11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/*- 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * All rights reserved. 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Redistribution and use in source and binary forms, with or without 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * modification, are permitted provided that the following conditions 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * are met: 81dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1. Redistributions of source code must retain the above copyright 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * notice, this list of conditions and the following disclaimer. 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. Redistributions in binary form must reproduce the above copyright 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * notice, this list of conditions and the following disclaimer in the 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * documentation and/or other materials provided with the distribution. 131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * SUCH DAMAGE. 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include <sys/cdefs.h> 28a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 29a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 30a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <float.h> 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define TBLBITS 4 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#define TBLSIZE (1 << TBLBITS) 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const float 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project redux = 0x1.8p23f / TBLSIZE, 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P1 = 0x1.62e430p-1f, 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P2 = 0x1.ebfbe0p-3f, 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P3 = 0x1.c6b348p-5f, 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P4 = 0x1.3b2c9cp-7f; 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 4578419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughesstatic volatile float 4678419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes huge = 0x1p100f, 4778419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes twom100 = 0x1p-100f; 48a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double exp2ft[TBLSIZE] = { 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.6a09e667f3bcdp-1, 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.7a11473eb0187p-1, 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.8ace5422aa0dbp-1, 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.9c49182a3f090p-1, 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.ae89f995ad3adp-1, 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.c199bdd85529cp-1, 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.d5818dcfba487p-1, 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.ea4afa2a490dap-1, 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.0000000000000p+0, 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.0b5586cf9890fp+0, 601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.172b83c7d517bp+0, 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.2387a6e756238p+0, 621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.306fe0a31b715p+0, 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.3dea64c123422p+0, 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.4bfdad5362a27p+0, 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 0x1.5ab07dd485429p+0, 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project}; 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * exp2f(x): compute the base 2 exponential of x 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Method: (equally-spaced tables) 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Reduce x: 761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * x = 2**k + y, for integer k and |y| <= 1/2. 771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Thus we have exp2f(x) = 2**k * exp2(y). 781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Reduce y: 801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * with |z| <= 2**-(TBLSIZE+1). 831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 86a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * Using double precision for everything except the reduction makes 87a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes * roundoff error insignificant and simplifies the scaling step. 881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * This method is due to Tang, but I do not use his suggested parameters: 901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Tang, P. Table-driven Implementation of the Exponential Function 921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectfloat 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectexp2f(float x) 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 97a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes double tv, twopk, u, z; 98a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes float t; 99a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes uint32_t hx, ix, i0; 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t k; 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* Filter out exceptional cases. */ 103a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes GET_FLOAT_WORD(hx, x); 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx & 0x7fffffff; /* high word of |x| */ 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix >= 0x43000000) { /* |x| >= 128 */ 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix >= 0x7f800000) { 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 108a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return (x + x); /* x is NaN or +Inf */ 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (0.0); /* x is -Inf */ 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(x >= 0x1.0p7f) 1131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (huge * huge); /* overflow */ 1141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(x <= -0x1.2cp7f) 1151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (twom100 * twom100); /* underflow */ 1161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 1171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (1.0f + x); 1181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* Reduce x, computing z, i0, and k. */ 121a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes STRICT_ASSIGN(float, t, x + redux); 1221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_FLOAT_WORD(i0, t); 1231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project i0 += TBLSIZE / 2; 124a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes k = (i0 >> TBLBITS) << 20; 1251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project i0 &= TBLSIZE - 1; 1261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t -= redux; 1271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x - t; 128a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes INSERT_WORDS(twopk, 0x3ff00000 + k, 0); 1291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 1311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project tv = exp2ft[i0]; 132a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes u = tv * z; 133a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4); 1341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 135a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes /* Scale by 2**(k>>20). */ 136a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return (tv * twopk); 1371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 138