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-rw-r--r--src/math/exp.c121
1 files changed, 49 insertions, 72 deletions
diff --git a/src/math/exp.c b/src/math/exp.c
index 29bf9609..5c0edee4 100644
--- a/src/math/exp.c
+++ b/src/math/exp.c
@@ -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);
}