float
__ieee754_atanhf (float x)
{
float xa = fabsf (x);
float t;
if (isless (xa, 0.5f))
{
if (__glibc_unlikely (xa < 0x1.0p-28f))
{
math_force_eval (huge + x);
if (fabsf (x) < FLT_MIN)
{
float force_underflow = x * x;
math_force_eval (force_underflow);
}
return x;
}
t = xa + xa;
t = 0.5f * __log1pf (t + t * xa / (1.0f - xa));
}
else if (__glibc_likely (isless (xa, 1.0f)))
t = 0.5f * __log1pf ((xa + xa) / (1.0f - xa));
else
{
if (isgreater (xa, 1.0f))
return (x - x) / (x - x);
return x / 0.0f;
}
return __copysignf (t, x);
}
float
__ieee754_atanhf (float x)
{
float xa = fabsf (x);
float t;
if (isless (xa, 0.5f))
{
if (__builtin_expect (xa < 0x1.0p-28f, 0))
{
math_force_eval (huge + x);
return x;
}
t = xa + xa;
t = 0.5f * __log1pf (t + t * xa / (1.0f - xa));
}
else if (__builtin_expect (isless (xa, 1.0f), 1))
t = 0.5f * __log1pf ((xa + xa) / (1.0f - xa));
else
{
if (isgreater (xa, 1.0f))
return (x - x) / (x - x);
return x / 0.0f;
}
return __copysignf (t, x);
}
float
__asinhf(float x)
{
float w;
int32_t hx,ix;
GET_FLOAT_WORD(hx,x);
ix = hx&0x7fffffff;
if(__builtin_expect(ix< 0x38000000, 0)) { /* |x|<2**-14 */
math_check_force_underflow (x);
if(huge+x>one) return x; /* return x inexact except 0 */
}
if(__builtin_expect(ix>0x47000000, 0)) { /* |x| > 2**14 */
if(ix>=0x7f800000) return x+x; /* x is inf or NaN */
w = __ieee754_logf(fabsf(x))+ln2;
} else {
float xa = fabsf(x);
if (ix>0x40000000) { /* 2**14 > |x| > 2.0 */
w = __ieee754_logf(2.0f*xa+one/(__ieee754_sqrtf(xa*xa+one)+xa));
} else { /* 2.0 > |x| > 2**-14 */
float t = xa*xa;
w =__log1pf(xa+t/(one+__ieee754_sqrtf(one+t)));
}
}
return __copysignf(w, x);
}
float log10f(float x)
{
float f,hfsq,hi,lo,r,y;
int32_t i,k,hx;
GET_FLOAT_WORD(hx, x);
k = 0;
if (hx < 0x00800000) { /* x < 2**-126 */
if ((hx&0x7fffffff) == 0)
return -two25/0.0f; /* log(+-0)=-inf */
if (hx < 0)
return (x-x)/0.0f; /* log(-#) = NaN */
/* subnormal number, scale up x */
k -= 25;
x *= two25;
GET_FLOAT_WORD(hx, x);
}
if (hx >= 0x7f800000)
return x+x;
if (hx == 0x3f800000)
return 0.0f; /* log(1) = +0 */
k += (hx>>23) - 127;
hx &= 0x007fffff;
i = (hx+(0x4afb0d))&0x800000;
SET_FLOAT_WORD(x, hx|(i^0x3f800000)); /* normalize x or x/2 */
k += i>>23;
y = (float)k;
f = x - 1.0f;
hfsq = 0.5f * f * f;
r = __log1pf(f);
// FIXME
// /* See log2f.c and log2.c for details. */
// if (sizeof(float_t) > sizeof(float))
// return (r - hfsq + f) * ((float_t)ivln10lo + ivln10hi) +
// y * ((float_t)log10_2lo + log10_2hi);
hi = f - hfsq;
GET_FLOAT_WORD(hx, hi);
SET_FLOAT_WORD(hi, hx&0xfffff000);
lo = (f - hi) - hfsq + r;
return y*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi +
hi*ivln10hi + y*log10_2hi;
}
__complex__ float
__clog10f (__complex__ float x)
{
__complex__ float 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_LOG10Ef : 0.0;
__imag__ result = __copysignf (__imag__ result, __imag__ x);
/* Yes, the following line raises an exception. */
__real__ result = -1.0 / fabsf (__real__ x);
}
else if (__glibc_likely (rcls != FP_NAN && icls != FP_NAN))
{
/* Neither real nor imaginary part is NaN. */
float absx = fabsf (__real__ x), absy = fabsf (__imag__ x);
int scale = 0;
if (absx < absy)
{
float t = absx;
absx = absy;
absy = t;
}
if (absx > FLT_MAX / 2.0f)
{
scale = -1;
absx = __scalbnf (absx, scale);
