mirror of
git://git.musl-libc.org/musl
synced 2025-03-11 06:07:29 +00:00
math: fix underflow in exp*.c and long double handling in exp2l
* don't care about inexact flag * use double_t and float_t (faster, smaller, more precise on x86) * exp: underflow when result is zero or subnormal and not -inf * exp2: underflow when result is zero or subnormal and not exact * expm1: underflow when result is zero or subnormal * expl: don't underflow on -inf * exp2: fix incorrect comment * expm1: simplify special case handling and overflow properly * expm1: cleanup final scaling and fix negative left shift ub (twopk)
This commit is contained in:
Szabolcs Nagy
parent
ea9bb95a5b
commit
39c910fb06
@ -80,7 +80,7 @@ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
|
||||
|
||||
double exp(double x)
|
||||
{
|
||||
double hi, lo, c, xx;
|
||||
double_t hi, lo, c, xx, y;
|
||||
int k, sign;
|
||||
uint32_t hx;
|
||||
|
||||
@ -89,20 +89,19 @@ double exp(double x)
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* special cases */
|
||||
if (hx >= 0x40862e42) { /* if |x| >= 709.78... */
|
||||
if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
|
||||
if (isnan(x))
|
||||
return x;
|
||||
if (hx == 0x7ff00000 && sign) /* -inf */
|
||||
return 0;
|
||||
if (x > 709.782712893383973096) {
|
||||
/* overflow if x!=inf */
|
||||
STRICT_ASSIGN(double, x, 0x1p1023 * x);
|
||||
return x;
|
||||
}
|
||||
if (x < -745.13321910194110842) {
|
||||
/* underflow */
|
||||
STRICT_ASSIGN(double, x, 0x1p-1000 * 0x1p-1000);
|
||||
return x;
|
||||
if (x < -708.39641853226410622) {
|
||||
/* underflow if x!=-inf */
|
||||
FORCE_EVAL((float)(-0x1p-149/x));
|
||||
if (x < -745.13321910194110842)
|
||||
return 0;
|
||||
}
|
||||
}
|
||||
|
||||
@ -128,8 +127,8 @@ double exp(double x)
|
||||
/* x is now in primary range */
|
||||
xx = x*x;
|
||||
c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
|
||||
x = 1 + (x*c/(2-c) - lo + hi);
|
||||
y = 1 + (x*c/(2-c) - lo + hi);
|
||||
if (k == 0)
|
||||
return x;
|
||||
return scalbn(x, k);
|
||||
return y;
|
||||
return scalbn(y, k);
|
||||
}
|
||||
|
||||
@ -305,7 +305,7 @@ static const double tbl[TBLSIZE * 2] = {
|
||||
* Method: (accurate tables)
|
||||
*
|
||||
* Reduce x:
|
||||
* x = 2**k + y, for integer k and |y| <= 1/2.
|
||||
* x = k + y, for integer k and |y| <= 1/2.
|
||||
* Thus we have exp2(x) = 2**k * exp2(y).
|
||||
*
|
||||
* Reduce y:
|
||||
@ -330,41 +330,41 @@ static const double tbl[TBLSIZE * 2] = {
|
||||
*/
|
||||
double exp2(double x)
|
||||
{
|
||||
double r, t, z;
|
||||
uint32_t hx, ix, i0;
|
||||
double_t r, t, z;
|
||||
uint32_t ix, i0;
|
||||
union {double f; uint64_t i;} u = {x};
|
||||
union {uint32_t u; int32_t i;} k;
|
||||
|
||||
/* Filter out exceptional cases. */
|
||||
GET_HIGH_WORD(hx, x);
|
||||
ix = hx & 0x7fffffff;
|
||||
if (ix >= 0x40900000) { /* |x| >= 1024 */
|
||||
