,
    0x3ff1ed5022fcd938, 0x3d01900000000000,
    0x3ff1f9c18438cdf7, 0xbd1b780000000000,
    0x3ff2063b88628d8f, 0x3d2d940000000000,
    0x3ff212be3578a81e, 0x3cd8000000000000,
    0x3ff21f49917ddd41, 0x3d2b340000000000,
    0x3ff22bdda2791323, 0x3d19f80000000000,
    0x3ff2387a6e7561e7, 0xbd19c80000000000,
    0x3ff2451ffb821427, 0x3d02300000000000,
    0x3ff251ce4fb2a602, 0xbd13480000000000,
    0x3ff25e85711eceb0, 0x3d12700000000000,
    0x3ff26b4565e27d16, 0x3d11d00000000000,
    0x3ff2780e341de00f, 0x3d31ee0000000000,
    0x3ff284dfe1f5633e, 0xbd14c00000000000,
    0x3ff291ba7591bb30, 0xbd13d80000000000,
    0x3ff29e9df51fdf09, 0x3d08b00000000000,
    0x3ff2ab8a66d10e9b, 0xbd227c0000000000,
    0x3ff2b87fd0dada3a, 0x3d2a340000000000,
    0x3ff2c57e39771af9, 0xbd10800000000000,
    0x3ff2d285a6e402d9, 0xbd0ed00000000000,
    0x3ff2df961f641579, 0xbcf4200000000000,
    0x3ff2ecafa93e2ecf, 0xbd24980000000000,
    0x3ff2f9d24abd8822, 0xbd16300000000000,
    0x3ff306fe0a31b625, 0xbd32360000000000,
    0x3ff31432edeea50b, 0xbd70df8000000000,
    0x3ff32170fc4cd7b8, 0xbd22480000000000,
    0x3ff32eb83ba8e9a2, 0xbd25980000000000,
    0x3ff33c08b2641766, 0x3d1ed00000000000,
    0x3ff3496266e3fa27, 0xbcdc000000000000,
    0x3ff356c55f929f0f, 0xbd30d80000000000,
    0x3ff36431a2de88b9, 0x3d22c80000000000,
    0x3ff371a7373aaa39, 0x3d20600000000000,
    0x3ff37f26231e74fe, 0xbd16600000000000,
    0x3ff38cae6d05d838, 0xbd0ae00000000000,
    0x3ff39a401b713ec3, 0xbd44720000000000,
    0x3ff3a7db34e5a020, 0x3d08200000000000,
    0x3ff3b57fbfec6e95, 0x3d3e800000000000,
    0x3ff3c32dc313a8f2, 0x3cef800000000000,
    0x3ff3d0e544ede122, 0xbd17a00000000000,
    0x3ff3dea64c1234bb, 0x3d26300000000000,
    0x3ff3ec70df1c4ecc, 0xbd48a60000000000,
    0x3ff3fa4504ac7e8c, 0xbd3cdc0000000000,
    0x3ff40822c367a0bb, 0x3d25b80000000000,
    0x3ff4160a21f72e95, 0x3d1ec00000000000,
    0x3ff423fb27094646, 0xbd13600000000000,
    0x3ff431f5d950a920, 0x3d23980000000000,
    0x3ff43ffa3f84b9eb, 0x3cfa000000000000,
    0x3ff44e0860618919, 0xbcf6c00000000000,
    0x3ff45c2042a7d201, 0xbd0bc00000000000,
    0x3ff46a41ed1d0016, 0xbd12800000000000,
    0x3ff4786d668b3326, 0x3d30e00000000000,
    0x3ff486a2b5c13c00, 0xbd2d400000000000,
    0x3ff494e1e192af04, 0x3d0c200000000000,
    0x3ff4a32af0d7d372, 0xbd1e500000000000,
    0x3ff4b17dea6db801, 0x3d07800000000000,
    0x3ff4bfdad53629e1, 0xbd13800000000000,
    0x3ff4ce41b817c132, 0x3d00800000000000,
    0x3ff4dcb299fddddb, 0x3d2c700000000000,
    0x3ff4eb2d81d8ab96, 0xbd1ce00000000000,
    0x3ff4f9b2769d2d02, 0x3d19200000000000,
    0x3ff508417f4531c1, 0xbd08c00000000000,
    0x3ff516daa2cf662a, 0xbcfa000000000000,
    0x3ff5257de83f51ea, 0x3d4a080000000000,
    0x3ff5342b569d4eda, 0xbd26d80000000000,
    0x3ff542e2f4f6ac1a, 0xbd32440000000000,
    0x3ff551a4ca5d94db, 0x3d483c0000000000,
    0x3ff56070dde9116b, 0x3d24b00000000000,
    0x3ff56f4736b529de, 0x3d415a0000000000,
    0x3ff57e27dbe2c40e, 0xbd29e00000000000,
    0x3ff58d12d497c76f, 0xbd23080000000000,
    0x3ff59c0827ff0b4c, 0x3d4dec0000000000,
    0x3ff5ab07dd485427, 0xbcc4000000000000,
    0x3ff5ba11fba87af4, 0x3d30080000000000,
    0x3ff5c9268a59460b, 0xbd26c80000000000,
    0x3ff5d84590998e3f, 0x3d469a0000000000,
    0x3ff5e76f15ad20e1, 0xbd1b400000000000,
    0x3ff5f6a320dcebca, 0x3d17700000000000,
    0x3ff605e1b976dcb8, 0x3d26f80000000000,
    0x3ff6152ae6cdf715, 0x3d01000000000000,
    0x3ff6247eb03a5531, 0xbd15d00000000000,
    0x3ff633dd1d1929b5, 0xbd12d00000000000,
    0x3ff6434634ccc313, 0xbcea800000000000,
    0x3ff652b9febc8efa, 0xbd28600000000000,
    0x3ff6623882553397, 0x3d71fe0000000000,
    0x3ff671c1c708328e, 0xbd37200000000000,
    0x3ff68155d44ca97e, 0x3ce6800000000000,
    0x3ff690f4b19e9471, 0xbd29780000000000,
];