absy = (absy >= FLT_MIN * 2.0f ? __scalbnf (absy, scale) : 0.0f);
}
else if (absx < FLT_MIN && absy < FLT_MIN)
{
scale = FLT_MANT_DIG;
absx = __scalbnf (absx, scale);
absy = __scalbnf (absy, scale);
}
if (absx == 1.0f && scale == 0)
{
float absy2 = absy * absy;
if (absy2 <= FLT_MIN * 2.0f * (float) M_LN10)
{
float force_underflow = absy2 * absy2;
__real__ result = absy2 * ((float) M_LOG10E / 2.0f);
math_force_eval (force_underflow);
}
else
__real__ result = __log1pf (absy2) * ((float) M_LOG10E / 2.0f);
}
else if (absx > 1.0f && absx < 2.0f && absy < 1.0f && scale == 0)
{
float d2m1 = (absx - 1.0f) * (absx + 1.0f);
if (absy >= FLT_EPSILON)
d2m1 += absy * absy;
__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
}
else if (absx < 1.0f
&& absx >= 0.75f
&& absy < FLT_EPSILON / 2.0f
&& scale == 0)
{
float d2m1 = (absx - 1.0f) * (absx + 1.0f);
__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
}
else if (absx < 1.0f && (absx >= 0.75f || absy >= 0.5f) && scale == 0)
{
float d2m1 = __x2y2m1f (absx, absy);
__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
}
else
{
float d = __ieee754_hypotf (absx, absy);
__real__ result = __ieee754_log10f (d) - scale * M_LOG10_2f;
}
__imag__ result = M_LOG10E * __ieee754_atan2f (__imag__ x, __real__ x);
}
else
{
__imag__ result = __nanf ("");
if (rcls == FP_INFINITE || icls == FP_INFINITE)
/* Real or imaginary part is infinite. */
__real__ result = HUGE_VALF;
else
__real__ result = __nanf ("");
}
return result;
}
__complex__ float
__clogf (__complex__ float x)
{
__complex__ float 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 = __copysignf (__imag__ result, __imag__ x);
/* Yes, the following line raises an exception. */
__real__ result = -1.0 / fabsf (__real__ x);
}
else if (__builtin_expect (rcls != FP_NAN && icls != FP_NAN, 1))
{
/* Neither real nor imaginary part is NaN. */
float absx = fabsf (__real__ x), absy = fabsf (__imag__ x);
int scale = 0;
if (absx < absy)
{
float t = absx;
absx = absy;
absy = t;
}
if (absx > FLT_MAX / 2.0f)
{
scale = -1;
absx = __scalbnf (absx, scale);
absy = (absy >= FLT_MIN * 2.0f ? __scalbnf (absy, scale) : 0.0f);
}
else if (absx < FLT_MIN && absy < FLT_MIN)
{
scale = FLT_MANT_DIG;
absx = __scalbnf (absx, scale);
absy = __scalbnf (absy, scale);
}
if (absx == 1.0f && scale == 0)
{
float absy2 = absy * absy;
if (absy2 <= FLT_MIN * 2.0f)
{
#if __FLT_EVAL_METHOD__ == 0
__real__ result = absy2 / 2.0f - absy2 * absy2 / 4.0f;
#else
volatile float force_underflow = absy2 * absy2 / 4.0f;
__real__ result = absy2 / 2.0f - force_underflow;
#endif
}
else
__real__ result = __log1pf (absy2) / 2.0f;
}
else if (absx > 1.0f && absx < 2.0f && absy < 1.0f && scale == 0)
{
float d2m1 = (absx - 1.0f) * (absx + 1.0f);
if (absy >= FLT_EPSILON)
d2m1 += absy * absy;
__real__ result = __log1pf (d2m1) / 2.0f;
}
else if (absx < 1.0f
&& absx >= 0.75f
&& absy < FLT_EPSILON / 2.0f
&& scale == 0)
{
float d2m1 = (absx - 1.0f) * (absx + 1.0f);
__real__ result = __log1pf (d2m1) / 2.0f;
}
else if (absx < 1.0f && (absx >= 0.75f || absy >= 0.5f) && scale == 0)
{
float d2m1 = __x2y2m1f (absx, absy);
__real__ result = __log1pf (d2m1) / 2.0f;
}
else
{
float d = __ieee754_hypotf (absx, absy);
__real__ result = __ieee754_logf (d) - scale * (float) M_LN2;
}