if (ix >= 0x7ff00000) {
|
||||
GET_LOW_WORD(ix, x);
|
||||
if (hx == 0xfff00000 && ix == 0) /* -inf */
|
||||
return 0;
|
||||
return x;
|
||||
}
|
||||
if (x >= 1024) {
|
||||
ix = u.i>>32 & 0x7fffffff;
|
||||
if (ix >= 0x408ff000) { /* |x| >= 1022 or nan */
|
||||
if (ix >= 0x40900000 && u.i>>63 == 0) { /* x >= 1024 or nan */
|
||||
/* overflow */
|
||||
STRICT_ASSIGN(double, x, x * 0x1p1023);
|
||||
return x;
|
||||
}
|
||||
if (x <= -1075) {
|
||||
STRICT_ASSIGN(double, x, 0x1p-1000*0x1p-1000);
|
||||
return x;
|
||||
if (ix >= 0x7ff00000) /* -inf or -nan */
|
||||
return -1/x;
|
||||
if (u.i>>63) { /* x <= -1022 */
|
||||
/* underflow */
|
||||
if (x <= -1075 || x - 0x1p52 + 0x1p52 != x)
|
||||
FORCE_EVAL((float)(-0x1p-149/x));
|
||||
if (x <= -1075)
|
||||
return 0;
|
||||
}
|
||||
} else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */
|
||||
return 1.0 + x;
|
||||
}
|
||||
|
||||
/* Reduce x, computing z, i0, and k. */
|
||||
STRICT_ASSIGN(double, t, x + redux);
|
||||
GET_LOW_WORD(i0, t);
|
||||
u.f = x + redux;
|
||||
i0 = u.i;
|
||||
i0 += TBLSIZE / 2;
|
||||
k.u = i0 / TBLSIZE * TBLSIZE;
|
||||
k.i /= TBLSIZE;
|
||||
i0 %= TBLSIZE;
|
||||
t -= redux;
|
||||
z = x - t;
|
||||
u.f -= redux;
|
||||
z = x - u.f;
|
||||
|
||||
/* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
|
||||
t = tbl[2*i0]; /* exp2t[i0] */
|
||||
|
||||
@ -63,7 +63,7 @@ static const double exp2ft[TBLSIZE] = {
|
||||
* Method: (equally-spaced tables)
|
||||
*
|
||||
* Reduce x:
|
||||
* x = 2**k + y, for integer k and |y| <= 1/2.
|
||||
* x = k + y, for integer k and |y| <= 1/2.
|
||||
* Thus we have exp2f(x) = 2**k * exp2(y).
|
||||
*
|
||||
* Reduce y:
|
||||
@ -83,46 +83,42 @@ static const double exp2ft[TBLSIZE] = {
|
||||
*/
|
||||
float exp2f(float x)
|
||||
{
|
||||
double tv, twopk, u, z;
|
||||
float t;
|
||||
uint32_t hx, ix, i0, k;
|
||||
double_t t, r, z;
|
||||
union {float f; uint32_t i;} u = {x};
|
||||
union {double f; uint64_t i;} uk;
|
||||
uint32_t ix, i0, k;
|
||||
|
||||
/* Filter out exceptional cases. */
|
||||
GET_FLOAT_WORD(hx, x);
|
||||
ix = hx & 0x7fffffff;
|
||||
if (ix >= 0x43000000) { /* |x| >= 128 */
|
||||
if (ix >= 0x7f800000) {
|
||||
if (hx == 0xff800000) /* -inf */
|
||||
return 0;
|
||||
return x;
|
||||
}
|
||||
if (x >= 128) {
|
||||
ix = u.i & 0x7fffffff;
|
||||
if (ix > 0x42fc0000) { /* |x| > 126 */
|
||||
if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */
|
||||
STRICT_ASSIGN(float, x, x * 0x1p127f);
|
||||
return x;
|
||||
}
|
||||
if (x <= -150) {
|
||||
STRICT_ASSIGN(float, x, 0x1p-100f*0x1p-100f);
|
||||
return x;
|
||||
if (u.i >= 0x80000000) { /* x < -126 */
|
||||
if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
|
||||
FORCE_EVAL(-0x1p-149f/x);
|
||||
if (u.i >= 0xc3160000) /* x <= -150 */
|
||||
return 0;
|
||||
}
|
||||
} else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
|
||||
