/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│ │ vi: set noet ft=c ts=8 sw=8 fenc=utf-8 :vi │ ╚──────────────────────────────────────────────────────────────────────────────╝ │ │ │ Copyright (c) 1992-2024 The FreeBSD Project │ │ Copyright (c) 1993 Sun Microsystems, Inc. │ │ 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. │ │ │ ╚─────────────────────────────────────────────────────────────────────────────*/ #include "libc/tinymath/freebsd.internal.h" __static_yoink("freebsd_libm_notice"); __static_yoink("fdlibm_notice"); /* double log1p(double x) * * Method : * 1. Argument Reduction: find k and f such that * 1+x = 2^k * (1+f), * where sqrt(2)/2 < 1+f < sqrt(2) . * * Note. If k=0, then f=x is exact. However, if k!=0, then f * may not be representable exactly. In that case, a correction * term is need. Let u=1+x rounded. Let c = (1+x)-u, then * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), * and add back the correction term c/u. * (Note: when x > 2**53, one can simply return log(x)) * * 2. Approximation of log1p(f). * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) * = 2s + 2/3 s**3 + 2/5 s**5 + ....., * = 2s + s*R * We use a special Reme algorithm on [0,0.1716] to generate * a polynomial of degree 14 to approximate R The maximum error * of this polynomial approximation is bounded by 2**-58.45. In * other words, * 2 4 6 8 10 12 14 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s * (the values of Lp1 to Lp7 are listed in the program) * and * | 2 14 | -58.45 * | Lp1*s +...+Lp7*s - R(z) | <= 2 * | | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. * In order to guarantee error in log below 1ulp, we compute log * by * log1p(f) = f - (hfsq - s*(hfsq+R)). * * 3. Finally, log1p(x) = k*ln2 + log1p(f). * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) * Here ln2 is split into two floating point number: * ln2_hi + ln2_lo, * where n*ln2_hi is always exact for |n| < 2000. * * Special cases: * log1p(x) is NaN with signal if x < -1 (including -INF) ; * log1p(+INF) is +INF; log1p(-1) is -INF with signal; * log1p(NaN) is that NaN with no signal. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. * * Note: Assuming log() return accurate answer, the following * algorithm can be used to compute log1p(x) to within a few ULP: * * u = 1+x; * if(u==1.0) return x ; else * return log(u)*(x/(u-1.0)); * * See HP-15C Advanced Functions Handbook, p.193. */ static const double ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ static const double zero = 0.0; static volatile double vzero = 0.0; /** * Returns log(1 + x). */ double log1p(double x) { double hfsq,f,c,s,z,R,u; int32_t k,hx,hu,ax; GET_HIGH_WORD(hx,x); ax = hx&0x7fffffff; k = 1; if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ if(ax>=0x3ff00000) { /* x <= -1.0 */ if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ } if(ax<0x3e200000) { /* |x| < 2**-29 */ if(two54+x>zero /* raise inexact */ &&ax<0x3c900000) /* |x| < 2**-54 */ return x; else return x - x*x*0.5; } if(hx>0||hx<=((int32_t)0xbfd2bec4)) { k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ } if (hx >= 0x7ff00000) return x+x; if(k!=0) { if(hx<0x43400000) { STRICT_ASSIGN(double,u,1.0+x); GET_HIGH_WORD(hu,u); k = (hu>>20)-1023; c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ c /= u; } else { u = x; GET_HIGH_WORD(hu,u); k = (hu>>20)-1023; c = 0; } hu &= 0x000fffff; /* * The approximation to sqrt(2) used in thresholds is not * critical. However, the ones used above must give less * strict bounds than the one here so that the k==0 case is * never reached from here, since here we have committed to * using the correction term but don't use it if k==0. */ if(hu<0x6a09e) { /* u ~< sqrt(2) */ SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ } else { k += 1; SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ hu = (0x00100000-hu)>>2; } f = u-1.0; } hfsq=0.5*f*f; if(hu==0) { /* |f| < 2**-20 */ if(f==zero) { if(k==0) { return zero; } else { c += k*ln2_lo; return k*ln2_hi+c; } } R = hfsq*(1.0-0.66666666666666666*f); if(k==0) return f-R; else return k*ln2_hi-((R-(k*ln2_lo+c))-f); } s = f/(2.0+f); z = s*s; R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); if(k==0) return f-(hfsq-s*(hfsq+R)); else return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); } #if (LDBL_MANT_DIG == 53) __weak_reference(log1p, log1pl); #endif