__imag__ result = __ieee754_atan2f (__imag__ x, __real__ x);
}
else
{
__imag__ result = __nanf ("");
if (rcls == FP_INFINITE || icls == FP_INFINITE)
/* Real or imaginary part is infinite. */
__real__ result = HUGE_VALF;
else
__real__ result = __nanf ("");
}
return result;
}
__complex__ float
__catanf (__complex__ float x)
{
__complex__ float res;
int rcls = fpclassify (__real__ x);
int icls = fpclassify (__imag__ x);
if (__glibc_unlikely (rcls <= FP_INFINITE || icls <= FP_INFINITE))
{
if (rcls == FP_INFINITE)
{
__real__ res = __copysignf (M_PI_2, __real__ x);
__imag__ res = __copysignf (0.0, __imag__ x);
}
else if (icls == FP_INFINITE)
{
if (rcls >= FP_ZERO)
__real__ res = __copysignf (M_PI_2, __real__ x);
else
__real__ res = __nanf ("");
__imag__ res = __copysignf (0.0, __imag__ x);
}
else if (icls == FP_ZERO || icls == FP_INFINITE)
{
__real__ res = __nanf ("");
__imag__ res = __copysignf (0.0, __imag__ x);
}
else
{
__real__ res = __nanf ("");
__imag__ res = __nanf ("");
}
}
else if (__glibc_unlikely (rcls == FP_ZERO && icls == FP_ZERO))
{
res = x;
}
else
{
if (fabsf (__real__ x) >= 16.0f / FLT_EPSILON
|| fabsf (__imag__ x) >= 16.0f / FLT_EPSILON)
{
__real__ res = __copysignf ((float) M_PI_2, __real__ x);
if (fabsf (__real__ x) <= 1.0f)
__imag__ res = 1.0f / __imag__ x;
else if (fabsf (__imag__ x) <= 1.0f)
__imag__ res = __imag__ x / __real__ x / __real__ x;
else
{
float h = __ieee754_hypotf (__real__ x / 2.0f,
__imag__ x / 2.0f);
__imag__ res = __imag__ x / h / h / 4.0f;
}
}
else
{
float den, absx, absy;
absx = fabsf (__real__ x);
absy = fabsf (__imag__ x);
if (absx < absy)
{
float t = absx;
absx = absy;
absy = t;
}
if (absy < FLT_EPSILON / 2.0f)
{
den = (1.0f - absx) * (1.0f + absx);
if (den == -0.0f)
den = 0.0f;
}
else if (absx >= 1.0f)
den = (1.0f - absx) * (1.0f + absx) - absy * absy;
else if (absx >= 0.75f || absy >= 0.5f)
den = -__x2y2m1f (absx, absy);
else
den = (1.0f - absx) * (1.0f + absx) - absy * absy;
__real__ res = 0.5f * __ieee754_atan2f (2.0f * __real__ x, den);
if (fabsf (__imag__ x) == 1.0f
&& fabsf (__real__ x) < FLT_EPSILON * FLT_EPSILON)
__imag__ res = (__copysignf (0.5f, __imag__ x)
* ((float) M_LN2
- __ieee754_logf (fabsf (__real__ x))));
else
{
float r2 = 0.0f, num, f;
if (fabsf (__real__ x) >= FLT_EPSILON * FLT_EPSILON)
r2 = __real__ x * __real__ x;
num = __imag__ x + 1.0f;
num = r2 + num * num;
den = __imag__ x - 1.0f;
den = r2 + den * den;
f = num / den;
if (f < 0.5f)
__imag__ res = 0.25f * __ieee754_logf (f);
else
{
num = 4.0f * __imag__ x;
__imag__ res = 0.25f * __log1pf (num / den);
}
}
}
if (fabsf (__real__ res) < FLT_MIN)
{
volatile float force_underflow = __real__ res * __real__ res;
(void) force_underflow;
}
if (fabsf (__imag__ res) < FLT_MIN)
{
volatile float force_underflow = __imag__ res * __imag__ res;
(void) force_underflow;
}
}
return res;
}
__complex__ float
__kernel_casinhf (__complex__ float x, int adj)
{
__complex__ float res;
float rx, ix;
__complex__ float y;
/* Avoid cancellation by reducing to the first quadrant. */
rx = fabsf (__real__ x);
ix = fabsf (__imag__ x);
if (rx >= 1.0f / FLT_EPSILON || ix >= 1.0f / FLT_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)
{
float t = __real__ y;
__real__ y = __copysignf (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clogf (y);
__real__ res += (float) M_LN2;
}