return 1.0f + x;
|
||||
}
|
||||
|
||||
/* Reduce x, computing z, i0, and k. */
|
||||
STRICT_ASSIGN(float, t, x + redux);
|
||||
GET_FLOAT_WORD(i0, t);
|
||||
u.f = x + redux;
|
||||
i0 = u.i;
|
||||
i0 += TBLSIZE / 2;
|
||||
k = (i0 / TBLSIZE) << 20;
|
||||
k = i0 / TBLSIZE;
|
||||
uk.i = (uint64_t)(0x3ff + k)<<52;
|
||||
i0 &= TBLSIZE - 1;
|
||||
t -= redux;
|
||||
z = x - t;
|
||||
INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
|
||||
|
||||
u.f -= redux;
|
||||
z = x - u.f;
|
||||
/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
|
||||
tv = exp2ft[i0];
|
||||
u = tv * z;
|
||||
tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
|
||||
r = exp2ft[i0];
|
||||
t = r * z;
|
||||
r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
|
||||
|
||||
/* Scale by 2**(k>>20). */
|
||||
return tv * twopk;
|
||||
/* Scale by 2**k */
|
||||
return r * uk.f;
|
||||
}
|
||||
|
||||
@ -33,13 +33,9 @@ long double exp2l(long double x)
|
||||
return exp2(x);
|
||||
}
|
||||
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
|
||||
|
||||
#define TBLBITS 7
|
||||
#define TBLSIZE (1 << TBLBITS)
|
||||
|
||||
#define BIAS (LDBL_MAX_EXP - 1)
|
||||
#define EXPMASK (BIAS + LDBL_MAX_EXP)
|
||||
|
||||
static const double
|
||||
redux = 0x1.8p63 / TBLSIZE,
|
||||
P1 = 0x1.62e42fefa39efp-1,
|
||||
@ -203,29 +199,29 @@ static const double tbl[TBLSIZE * 2] = {
|
||||
*/
|
||||
long double exp2l(long double x)
|
||||
{
|
||||
union IEEEl2bits u, v;
|
||||
union ldshape u = {x};
|
||||
int e = u.i.se & 0x7fff;
|
||||
long double r, z;
|
||||
uint32_t hx, ix, i0;
|
||||
uint32_t i0;
|
||||
union {uint32_t u; int32_t i;} k;
|
||||
|
||||
/* Filter out exceptional cases. */
|
||||
u.e = x;
|
||||
hx = u.xbits.expsign;
|
||||
ix = hx & EXPMASK;
|
||||
if (ix >= BIAS + 14) { /* |x| >= 16384 or x is NaN */
|
||||
if (ix == EXPMASK) {
|
||||
if (u.xbits.man == 1ULL << 63 && hx == 0xffff) /* -inf */
|
||||
if (e >= 0x3fff + 13) { /* |x| >= 8192 or x is NaN */
|
||||
if (u.i.se >= 0x3fff + 14 && u.i.se >> 15 == 0)
|
||||
/* overflow */
|
||||
return x * 0x1p16383L;
|
||||
if (e == 0x7fff) /* -inf or -nan */
|
||||
return -1/x;
|
||||
if (x < -16382) {
|
||||
if (x <= -16446 || x - 0x1p63 + 0x1p63 != x)
|
||||
/* underflow */
|
||||
FORCE_EVAL((float)(-0x1p-149/x));
|
||||
if (x <= -16446)
|
||||
return 0;
|
||||
return x;
|
||||
}
|
||||
if (x >= 16384) {
|
||||
x *= 0x1p16383L;
|
||||
return x;
|
||||
}
|
||||
if (x <= -16446)
|
||||
return 0x1p-10000L*0x1p-10000L;
|
||||
} else if (ix < BIAS - 64) /* |x| < 0x1p-64 */
|
||||
} else if (e < 0x3fff - 64) {
|
||||
return 1 + x;
|
||||
}
|
||||
|
||||
/*
|
||||
* Reduce x, computing z, i0, and k. The low bits of x + redux
|
||||
@ -238,13 +234,13 @@ long double exp2l(long double x)
|
||||
* We split this into k = 0xabc and i0 = 0x12 (adjusted to
|
||||
* index into the table), then we compute z = 0x0.003456p0.
