double __ieee754_atanh (double x) { double xa = fabs (x); double t; if (isless (xa, 0.5)) { if (__builtin_expect (xa < 0x1.0p-28, 0)) { math_force_eval (huge + x); return x; } t = xa + xa; t = 0.5 * __log1p (t + t * xa / (1.0 - xa)); } else if (__builtin_expect (isless (xa, 1.0), 1)) t = 0.5 * __log1p ((xa + xa) / (1.0 - xa)); else { if (isgreater (xa, 1.0)) return (x - x) / (x - x); return x / 0.0; } return __copysign (t, x); }
double __ieee754_atanh (double x) { double xa = fabs (x); double t; if (isless (xa, 0.5)) { if (__glibc_unlikely (xa < 0x1.0p-28)) { math_force_eval (huge + x); math_check_force_underflow (x); return x; } t = xa + xa; t = 0.5 * __log1p (t + t * xa / (1.0 - xa)); } else if (__glibc_likely (isless (xa, 1.0))) t = 0.5 * __log1p ((xa + xa) / (1.0 - xa)); else { if (isgreater (xa, 1.0)) return (x - x) / (x - x); return x / 0.0; } return __copysign (t, x); }
double __ieee754_acosh (double x) { int64_t hx; EXTRACT_WORDS64 (hx, x); if (hx > INT64_C (0x4000000000000000)) { if (__builtin_expect (hx >= INT64_C (0x41b0000000000000), 0)) { /* x > 2**28 */ if (hx >= INT64_C (0x7ff0000000000000)) /* x is inf of NaN */ return x + x; else return __ieee754_log (x) + ln2;/* acosh(huge)=log(2x) */ } /* 2**28 > x > 2 */ double t = x * x; return __ieee754_log (2.0 * x - one / (x + __ieee754_sqrt (t - one))); } else if (__builtin_expect (hx > INT64_C (0x3ff0000000000000), 1)) { /* 1<x<2 */ double t = x - one; return __log1p (t + __ieee754_sqrt (2.0 * t + t * t)); } else if (__builtin_expect (hx == INT64_C (0x3ff0000000000000), 1)) return 0.0; /* acosh(1) = 0 */ else /* x < 1 */ return (x - x) / (x - x); }
double __asinh (double x) { double w; int32_t hx, ix; GET_HIGH_WORD (hx, x); ix = hx & 0x7fffffff; if (__glibc_unlikely (ix < 0x3e300000)) /* |x|<2**-28 */ { if (huge + x > one) return x; /* return x inexact except 0 */ } if (__glibc_unlikely (ix > 0x41b00000)) /* |x| > 2**28 */ { if (ix >= 0x7ff00000) return x + x; /* x is inf or NaN */ w = __ieee754_log (fabs (x)) + ln2; } else { double xa = fabs (x); if (ix > 0x40000000) /* 2**28 > |x| > 2.0 */ { w = __ieee754_log (2.0 * xa + one / (__ieee754_sqrt (xa * xa + one) + xa)); } else /* 2.0 > |x| > 2**-28 */ { double t = xa * xa; w = __log1p (xa + t / (one + __ieee754_sqrt (one + t))); } } return __copysign (w, x); }
double log10(double x) { double f,hfsq,hi,lo,r,val_hi,val_lo,w,y,y2; int32_t i,k,hx; uint32_t lx; EXTRACT_WORDS(hx, lx, x); k = 0; if (hx < 0x00100000) { /* x < 2**-1022 */ if (((hx&0x7fffffff)|lx) == 0) return -two54/0.0; /* log(+-0)=-inf */ if (hx<0) return (x-x)/0.0; /* log(-#) = NaN */ /* subnormal number, scale up x */ k -= 54; x *= two54; GET_HIGH_WORD(hx, x); } if (hx >= 0x7ff00000) return x+x; if (hx == 0x3ff00000 && lx == 0) return 0.0; /* log(1) = +0 */ k += (hx>>20) - 1023; hx &= 0x000fffff; i = (hx+0x95f64)&0x100000; SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */ k += i>>20; y = (double)k; f = x - 1.0; hfsq = 0.5*f*f; r = __log1p(f); /* See log2.c for details. */ hi = f - hfsq; SET_LOW_WORD(hi, 0); lo = (f - hi) - hfsq + r; val_hi = hi*ivln10hi; y2 = y*log10_2hi; val_lo = y*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi; /* * Extra precision in for adding y*log10_2hi is not strictly needed * since there is no very large cancellation near x = sqrt(2) or * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs * with some parallelism and it reduces the error for many args. */ w = y2 + val_hi; val_lo += (y2 - w) + val_hi; val_hi = w; return val_lo + val_hi; }