else if (rx >= 0.5f && ix < FLT_EPSILON / 8.0f)
{
float s = __ieee754_hypotf (1.0f, rx);
__real__ res = __ieee754_logf (rx + s);
if (adj)
__imag__ res = __ieee754_atan2f (s, __imag__ x);
else
__imag__ res = __ieee754_atan2f (ix, s);
}
else if (rx < FLT_EPSILON / 8.0f && ix >= 1.5f)
{
float s = __ieee754_sqrtf ((ix + 1.0f) * (ix - 1.0f));
__real__ res = __ieee754_logf (ix + s);
if (adj)
__imag__ res = __ieee754_atan2f (rx, __copysignf (s, __imag__ x));
else
__imag__ res = __ieee754_atan2f (s, rx);
}
else if (ix > 1.0f && ix < 1.5f && rx < 0.5f)
{
if (rx < FLT_EPSILON * FLT_EPSILON)
{
float ix2m1 = (ix + 1.0f) * (ix - 1.0f);
float s = __ieee754_sqrtf (ix2m1);
__real__ res = __log1pf (2.0f * (ix2m1 + ix * s)) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (rx, __copysignf (s, __imag__ x));
else
__imag__ res = __ieee754_atan2f (s, rx);
}
else
{
float ix2m1 = (ix + 1.0f) * (ix - 1.0f);
float rx2 = rx * rx;
float f = rx2 * (2.0f + rx2 + 2.0f * ix * ix);
float d = __ieee754_sqrtf (ix2m1 * ix2m1 + f);
float dp = d + ix2m1;
float dm = f / dp;
float r1 = __ieee754_sqrtf ((dm + rx2) / 2.0f);
float r2 = rx * ix / r1;
__real__ res
= __log1pf (rx2 + dp + 2.0f * (rx * r1 + ix * r2)) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (rx + r1, __copysignf (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2f (ix + r2, rx + r1);
}
}
else if (ix == 1.0f && rx < 0.5f)
{
if (rx < FLT_EPSILON / 8.0f)
{
__real__ res = __log1pf (2.0f * (rx + __ieee754_sqrtf (rx))) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (__ieee754_sqrtf (rx),
__copysignf (1.0f, __imag__ x));
else
__imag__ res = __ieee754_atan2f (1.0f, __ieee754_sqrtf (rx));
}
else
{
float d = rx * __ieee754_sqrtf (4.0f + rx * rx);
float s1 = __ieee754_sqrtf ((d + rx * rx) / 2.0f);
float s2 = __ieee754_sqrtf ((d - rx * rx) / 2.0f);
__real__ res = __log1pf (rx * rx + d + 2.0f * (rx * s1 + s2)) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (rx + s1,
__copysignf (1.0f + s2,
__imag__ x));
else
__imag__ res = __ieee754_atan2f (1.0f + s2, rx + s1);
}
}
else if (ix < 1.0f && rx < 0.5f)
{
if (ix >= FLT_EPSILON)
{
if (rx < FLT_EPSILON * FLT_EPSILON)
{
float onemix2 = (1.0f + ix) * (1.0f - ix);
float s = __ieee754_sqrtf (onemix2);
__real__ res = __log1pf (2.0f * rx / s) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (s, __imag__ x);
else
__imag__ res = __ieee754_atan2f (ix, s);
}
else
{
float onemix2 = (1.0f + ix) * (1.0f - ix);
float rx2 = rx * rx;
float f = rx2 * (2.0f + rx2 + 2.0f * ix * ix);
float d = __ieee754_sqrtf (onemix2 * onemix2 + f);
float dp = d + onemix2;
float dm = f / dp;
float r1 = __ieee754_sqrtf ((dp + rx2) / 2.0f);
float r2 = rx * ix / r1;
__real__ res
= __log1pf (rx2 + dm + 2.0f * (rx * r1 + ix * r2)) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (rx + r1,
__copysignf (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2f (ix + r2, rx + r1);
}
}
else
{
float s = __ieee754_hypotf (1.0f, rx);
__real__ res = __log1pf (2.0f * rx * (rx + s)) / 2.0f;
if (adj)
__imag__ res = __ieee754_atan2f (s, __imag__ x);
else
__imag__ res = __ieee754_atan2f (ix, s);
}
if (__real__ res < FLT_MIN)
{
volatile float force_underflow = __real__ res * __real__ res;
(void) force_underflow;
}
}
else
{