|
||||
*/
|
||||
u.e = x + redux;
|
||||
i0 = u.bits.manl + TBLSIZE / 2;
|
||||
u.f = x + redux;
|
||||
i0 = u.i.m + TBLSIZE / 2;
|
||||
k.u = i0 / TBLSIZE * TBLSIZE;
|
||||
k.i /= TBLSIZE;
|
||||
i0 %= TBLSIZE;
|
||||
u.e -= redux;
|
||||
z = x - u.e;
|
||||
u.f -= redux;
|
||||
z = x - u.f;
|
||||
|
||||
/* Compute r = exp2l(y) = exp2lt[i0] * p(z). */
|
||||
long double t_hi = tbl[2*i0];
|
||||
|
||||
@ -29,7 +29,7 @@ P2 = -2.7667332906e-3f; /* -0xb55215.0p-32 */
|
||||
|
||||
float expf(float x)
|
||||
{
|
||||
float hi, lo, c, xx;
|
||||
float_t hi, lo, c, xx, y;
|
||||
int k, sign;
|
||||
uint32_t hx;
|
||||
|
||||
@ -38,20 +38,17 @@ float expf(float x)
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* special cases */
|
||||
if (hx >= 0x42b17218) { /* if |x| >= 88.722839f or NaN */
|
||||
if (hx > 0x7f800000) /* NaN */
|
||||
return x;
|
||||
if (!sign) {
|
||||
/* overflow if x!=inf */
|
||||
if (hx >= 0x42aeac50) { /* if |x| >= -87.33655f or NaN */
|
||||
if (hx >= 0x42b17218 && !sign) { /* x >= 88.722839f */
|
||||
/* overflow */
|
||||
STRICT_ASSIGN(float, x, x * 0x1p127f);
|
||||
return x;
|
||||
}
|
||||
if (hx == 0x7f800000) /* -inf */
|
||||
return 0;
|
||||
if (hx >= 0x42cff1b5) { /* x <= -103.972084f */
|
||||
if (sign) {
|
||||
/* underflow */
|
||||
STRICT_ASSIGN(float, x, 0x1p-100f*0x1p-100f);
|
||||
return x;
|
||||
FORCE_EVAL(-0x1p-149f/x);
|
||||
if (hx >= 0x42cff1b5) /* x <= -103.972084f */
|
||||
return 0;
|
||||
}
|
||||
}
|
||||
|
||||
@ -77,8 +74,8 @@ float expf(float x)
|
||||
/* x is now in primary range */
|
||||
xx = x*x;
|
||||
c = x - xx*(P1+xx*P2);
|
||||
x = 1 + (x*c/(2-c) - lo + hi);
|
||||
y = 1 + (x*c/(2-c) - lo + hi);
|
||||
if (k == 0)
|
||||
return x;
|
||||
return scalbnf(x, k);
|
||||
return y;
|
||||
return scalbnf(y, k);
|
||||
}
|
||||
|
||||
@ -100,7 +100,7 @@ long double expl(long double x)
|
||||
if (x > 11356.5234062941439488L) /* x > ln(2^16384 - 0.5) */
|
||||
return x * 0x1p16383L;
|
||||
if (x < -11399.4985314888605581L) /* x < ln(2^-16446) */
|
||||
return 0x1p-10000L * 0x1p-10000L;
|
||||
return -0x1p-16445L/x;
|
||||
|
||||
/* Express e**x = e**f 2**k
|
||||
* = e**(f + k ln(2))
|
||||
|
||||
@ -31,7 +31,7 @@
|
||||
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
||||
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
||||
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
||||
* We use a special Reme algorithm on [0,0.347] to generate
|
||||
* We use a special Remez algorithm on [0,0.347] to generate
|
||||
* a polynomial of degree 5 in r*r to approximate R1. The
|
||||
* maximum error of this polynomial approximation is bounded
|
||||
* by 2**-61. In other words,
|
||||
@ -107,8 +107,6 @@
|
||||
#include "libm.h"
|
||||
|
||||
static const double
|
||||
huge = 1.0e+300,
|
||||
tiny = 1.0e-300,
|
||||
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
|
||||
ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
||||
ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
||||
@ -122,39 +120,27 @@ Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
||||
|
||||
double expm1(double x)
|
||||
{
|
||||
double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
|
||||
int32_t k,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_HIGH_WORD(hx, x);
|
||||
xsb = hx&0x80000000; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
|
||||