Err mathlib_log1p(UInt16 refnum, double x, double *result) { #pragma unused(refnum) *result = __log1p(x); return mlErrNone; }
double log2(double x) { double f,hfsq,hi,lo,r,val_hi,val_lo,w,y; int32_t i,k,hx; uint32_t lx; EXTRACT_WORDS(hx, lx, x); k = 0; if (hx < 0x00100000) { /* x < 2**-1022 */ if (((hx&0x7fffffff)|lx) == 0) return -two54/0.0; /* log(+-0)=-inf */ if (hx < 0) return (x-x)/0.0; /* log(-#) = NaN */ /* subnormal number, scale up x */ k -= 54; x *= two54; GET_HIGH_WORD(hx, x); } if (hx >= 0x7ff00000) return x+x; if (hx == 0x3ff00000 && lx == 0) return 0.0; /* log(1) = +0 */ k += (hx>>20) - 1023; hx &= 0x000fffff; i = (hx+0x95f64) & 0x100000; SET_HIGH_WORD(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */ k += i>>20; y = (double)k; f = x - 1.0; hfsq = 0.5*f*f; r = __log1p(f); /* * f-hfsq must (for args near 1) be evaluated in extra precision * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). * This is fairly efficient since f-hfsq only depends on f, so can * be evaluated in parallel with R. Not combining hfsq with R also * keeps R small (though not as small as a true `lo' term would be), * so that extra precision is not needed for terms involving R. * * Compiler bugs involving extra precision used to break Dekker's * theorem for spitting f-hfsq as hi+lo, unless double_t was used * or the multi-precision calculations were avoided when double_t * has extra precision. These problems are now automatically * avoided as a side effect of the optimization of combining the * Dekker splitting step with the clear-low-bits step. * * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra * precision to avoid a very large cancellation when x is very near * these values. Unlike the above cancellations, this problem is * specific to base 2. It is strange that adding +-1 is so much * harder than adding +-ln2 or +-log10_2. * * This uses Dekker's theorem to normalize y+val_hi, so the * compiler bugs are back in some configurations, sigh. And I * don't want to used double_t to avoid them, since that gives a * pessimization and the support for avoiding the pessimization * is not yet available. * * The multi-precision calculations for the multiplications are * routine. */ hi = f - hfsq; SET_LOW_WORD(hi, 0); lo = (f - hi) - hfsq + r; val_hi = hi*ivln2hi; val_lo = (lo+hi)*ivln2lo + lo*ivln2hi; /* spadd(val_hi, val_lo, y), except for not using double_t: */ w = y + val_hi; val_lo += (y - w) + val_hi; val_hi = w; return val_lo + val_hi; }
double log1p( double x ) { return __log1p( x ); }
__complex__ double __kernel_casinh (__complex__ double x, int adj) { __complex__ double res; double rx, ix; __complex__ double y; /* Avoid cancellation by reducing to the first quadrant. */ rx = fabs (__real__ x); ix = fabs (__imag__ x); if (rx >= 1.0 / DBL_EPSILON || ix >= 1.0 / DBL_EPSILON) { /* For large x in the first quadrant, x + csqrt (1 + x * x) is sufficiently close to 2 * x to make no significant difference to the result; avoid possible overflow from the squaring and addition. */ __real__ y = rx; __imag__ y = ix; if (adj) { double t = __real__ y; __real__ y = __copysign (__imag__ y, __imag__ x); __imag__ y = t; } res = __clog (y); __real__ res += M_LN2; } else if (rx >= 0.5 && ix < DBL_EPSILON / 8.0) { double s = __ieee754_hypot (1.0, rx); __real__ res = __ieee754_log (rx + s); if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } else if (rx < DBL_EPSILON / 8.0 && ix >= 1.5) { double s = __ieee754_sqrt ((ix + 1.0) * (ix - 1.0)); __real__ res = __ieee754_log (ix + s); if (adj) __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x)); else __imag__ res = __ieee754_atan2 (s, rx); } else if (ix > 1.0 && ix < 1.5 && rx < 0.5) { if (rx < DBL_EPSILON * DBL_EPSILON) { double ix2m1 = (ix + 1.0) * (ix - 1.0); double s = __ieee754_sqrt (ix2m1); __real__ res = __log1p (2.0 * (ix2m1 + ix * s)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x)); else __imag__ res = __ieee754_atan2 (s, rx); } else { double ix2m1 = (ix + 1.0) * (ix - 1.0); double rx2 = rx * rx; double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix); double d = __ieee754_sqrt (ix2m1 * ix2m1 + f); double dp = d + ix2m1; double dm = f / dp; double r1 = __ieee754_sqrt ((dm + rx2) / 2.0); double r2 = rx * ix / r1; __real__ res = __log1p (rx2 + dp + 2.0 * (rx * r1 + ix * r2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2, __imag__ x)); else __imag__ res = __ieee754_atan2 (ix + r2, rx + r1); } } else if (ix == 1.0 && rx < 0.5) { if (rx < DBL_EPSILON / 8.0) { __real__ res = __log1p (2.0 * (rx + __ieee754_sqrt (rx))) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (__ieee754_sqrt (rx), __copysign (1.0, __imag__ x)); else __imag__ res = __ieee754_atan2 (1.0, __ieee754_sqrt (rx)); } else { double d = rx * __ieee754_sqrt (4.0 + rx * rx); double s1 = __ieee754_sqrt ((d + rx * rx) / 2.0); double s2 = __ieee754_sqrt ((d - rx * rx) / 2.0); __real__ res = __log1p (rx * rx + d + 2.0 * (rx * s1 + s2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + s1, __copysign (1.0 + s2, __imag__ x)); else __imag__ res = __ieee754_atan2 (1.0 + s2, rx + s1); } } else if (ix < 1.0 && rx < 0.5) { if (ix >= DBL_EPSILON) { if (rx < DBL_EPSILON * DBL_EPSILON) { double onemix2 = (1.0 + ix) * (1.0 - ix); double s = __ieee754_sqrt (onemix2); __real__ res = __log1p (2.0 * rx / s) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } else { double onemix2 = (1.0 + ix) * (1.0 - ix); double rx2 = rx * rx; double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix); double d = __ieee754_sqrt (onemix2 * onemix2 + f); double dp = d + onemix2; double dm = f / dp; double r1 = __ieee754_sqrt ((dp + rx2) / 2.0); double r2 = rx * ix / r1; __real__ res = __log1p (rx2 + dm + 2.0 * (rx * r1 + ix * r2)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2, __imag__ x)); else __imag__ res = __ieee754_atan2 (ix + r2, rx + r1); } } else { double s = __ieee754_hypot (1.0, rx); __real__ res = __log1p (2.0 * rx * (rx + s)) / 2.0; if (adj) __imag__ res = __ieee754_atan2 (s, __imag__ x); else __imag__ res = __ieee754_atan2 (ix, s); } math_check_force_underflow_nonneg (__real__ res); } else { __real__ y = (rx - ix) * (rx + ix) + 1.0; __imag__ y = 2.0 * rx * ix; y = __csqrt (y); __real__ y += rx; __imag__ y += ix; if (adj) { double t = __real__ y; __real__ y = __copysign (__imag__ y, __imag__ x); __imag__ y = t; } res = __clog (y); } /* Give results the correct sign for the original argument. */ __real__ res = __copysign (__real__ res, __real__ x); __imag__ res = __copysign (__imag__ res, (adj ? 