| /* @(#)s_log1p.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
| for performance improvement on pipelined processors. |
| */ |
| |
| /* 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 <math.h> |
| #include <math_private.h> |
| |
| static const double |
| ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
| ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
| two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
| Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
| 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
| 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
| 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
| 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
| 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
| 1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */ |
| |
| static const double zero = 0.0; |
| |
| double |
| __log1p (double x) |
| { |
| double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4; |
| int32_t k, hx, hu, ax; |
| |
| GET_HIGH_WORD (hx, x); |
| ax = hx & 0x7fffffff; |
| |
| k = 1; |
| if (hx < 0x3FDA827A) /* x < 0.41422 */ |
| { |
| if (__builtin_expect (ax >= 0x3ff00000, 0)) /* x <= -1.0 */ |
| { |
| if (x == -1.0) |
| return -two54 / (x - x); /* log1p(-1)=+inf */ |
| else |
| return (x - x) / (x - x); /* log1p(x<-1)=NaN */ |
| } |
| if (__builtin_expect (ax < 0x3e200000, 0)) /* |x| < 2**-29 */ |
| { |
| math_force_eval (two54 + x); /* raise inexact */ |
| if (ax < 0x3c900000) /* |x| < 2**-54 */ |
| return x; |
| else |
| return x - x * x * 0.5; |
| } |
| if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3)) |
| { |
| k = 0; f = x; hu = 1; |
| } /* -0.2929<x<0.41422 */ |
| } |
| else if (__builtin_expect (hx >= 0x7ff00000, 0)) |
| return x + x; |
| if (k != 0) |
| { |
| if (hx < 0x43400000) |
| { |
| 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; |
| if (hu < 0x6a09e) |
| { |
| 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; |
| R1 = z * Lp[1]; z2 = z * z; |
| R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2; |
| R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2; |
| R4 = Lp[6] + z * Lp[7]; |
| R = R1 + z2 * R2 + z4 * R3 + z6 * R4; |
| if (k == 0) |
| return f - (hfsq - s * (hfsq + R)); |
| else |
| return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f); |
| } |
| weak_alias (__log1p, log1p) |
| #ifdef NO_LONG_DOUBLE |
| strong_alias (__log1p, __log1pl) |
| weak_alias (__log1p, log1pl) |
| #endif |