union {double f; uint64_t i;} u = {x};
|
||||
uint32_t hx = u.i>>32 & 0x7fffffff;
|
||||
int k, sign = u.i>>63;
|
||||
|
||||
/* filter out huge and non-finite argument */
|
||||
if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
||||
if (hx >= 0x40862E42) { /* if |x|>=709.78... */
|
||||
if (hx >= 0x7ff00000) {
|
||||
uint32_t low;
|
||||
|
||||
GET_LOW_WORD(low, x);
|
||||
if (((hx&0xfffff)|low) != 0) /* NaN */
|
||||
return x+x;
|
||||
return xsb==0 ? x : -1.0; /* exp(+-inf)={inf,-1} */
|
||||
}
|
||||
if(x > o_threshold)
|
||||
return huge*huge; /* overflow */
|
||||
}
|
||||
if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
|
||||
/* raise inexact */
|
||||
if(x+tiny<0.0)
|
||||
return tiny-1.0; /* return -1 */
|
||||
if (isnan(x))
|
||||
return x;
|
||||
if (sign)
|
||||
return -1;
|
||||
if (x > o_threshold) {
|
||||
x *= 0x1p1023;
|
||||
return x;
|
||||
}
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
||||
if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
||||
if (xsb == 0) {
|
||||
if (!sign) {
|
||||
hi = x - ln2_hi;
|
||||
lo = ln2_lo;
|
||||
k = 1;
|
||||
@ -164,7 +150,7 @@ double expm1(double x)
|
||||
k = -1;
|
||||
}
|
||||
} else {
|
||||
k = invln2*x + (xsb==0 ? 0.5 : -0.5);
|
||||
k = invln2*x + (sign ? -0.5 : 0.5);
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
||||
lo = t*ln2_lo;
|
||||
@ -172,9 +158,9 @@ double expm1(double x)
|
||||
STRICT_ASSIGN(double, x, hi - lo);
|
||||
c = (hi-x)-lo;
|
||||
} else if (hx < 0x3c900000) { /* |x| < 2**-54, return x */
|
||||
/* raise inexact flags when x != 0 */
|
||||
t = huge+x;
|
||||
return x - (t-(huge+x));
|
||||
if (hx < 0x00100000)
|
||||
FORCE_EVAL((float)x);
|
||||
return x;
|
||||
} else
|
||||
k = 0;
|
||||
|
||||
@ -186,9 +172,9 @@ double expm1(double x)
|
||||
e = hxs*((r1-t)/(6.0 - x*t));
|
||||
if (k == 0) /* c is 0 */
|
||||
return x - (x*e-hxs);
|
||||
INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0); /* 2^k */
|
||||
e = x*(e-c) - c;
|
||||
e -= hxs;
|
||||
/* exp(x) ~ 2^k (x_reduced - e + 1) */
|
||||
if (k == -1)
|
||||
return 0.5*(x-e) - 0.5;
|
||||
if (k == 1) {
|
||||
@ -196,24 +182,20 @@ double expm1(double x)
|
||||
return -2.0*(e-(x+0.5));
|
||||
return 1.0+2.0*(x-e);
|
||||
}
|
||||
if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
|
||||
y = 1.0 - (e-x);
|
||||
u.i = (uint64_t)(0x3ff + k)<<52; /* 2^k */
|
||||
twopk = u.f;
|
||||
if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
|
||||
y = x - e + 1.0;
|
||||
if (k == 1024)
|
||||
y = y*2.0*0x1p1023;
|
||||
else
|
||||
y = y*twopk;
|
||||
return y - 1.0;
|
||||
}
|
||||
t = 1.0;
|
||||
if (k < 20) {
|
||||
SET_HIGH_WORD(t, 0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
|
||||
y = t-(e-x);
|
||||
y = y*twopk;
|
||||
} else {
|
||||
SET_HIGH_WORD(t, ((0x3ff-k)<<20)); /* 2^-k */
|
||||
y = x-(e+t);
|
||||
y += 1.0;
|
||||
y = y*twopk;
|
||||
}
|
||||
u.i = (uint64_t)(0x3ff - k)<<52; /* 2^-k */
|
||||
if (k < 20)
|
||||
y = (x-e+(1-u.f))*twopk;
|
||||
else
|
||||
y = (x-(e+u.f)+1)*twopk;
|
||||
return y;
|
||||
}
|
||||
|
||||
@ -16,8 +16,6 @@
|
||||
#include "libm.h"
|
||||
|
||||
static const float
|
||||
huge = 1.0e+30,
|
||||
tiny = 1.0e-30,
|
||||
o_threshold = 8.8721679688e+01, /* 0x42b17180 */
|
||||
ln2_hi = 6.9313812256e-01, /* 0x3f317180 */
|
||||
ln2_lo = 9.0580006145e-06, /* 0x3717f7d1 */
|
||||
@ -32,35 +30,27 @@ Q2 = 1.5807170421e-3; /* 0xcf3010.0p-33 */