1.0 : __imag__ x)); return res; }
__complex__ double __catanh (__complex__ double x) { __complex__ double res; int rcls = fpclassify (__real__ x); int icls = fpclassify (__imag__ x); if (__glibc_unlikely (rcls <= FP_INFINITE || icls <= FP_INFINITE)) { if (icls == FP_INFINITE) { __real__ res = __copysign (0.0, __real__ x); __imag__ res = __copysign (M_PI_2, __imag__ x); } else if (rcls == FP_INFINITE || rcls == FP_ZERO) { __real__ res = __copysign (0.0, __real__ x); if (icls >= FP_ZERO) __imag__ res = __copysign (M_PI_2, __imag__ x); else __imag__ res = __nan (""); } else { __real__ res = __nan (""); __imag__ res = __nan (""); } } else if (__glibc_unlikely (rcls == FP_ZERO && icls == FP_ZERO)) { res = x; } else { if (fabs (__real__ x) >= 16.0 / DBL_EPSILON || fabs (__imag__ x) >= 16.0 / DBL_EPSILON) { __imag__ res = __copysign (M_PI_2, __imag__ x); if (fabs (__imag__ x) <= 1.0) __real__ res = 1.0 / __real__ x; else if (fabs (__real__ x) <= 1.0) __real__ res = __real__ x / __imag__ x / __imag__ x; else { double h = __ieee754_hypot (__real__ x / 2.0, __imag__ x / 2.0); __real__ res = __real__ x / h / h / 4.0; } } else { if (fabs (__real__ x) == 1.0 && fabs (__imag__ x) < DBL_EPSILON * DBL_EPSILON) __real__ res = (__copysign (0.5, __real__ x) * (M_LN2 - __ieee754_log (fabs (__imag__ x)))); else { double i2 = 0.0; if (fabs (__imag__ x) >= DBL_EPSILON * DBL_EPSILON) i2 = __imag__ x * __imag__ x; double num = 1.0 + __real__ x; num = i2 + num * num; double den = 1.0 - __real__ x; den = i2 + den * den; double f = num / den; if (f < 0.5) __real__ res = 0.25 * __ieee754_log (f); else { num = 4.0 * __real__ x; __real__ res = 0.25 * __log1p (num / den); } } double absx, absy, den; absx = fabs (__real__ x); absy = fabs (__imag__ x); if (absx < absy) { double t = absx; absx = absy; absy = t; } if (absy < DBL_EPSILON / 2.0) { den = (1.0 - absx) * (1.0 + absx); if (den == -0.0) den = 0.0; } else if (absx >= 1.0) den = (1.0 - absx) * (1.0 + absx) - absy * absy; else if (absx >= 0.75 || absy >= 0.5) den = -__x2y2m1 (absx, absy); else den = (1.0 - absx) * (1.0 + absx) - absy * absy; __imag__ res = 0.5 * __ieee754_atan2 (2.0 * __imag__ x, den); } math_check_force_underflow_complex (res); } return res; }
__complex__ double __clog (__complex__ double x) { __complex__ double result; int rcls = fpclassify (__real__ x); int icls = fpclassify (__imag__ x); if (__glibc_unlikely (rcls == FP_ZERO && icls == FP_ZERO)) { /* Real and imaginary part are 0.0. */ __imag__ result = signbit (__real__ x) ? M_PI : 0.0; __imag__ result = __copysign (__imag__ result, __imag__ x); /* Yes, the following line raises an exception. */ __real__ result = -1.0 / fabs (__real__ x); } else if (__glibc_likely (rcls != FP_NAN && icls != FP_NAN)) { /* Neither real nor imaginary part is NaN. */ double absx = fabs (__real__ x), absy = fabs (__imag__ x); int scale = 0; if (absx < absy) { double t = absx; absx = absy; absy = t; } if (absx > DBL_MAX / 2.0) { scale = -1; absx = __scalbn (absx, scale); absy = (absy >= DBL_MIN * 2.0 ? __scalbn (absy, scale) : 0.0); } else if (absx < DBL_MIN && absy < DBL_MIN) { scale = DBL_MANT_DIG; absx = __scalbn (absx, scale); absy = __scalbn (absy, scale); } if (absx == 1.0 && scale == 