__real__ y = (rx - ix) * (rx + ix) + 1.0f;
__imag__ y = 2.0f * rx * ix;
y = __csqrtf (y);
__real__ y += rx;
__imag__ y += ix;
if (adj)
{
float t = __real__ y;
__real__ y = __copysignf (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clogf (y);
}
/* Give results the correct sign for the original argument. */
__real__ res = __copysignf (__real__ res, __real__ x);
__imag__ res = __copysignf (__imag__ res, (adj ? 1.0f : __imag__ x));
return res;
}
float
__lgamma_negf (float x, int *signgamp)
{
/* Determine the half-integer region X lies in, handle exact
integers and determine the sign of the result. */
int i = __floorf (-2 * x);
if ((i & 1) == 0 && i == -2 * x)
return 1.0f / 0.0f;
float xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
i -= 4;
*signgamp = ((i & 2) == 0 ? -1 : 1);
SET_RESTORE_ROUNDF (FE_TONEAREST);
/* Expand around the zero X0 = X0_HI + X0_LO. */
float x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
float xdiff = x - x0_hi - x0_lo;
/* For arguments in the range -3 to -2, use polynomial
approximations to an adjusted version of the gamma function. */
if (i < 2)
{
int j = __floorf (-8 * x) - 16;
float xm = (-33 - 2 * j) * 0.0625f;
float x_adj = x - xm;
size_t deg = poly_deg[j];
size_t end = poly_end[j];
float g = poly_coeff[end];
for (size_t j = 1; j <= deg; j++)
g = g * x_adj + poly_coeff[end - j];
return __log1pf (g * xdiff / (x - xn));
}
/* The result we want is log (sinpi (X0) / sinpi (X))
+ log (gamma (1 - X0) / gamma (1 - X)). */
float x_idiff = fabsf (xn - x), x0_idiff = fabsf (xn - x0_hi - x0_lo);
float log_sinpi_ratio;
if (x0_idiff < x_idiff * 0.5f)
/* Use log not log1p to avoid inaccuracy from log1p of arguments
close to -1. */
log_sinpi_ratio = __ieee754_logf (lg_sinpi (x0_idiff)
/ lg_sinpi (x_idiff));
else
{
/* Use log1p not log to avoid inaccuracy from log of arguments
close to 1. X0DIFF2 has positive sign if X0 is further from
XN than X is from XN, negative sign otherwise. */
float x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5f;
float sx0d2 = lg_sinpi (x0diff2);
float cx0d2 = lg_cospi (x0diff2);
log_sinpi_ratio = __log1pf (2 * sx0d2
* (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
}
float log_gamma_ratio;
float y0 = math_narrow_eval (1 - x0_hi);
float y0_eps = -x0_hi + (1 - y0) - x0_lo;
float y = math_narrow_eval (1 - x);
float y_eps = -x + (1 - y);
/* We now wish to compute LOG_GAMMA_RATIO
= log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
accurately approximates the difference Y0 + Y0_EPS - Y -
Y_EPS. Use Stirling's approximation. */
float log_gamma_high
= (xdiff * __log1pf ((y0 - e_hi - e_lo + y0_eps) / e_hi)
+ (y - 0.5f + y_eps) * __log1pf (xdiff / y));
/* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
float y0r = 1 / y0, yr = 1 / y;
float y0r2 = y0r * y0r, yr2 = yr * yr;
float rdiff = -xdiff / (y * y0);
float bterm[NCOEFF];
float dlast = rdiff, elast = rdiff * yr * (yr + y0r);
bterm[0] = dlast * lgamma_coeff[0];
for (size_t j = 1; j < NCOEFF; j++)
{
float dnext = dlast * y0r2 + elast;
float enext = elast * yr2;
bterm[j] = dnext * lgamma_coeff[j];
dlast = dnext;
elast = enext;
}
float log_gamma_low = 0;
for (size_t j = 0; j < NCOEFF; j++)
log_gamma_low += bterm[NCOEFF - 1 - j];
log_gamma_ratio = log_gamma_high + log_gamma_low;
return log_sinpi_ratio + log_gamma_ratio;
}