|
||||
|
||||
float expm1f(float x)
|
||||
{
|
||||
float y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
|
||||
int32_t k,xsb;
|
||||
uint32_t hx;
|
||||
|
||||
GET_FLOAT_WORD(hx, x);
|
||||
xsb = hx&0x80000000; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
float_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
|
||||
union {float f; uint32_t i;} u = {x};
|
||||
uint32_t hx = u.i & 0x7fffffff;
|
||||
int k, sign = u.i >> 31;
|
||||
|
||||
/* filter out huge and non-finite argument */
|
||||
if (hx >= 0x4195b844) { /* if |x|>=27*ln2 */
|
||||
if (hx >= 0x42b17218) { /* if |x|>=88.721... */
|
||||
if (hx > 0x7f800000) /* NaN */
|
||||
return x+x;
|
||||
if (hx == 0x7f800000) /* exp(+-inf)={inf,-1} */
|
||||
return xsb==0 ? x : -1.0;
|
||||
if (x > o_threshold)
|
||||
return huge*huge; /* overflow */
|
||||
}
|
||||
if (xsb != 0) { /* x < -27*ln2 */
|
||||
/* raise inexact */
|
||||
if (x+tiny < 0.0f)
|
||||
return tiny-1.0f; /* return -1 */
|
||||
if (hx > 0x7f800000) /* NaN */
|
||||
return x;
|
||||
if (sign)
|
||||
return -1;
|
||||
if (x > o_threshold) {
|
||||
x *= 0x1p127f;
|
||||
return x;
|
||||
}
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if (hx > 0x3eb17218) { /* if |x| > 0.5 ln2 */
|
||||
if (hx < 0x3F851592) { /* and |x| < 1.5 ln2 */
|
||||
if (xsb == 0) {
|
||||
if (!sign) {
|
||||
hi = x - ln2_hi;
|
||||
lo = ln2_lo;
|
||||
k = 1;
|
||||
@ -70,7 +60,7 @@ float expm1f(float x)
|
||||
k = -1;
|
||||
}
|
||||
} else {
|
||||
k = invln2*x + (xsb==0 ? 0.5f : -0.5f);
|
||||
k = invln2*x + (sign ? -0.5f : 0.5f);
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
||||
lo = t*ln2_lo;
|
||||
@ -78,8 +68,9 @@ float expm1f(float x)
|
||||
STRICT_ASSIGN(float, x, hi - lo);
|
||||
c = (hi-x)-lo;
|
||||
} else if (hx < 0x33000000) { /* when |x|<2**-25, return x */
|
||||
t = huge+x; /* return x with inexact flags when x!=0 */
|
||||
return x - (t-(huge+x));
|
||||
if (hx < 0x00800000)
|
||||
FORCE_EVAL(x*x);
|
||||
return x;
|
||||
} else
|
||||
k = 0;
|
||||
|
||||
@ -91,9 +82,9 @@ float expm1f(float x)
|
||||
e = hxs*((r1-t)/(6.0f - x*t));
|
||||
if (k == 0) /* c is 0 */
|
||||
return x - (x*e-hxs);
|
||||
SET_FLOAT_WORD(twopk, 0x3f800000+(k<<23)); /* 2^k */
|
||||
e = x*(e-c) - c;
|
||||
e -= hxs;
|
||||
/* exp(x) ~ 2^k (x_reduced - e + 1) */
|
||||
if (k == -1)
|
||||
return 0.5f*(x-e) - 0.5f;
|
||||
if (k == 1) {
|
||||
@ -101,24 +92,20 @@ float expm1f(float x)
|
||||
return -2.0f*(e-(x+0.5f));
|
||||
return 1.0f + 2.0f*(x-e);
|
||||
}
|
||||
if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
|
||||
y = 1.0f - (e - x);
|
||||
u.i = (0x7f+k)<<23; /* 2^k */
|
||||
twopk = u.f;
|
||||
if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
|
||||
y = x - e + 1.0f;
|
||||
if (k == 128)
|
||||
y = y*2.0f*0x1p127f;
|
||||
else
|
||||
y = y*twopk;
|
||||
return y - 1.0f;
|
||||
}
|
||||
t = 1.0f;
|
||||
if (k < 23) {
|
||||
SET_FLOAT_WORD(t, 0x3f800000 - (0x1000000>>k)); /* t=1-2^-k */
|
||||
y = t - (e - x);
|
||||
y = y*twopk;
|
||||
} else {
|
||||
SET_FLOAT_WORD(t, (0x7f-k)<<23); /* 2^-k */
|
||||
y = x - (e + t);
|
||||
y += 1.0f;
|
||||
y = y*twopk;
|
||||
}
|
||||
u.i = (0x7f-k)<<23; /* 2^-k */
|
||||
if (k < 23)
|
||||
y = (x-e+(1-u.f))*twopk;
|
||||
else
|
||||
y = (x-(e+u.f)+1)*twopk;
|
||||
return y;
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user