diff options
Diffstat (limited to 'src/math/log1p.c')
| -rw-r--r-- | src/math/log1p.c | 148 |
1 files changed, 49 insertions, 99 deletions
diff --git a/src/math/log1p.c b/src/math/log1p.c index a71ac423..00971349 100644 --- a/src/math/log1p.c +++ b/src/math/log1p.c @@ -10,6 +10,7 @@ * ==================================================== */ /* double log1p(double x) + * Return the natural logarithm of 1+x. * * Method : * 1. Argument Reduction: find k and f such that @@ -23,31 +24,9 @@ * 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)). + * 2. Approximation of log(1+f): See log.c * - * 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. + * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c * * Special cases: * log1p(x) is NaN with signal if x < -1 (including -INF) ; @@ -79,94 +58,65 @@ 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 */ +Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ +Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ +Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ +Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ +Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ +Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ +Lg7 = 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; + union {double f; uint64_t i;} u = {x}; + double_t hfsq,f,c,s,z,R,w,t1,t2,dk; + uint32_t hx,hu; + int k; + hx = u.i>>32; 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 (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */ + if (hx >= 0xbff00000) { /* x <= -1.0 */ + if (x == -1) + return x/0.0; /* log1p(-1) = -inf */ + return (x-x)/0.0; /* log1p(x<-1) = NaN */ } - if (ax < 0x3e200000) { /* |x| < 2**-29 */ - /* if 0x1p-1022 <= |x| < 0x1p-54, avoid raising underflow */ - if (ax < 0x3c900000 && ax >= 0x00100000) - return x; -#if FLT_EVAL_METHOD != 0 - FORCE_EVAL((float)x); -#endif - return x - x*x*0.5; + if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */ + /* underflow if subnormal */ + if ((hx&0x7ff00000) == 0) + FORCE_EVAL((float)x); + return x; } - if (hx > 0 || hx <= (int32_t)0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ + if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ k = 0; + c = 0; f = x; - hu = 1; } - } - if (hx >= 0x7ff00000) - return x+x; - if (k != 0) { - if (hx < 0x43400000) { - u = 1 + 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; + } else if (hx >= 0x7ff00000) + return x; + if (k) { + u.f = 1 + x; + hu = u.i>>32; + hu += 0x3ff00000 - 0x3fe6a09e; + k = (int)(hu>>20) - 0x3ff; + /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */ + if (k < 54) { + c = k >= 2 ? 1-(u.f-x) : x-(u.f-1); + c /= u.f; + } else 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; + /* reduce u into [sqrt(2)/2, sqrt(2)] */ + hu = (hu&0x000fffff) + 0x3fe6a09e; + u.i = (uint64_t)hu<<32 | (u.i&0xffffffff); + f = u.f - 1; } 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); + w = z*z; + t1 = w*(Lg2+w*(Lg4+w*Lg6)); + t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); + R = t2 + t1; + dk = k; + return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi; } |