/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* 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. */ #include "libm.h" 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 */ 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 -two54/0.0; /* log1p(-1)=+inf */ return (x-x)/(x-x); /* log1p(x<-1)=NaN */ } if (ax < 0x3e200000) { /* |x| < 2**-29 */ /* raise inexact */ if (two54 + x > 0.0 && ax < 0x3c900000) /* |x| < 2**-54 */ return x; return x - x*x*0.5; } if (hx > 0 || hx <= (int32_t)0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ k = 0; f = x; hu = 1; } } 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 == 0.0) { if(k == 0) return 0.0; c += k*ln2_lo; return k*ln2_hi + c; } R = hfsq*(1.0 - 0.66666666666666666*f); if (k == 0) return f - R; 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)); return k*ln2_hi - ((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); }