math: exp.c clean up

overflow and underflow was incorrect when the result was not stored.
an optimization for the 0.5*ln2 < |x| < 1.5*ln2 domain was removed.
did various cleanups around static constants and made the comments
consistent with the code.
This commit is contained in:
Szabolcs Nagy 2012-11-17 23:22:41 +01:00
parent a4db94ab78
commit bbbf045ce9
1 changed files with 49 additions and 72 deletions

View File

@ -25,7 +25,7 @@
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Remes algorithm on [0,0.34658] to generate
* We use a special Remez algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
@ -36,15 +36,15 @@
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + -------
* R - r
* r*R1(r)
* 2*r
* exp(r) = 1 + ----------
* R(r) - r
* r*c(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - R1(r)
* 2 - c(r)
* where
* 2 4 10
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
* 2 4 10
* c(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
@ -61,27 +61,16 @@
*
* Misc. info.
* For IEEE double
* if x > 7.09782712893383973096e+02 then exp(x) overflow
* if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
* if x > 709.782712893383973096 then exp(x) overflows
* if x < -745.133219101941108420 then exp(x) underflows
*/
#include "libm.h"
static const double
halF[2] = {0.5,-0.5,},
huge = 1.0e+300,
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */
half[2] = {0.5,-0.5},
ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
@ -89,68 +78,56 @@ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
static const volatile double
twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */
double exp(double x)
{
double y,hi=0.0,lo=0.0,c,t,twopk;
int32_t k=0,xsb;
double hi, lo, c, z;
int k, sign;
uint32_t hx;
GET_HIGH_WORD(hx, x);
xsb = (hx>>31)&1; /* sign bit of x */
sign = hx>>31;
hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */
if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
if (hx >= 0x7ff00000) {
uint32_t lx;
GET_LOW_WORD(lx,x);
if (((hx&0xfffff)|lx) != 0) /* NaN */
return x+x;
return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */
/* special cases */
if (hx >= 0x40862e42) { /* if |x| >= 709.78... */
if (isnan(x))
return x;
if (x > 709.782712893383973096) {
/* overflow if x!=inf */
STRICT_ASSIGN(double, x, 0x1p1023 * x);
return x;
}
if (x < -745.13321910194110842) {
/* underflow if x!=-inf */
STRICT_ASSIGN(double, x, 0x1p-1000 / -x * 0x1p-1000);
return x;
}
if (x > o_threshold)
return huge*huge; /* overflow */
if (x < u_threshold)
return twom1000*twom1000; /* underflow */
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
hi = x-ln2HI[xsb];
lo = ln2LO[xsb];
k = 1 - xsb - xsb;
} else {
k = (int)(invln2*x+halF[xsb]);
t = k;
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
lo = t*ln2LO[0];
}
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx < 0x3ff0a2b2) /* if |x| < 1.5 ln2 */
k = 1 - sign - sign; /* optimization */
else
k = (int)(invln2*x + half[sign]);
hi = x - k*ln2hi; /* k*ln2hi is exact here */
lo = k*ln2lo;
STRICT_ASSIGN(double, x, hi - lo);
} else if(hx < 0x3e300000) { /* |x| < 2**-28 */
/* raise inexact */
if (huge+x > 1.0)
return 1.0+x;
} else
} else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
k = 0;
hi = x;
lo = 0;
} else {
/* inexact if x!=0 */
FORCE_EVAL(0x1p1023 + x);
return 1 + x;
}
/* x is now in primary range */
t = x*x;
if (k >= -1021)
INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
else
INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
z = x*x;
c = x - z*(P1+z*(P2+z*(P3+z*(P4+z*P5))));
x = 1 + ((x*c/(2-c) - lo) + hi);
if (k == 0)
return 1.0 - ((x*c)/(c-2.0) - x);
y = 1.0-((lo-(x*c)/(2.0-c))-hi);
if (k < -1021)
return y*twopk*twom1000;
if (k == 1024)
return y*2.0*0x1p1023;
return y*twopk;
return x;
return scalbn(x, k);
}