0) { __real__ result = __log1p (absy * absy) / 2.0; math_check_force_underflow_nonneg (__real__ result); } else if (absx > 1.0 && absx < 2.0 && absy < 1.0 && scale == 0) { double d2m1 = (absx - 1.0) * (absx + 1.0); if (absy >= DBL_EPSILON) d2m1 += absy * absy; __real__ result = __log1p (d2m1) / 2.0; } else if (absx < 1.0 && absx >= 0.5 && absy < DBL_EPSILON / 2.0 && scale == 0) { double d2m1 = (absx - 1.0) * (absx + 1.0); __real__ result = __log1p (d2m1) / 2.0; } else if (absx < 1.0 && absx >= 0.5 && scale == 0 && absx * absx + absy * absy >= 0.5) { double d2m1 = __x2y2m1 (absx, absy); __real__ result = __log1p (d2m1) / 2.0; } else { double d = __ieee754_hypot (absx, absy); __real__ result = __ieee754_log (d) - scale * M_LN2; } __imag__ result = __ieee754_atan2 (__imag__ x, __real__ x); } else { __imag__ result = __nan (""); if (rcls == FP_INFINITE || icls == FP_INFINITE) /* Real or imaginary part is infinite. */ __real__ result = HUGE_VAL; else __real__ result = __nan (""); } return result; }
__complex__ double __clog10 (__complex__ double x) { __complex__ double result; int rcls = fpclassify (__real__ x); int icls = fpclassify (__imag__ x); if (__builtin_expect (rcls == FP_ZERO && icls == FP_ZERO, 0)) { /* Real and imaginary part are 0.0. */ __imag__ result = signbit (__real__ x) ? M_PI : 0.0; __imag__ result = __copysign (__imag__ result, __imag__ x); /* Yes, the following line raises an exception. */ __real__ result = -1.0 / fabs (__real__ x); } else if (__builtin_expect (rcls != FP_NAN && icls != FP_NAN, 1)) { /* Neither real nor imaginary part is NaN. */ double absx = fabs (__real__ x), absy = fabs (__imag__ x); int scale = 0; if (absx < absy) { double t = absx; absx = absy; absy = t; } if (absx > DBL_MAX / 2.0) { scale = -1; absx = __scalbn (absx, scale); absy = (absy >= DBL_MIN * 2.0 ? __scalbn (absy, scale) : 0.0); } else if (absx < DBL_MIN && absy < DBL_MIN) { scale = DBL_MANT_DIG; absx = __scalbn (absx, scale); absy = __scalbn (absy, scale); } if (absx == 1.0 && scale == 0) { double absy2 = absy * absy; if (absy2 <= DBL_MIN * 2.0 * M_LN10) { #if __FLT_EVAL_METHOD__ == 0 __real__ result = (absy2 / 2.0 - absy2 * absy2 / 4.0) * M_LOG10E; #else volatile double force_underflow = absy2 * absy2 / 4.0; __real__ result = (absy2 / 2.0 - force_underflow) * M_LOG10E; #endif } else __real__ result = __log1p (absy2) * (M_LOG10E / 2.0); } else if (absx > 1.0 && absx < 2.0 && absy < 1.0 && scale == 0) { double d2m1 = (absx - 1.0) * (absx + 1.0); if (absy >= DBL_EPSILON) d2m1 += absy * absy; __real__ result = __log1p (d2m1) * (M_LOG10E / 2.0); } else if (absx < 1.0 && absx >= 0.75 && absy < DBL_EPSILON / 2.0 && scale == 0) { double d2m1 = (absx - 1.0) * (absx + 1.0); __real__ result = __log1p (d2m1) * (M_LOG10E / 2.0); } else if (absx < 1.0 && (absx >= 0.75 || absy >= 0.5) && scale == 0) { double d2m1 = __x2y2m1 (absx, absy); __real__ result = __log1p (d2m1) * (M_LOG10E / 2.0); } else { double d = __ieee754_hypot (absx, absy); __real__ result = __ieee754_log10 (d) - scale * M_LOG10_2; } __imag__ result = M_LOG10E * __ieee754_atan2 (__imag__ x, __real__ x); } else { __imag__ result = __nan (""); if (rcls == FP_INFINITE || icls == FP_INFINITE) /* Real or imaginary part is infinite. */ __real__ result = HUGE_VAL; else __real__ result = __nan (""); } return result; }