exp.c File Reference

#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "exp_data.h"

Include dependency graph for exp.c:

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Macros
#define	N   (1 << EXP_TABLE_BITS)
#define	InvLn2N   __exp_data.invln2N
#define	NegLn2hiN   __exp_data.negln2hiN
#define	NegLn2loN   __exp_data.negln2loN
#define	Shift   __exp_data.shift
#define	T   __exp_data.tab
#define	C2   __exp_data.poly[5 - EXP_POLY_ORDER]
#define	C3   __exp_data.poly[6 - EXP_POLY_ORDER]
#define	C4   __exp_data.poly[7 - EXP_POLY_ORDER]
#define	C5   __exp_data.poly[8 - EXP_POLY_ORDER]

Functions
static double	specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
static uint32_t	top12 (double x)
double	exp (double x)

Macro Definition Documentation

C2

#define C2   __exp_data.poly[5 - EXP_POLY_ORDER]

Definition at line 33 of file exp.c.

C3

#define C3   __exp_data.poly[6 - EXP_POLY_ORDER]

Definition at line 34 of file exp.c.

C4

#define C4   __exp_data.poly[7 - EXP_POLY_ORDER]

Definition at line 35 of file exp.c.

C5

#define C5   __exp_data.poly[8 - EXP_POLY_ORDER]

Definition at line 36 of file exp.c.

InvLn2N

#define InvLn2N   __exp_data.invln2N

Definition at line 28 of file exp.c.

N

#define N   (1 << EXP_TABLE_BITS)

Definition at line 27 of file exp.c.

NegLn2hiN

#define NegLn2hiN   __exp_data.negln2hiN

Definition at line 29 of file exp.c.

NegLn2loN

#define NegLn2loN   __exp_data.negln2loN

Definition at line 30 of file exp.c.

Shift

#define Shift   __exp_data.shift

Definition at line 31 of file exp.c.

T

#define T   __exp_data.tab

Definition at line 32 of file exp.c.

Function Documentation

exp()

double exp

(

double

x

)

Definition at line 86 of file exp.c.

{
    uint32_t abstop;
    uint64_t ki, idx, top, sbits;
    double_t kd, z, r, r2, scale, tail, tmp;
 
    abstop = top12(x) & 0x7ff;
    if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
        if (abstop - top12(0x1p-54) >= 0x80000000)
            /* Avoid spurious underflow for tiny x.  */
            /* Note: 0 is common input.  */
            return WANT_ROUNDING ? 1.0 + x : 1.0;
        if (abstop >= top12(1024.0)) {
            if (asuint64(x) == asuint64(-INFINITY))
                return 0.0;
            if (abstop >= top12(INFINITY))
                return 1.0 + x;
            if (asuint64(x) >> 63)
                return __math_uflow(0);
            else
                return __math_oflow(0);
        }
        /* Large x is special cased below.  */
        abstop = 0;
    }
 
    /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
    /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
    z = InvLn2N * x;
#if TOINT_INTRINSICS
    kd = roundtoint(z);
    ki = converttoint(z);
#elif EXP_USE_TOINT_NARROW
    /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
    kd = eval_as_double(z + Shift);
    ki = asuint64(kd) >> 16;
    kd = (double_t)(int32_t)ki;
#else
    /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
    kd = eval_as_double(z + Shift);
    ki = asuint64(kd);
    kd -= Shift;
#endif
    r = x + kd * NegLn2hiN + kd * NegLn2loN;
    /* 2^(k/N) ~= scale * (1 + tail).  */
    idx = 2 * (ki % N);
    top = ki << (52 - EXP_TABLE_BITS);
    tail = asdouble(T[idx]);
    /* This is only a valid scale when -1023*N < k < 1024*N.  */
    sbits = T[idx + 1] + top;
    /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
    /* Evaluation is optimized assuming superscalar pipelined execution.  */
    r2 = r * r;
    /* Without fma the worst case error is 0.25/N ulp larger.  */
    /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
    tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
    if (predict_false(abstop == 0))
        return specialcase(tmp, sbits, ki);
    scale = asdouble(sbits);
    /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
       is no spurious underflow here even without fma.  */
    return eval_as_double(scale + scale * tmp);
}

specialcase()

static double specialcase ( double_t  tmp, uint64_t  sbits, uint64_t  ki  )

inlinestatic

Definition at line 45 of file exp.c.

{
    double_t scale, y;
 
    if ((ki & 0x80000000) == 0) {
        /* k > 0, the exponent of scale might have overflowed by <= 460.  */
        sbits -= 1009ull << 52;
        scale = asdouble(sbits);
        y = 0x1p1009 * (scale + scale * tmp);
        return eval_as_double(y);
    }
    /* k < 0, need special care in the subnormal range.  */
    sbits += 1022ull << 52;
    scale = asdouble(sbits);
    y = scale + scale * tmp;
    if (y < 1.0) {
        /* Round y to the right precision before scaling it into the subnormal
         range to avoid double rounding that can cause 0.5+E/2 ulp error where
         E is the worst-case ulp error outside the subnormal range.  So this
         is only useful if the goal is better than 1 ulp worst-case error.  */
        double_t hi, lo;
        lo = scale - y + scale * tmp;
        hi = 1.0 + y;
        lo = 1.0 - hi + y + lo;
        y = eval_as_double(hi + lo) - 1.0;
        /* Avoid -0.0 with downward rounding.  */
        if (WANT_ROUNDING && y == 0.0)
            y = 0.0;
        /* The underflow exception needs to be signaled explicitly.  */
        fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
    }
    y = 0x1p-1022 * y;
    return eval_as_double(y);
}

Referenced by exp().

top12()

static uint32_t top12 ( double  x )

inlinestatic

Definition at line 81 of file exp.c.

{
    return asuint64(x) >> 52;
}

Referenced by exp().