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math: fix underflow in exp*.c and long double handling in exp2l

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* 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
2013-09-04 07:51:11 +00:00
parent ea9bb95a5b
commit 39c910fb06
8 changed files with 139 additions and 182 deletions
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72
src/math/expm1.c
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@@ -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;
}