11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* @(#)s_log1p.c 5.1 93/09/24 */ 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Developed at SunPro, a Sun Microsystems, Inc. business. 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Permission to use, copy, modify, and distribute this 81dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * software is freely granted, provided that this notice 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is preserved. 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 13a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* double log1p(double x) 171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Method : 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1. Argument Reduction: find k and f such that 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1+x = 2^k * (1+f), 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where sqrt(2)/2 < 1+f < sqrt(2) . 221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note. If k=0, then f=x is exact. However, if k!=0, then f 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * may not be representable exactly. In that case, a correction 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * and add back the correction term c/u. 281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * (Note: when x > 2**53, one can simply return log(x)) 291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. Approximation of log1p(f). 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 2s + s*R 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * We use a special Reme algorithm on [0,0.1716] to generate 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * a polynomial of degree 14 to approximate R The maximum error 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * of this polynomial approximation is bounded by 2**-58.45. In 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * other words, 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2 4 6 8 10 12 14 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * (the values of Lp1 to Lp7 are listed in the program) 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * and 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * | 2 14 | -58.45 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * | Lp1*s +...+Lp7*s - R(z) | <= 2 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * | | 451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * In order to guarantee error in log below 1ulp, we compute log 471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * by 481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * log1p(f) = f - (hfsq - s*(hfsq+R)). 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3. Finally, log1p(x) = k*ln2 + log1p(f). 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Here ln2 is split into two floating point number: 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ln2_hi + ln2_lo, 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where n*ln2_hi is always exact for |n| < 2000. 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Special cases: 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * log1p(x) is NaN with signal if x < -1 (including -INF) ; 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * log1p(NaN) is that NaN with no signal. 601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Accuracy: 621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * according to an error analysis, the error is always less than 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1 ulp (unit in the last place). 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Constants: 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * The hexadecimal values are the intended ones for the following 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * constants. The decimal values may be used, provided that the 681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * compiler will convert from decimal to binary accurately enough 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * to produce the hexadecimal values shown. 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note: Assuming log() return accurate answer, the following 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * algorithm can be used to compute log1p(x) to within a few ULP: 731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * u = 1+x; 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * if(u==1.0) return x ; else 761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * return log(u)*(x/(u-1.0)); 771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * See HP-15C Advanced Functions Handbook, p.193. 791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 81a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <float.h> 82a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double 871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttwo54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectLp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double zero = 0.0; 9978419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughesstatic volatile double vzero = 0.0; 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectlog1p(double x) 1031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double hfsq,f,c,s,z,R,u; 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t k,hx,hu,ax; 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hx,x); 1081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ax = hx&0x7fffffff; 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project k = 1; 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ax>=0x3ff00000) { /* x <= -1.0 */ 11378419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ 1141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 1151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ax<0x3e200000) { /* |x| < 2**-29 */ 1171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(two54+x>zero /* raise inexact */ 1181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project &&ax<0x3c900000) /* |x| < 2**-54 */ 1191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return x; 1201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project else 1211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return x - x*x*0.5; 1221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>0||hx<=((int32_t)0xbfd2bec4)) { 1241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 1251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (hx >= 0x7ff00000) return x+x; 1271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(k!=0) { 1281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx<0x43400000) { 129a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes STRICT_ASSIGN(double,u,1.0+x); 1301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hu,u); 1311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project k = (hu>>20)-1023; 1321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 1331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project c /= u; 1341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 1351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project u = x; 1361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hu,u); 1371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project k = (hu>>20)-1023; 1381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project c = 0; 1391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project hu &= 0x000fffff; 1411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* 1421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * The approximation to sqrt(2) used in thresholds is not 1431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * critical. However, the ones used above must give less 1441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * strict bounds than the one here so that the k==0 case is 1451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * never reached from here, since here we have committed to 1461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * using the correction term but don't use it if k==0. 1471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hu<0x6a09e) { /* u ~< sqrt(2) */ 1491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 1501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 1511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project k += 1; 1521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 1531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project hu = (0x00100000-hu)>>2; 1541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project f = u-1.0; 1561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project hfsq=0.5*f*f; 1581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hu==0) { /* |f| < 2**-20 */ 159a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if(f==zero) { 160a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if(k==0) { 161a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return zero; 162a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes } else { 163a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes c += k*ln2_lo; 164a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return k*ln2_hi+c; 165a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes } 166a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes } 1671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R = hfsq*(1.0-0.66666666666666666*f); 1681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(k==0) return f-R; else 1691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return k*ln2_hi-((R-(k*ln2_lo+c))-f); 1701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = f/(2.0+f); 1721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = s*s; 1731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 1741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(k==0) return f-(hfsq-s*(hfsq+R)); else 1751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 1761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 17778419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes 17878419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes#if (LDBL_MANT_DIG == 53) 17978419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes__weak_reference(log1p, log1pl); 18078419467a2f88744ae2445fca5eb442877ebb1b0Elliott Hughes#endif 181