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Diffstat (limited to 'src/math/log1p.c')
| -rw-r--r-- | src/math/log1p.c | 171 |
1 files changed, 171 insertions, 0 deletions
diff --git a/src/math/log1p.c b/src/math/log1p.c new file mode 100644 index 00000000..f7154d0c --- /dev/null +++ b/src/math/log1p.c @@ -0,0 +1,171 @@ +/* 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 */ + +static const double zero = 0.0; + +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/zero; /* log1p(-1)=+inf */ + return (x-x)/(x-x); /* log1p(x<-1)=NaN */ + } + if (ax < 0x3e200000) { /* |x| < 2**-29 */ + /* raise inexact */ + if (two54 + x > zero && 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 == zero) { + if(k == 0) + return zero; + 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); +} |