// exp2(x): compute the base 2 exponential of x
//
// Accuracy: Peak error < 0.503 ulp for normalized results.
//
// Method: (accurate tables)
//
//   Reduce x:
//     x = k + y, for integer k and |y| <= 1/2.
//     Thus we have exp2(x) = 2**k * exp2(y).
//
//   Reduce y:
//     y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
//     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
//     with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
//
//   We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
//   a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
//   The values in exp2t[] and eps[] are chosen such that
//   exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
//   that exp2t[i] is accurate to 2**-64.
//
//   Note that the range of i is +-TBLSIZE/2, so we actually index the tables
//   by i0 = i + TBLSIZE/2.  For cache efficiency, exp2t[] and eps[] are
//   virtual tables, interleaved in the real table tbl[].
//
//   This method is due to Gal, with many details due to Gal and Bachelis:
//
//      Gal, S. and Bachelis, B.  An Accurate Elementary Mathematical Library
//      for the IEEE Floating Point Standard.  TOMS 17(1), 26-46 (1991).

/// Exponential, base 2 (f64)
///
/// Calculate `2^x`, that is, 2 raised to the power `x`.
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn exp2(mut x: f64) -> f64 {
    let redux = f64::from_bits(0x4338000000000000) / TBLSIZE as f64;
    let p1 = f64::from_bits(0x3fe62e42fefa39ef);
    let p2 = f64::from_bits(0x3fcebfbdff82c575);
    let p3 = f64::from_bits(0x3fac6b08d704a0a6);
    let p4 = f64::from_bits(0x3f83b2ab88f70400);
    let p5 = f64::from_bits(0x3f55d88003875c74);

    // double_t r, t, z;
    // uint32_t ix, i0;
    // union {double f; uint64_t i;} u = {x};
    // union {uint32_t u; int32_t i;} k;
    let x1p1023 = f64::from_bits(0x7fe0000000000000);
    let x1p52 = f64::from_bits(0x4330000000000000);
    let _0x1p_149 = f64::from_bits(0xb6a0000000000000);

    /* Filter out exceptional cases. */
    let ui = f64::to_bits(x);
    let ix = ui >> 32 & 0x7fffffff;
    if ix >= 0x408ff000 {
        /* |x| >= 1022 or nan */
        if ix >= 0x40900000 && ui >> 63 == 0 {
            /* x >= 1024 or nan */
            /* overflow */
            x *= x1p1023;
            return x;
        }
        if ix >= 0x7ff00000 {
            /* -inf or -nan */
            return -1.0 / x;
        }
        if ui >> 63 != 0 {
            /* x <= -1022 */
            /* underflow */
            if x <= -1075.0 || x - x1p52 + x1p52 != x {
                force_eval!((_0x1p_149 / x) as f32);
            }
            if x <= -1075.0 {
                return 0.0;
            }
        }
    } else if ix < 0x3c900000 {
        /* |x| < 0x1p-54 */
        return 1.0 + x;
    }

    /* Reduce x, computing z, i0, and k. */
    let ui = f64::to_bits(x + redux);
    let mut i0 = ui as u32;
    i0 = i0.wrapping_add(TBLSIZE as u32 / 2);
    let ku = i0 / TBLSIZE as u32 * TBLSIZE as u32;
    let ki = ku as i32 / TBLSIZE as i32;
    i0 %= TBLSIZE as u32;
    let uf = f64::from_bits(ui) - redux;
    let mut z = x - uf;

    /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
    let t = f64::from_bits(TBL[2 * i0 as usize]); /* exp2t[i0] */
    z -= f64::from_bits(TBL[2 * i0 as usize + 1]); /* eps[i0]   */
    let r = t + t * z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * p5))));

    scalbn(r, ki)
}

#[test]
fn i0_wrap_test() {
    let x = -3.0 / 256.0;
    assert_eq!(exp2(x), f64::from_bits(0x3fefbdba3692d514));
}