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v/vlib/math/exp.v
2021-10-08 17:44:55 +03:00

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module math
import math.internal
const (
f64_max_exp = f64(1024)
f64_min_exp = f64(-1021)
threshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
ln2_x56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
ln2_halfx3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
ln2_half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
ln2hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
ln2lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
inv_ln2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
// scaled coefficients related to expm1
expm1_q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
expm1_q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
expm1_q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
expm1_q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
expm1_q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
)
// exp returns e**x, the base-e exponential of x.
//
// special cases are:
// exp(+inf) = +inf
// exp(nan) = nan
// Very large values overflow to 0 or +inf.
// Very small values underflow to 1.
pub fn exp(x f64) f64 {
log2e := 1.44269504088896338700e+00
overflow := 7.09782712893383973096e+02
underflow := -7.45133219101941108420e+02
near_zero := 1.0 / (1 << 28) // 2**-28
// special cases
if is_nan(x) || is_inf(x, 1) {
return x
}
if is_inf(x, -1) {
return 0.0
}
if x > overflow {
return inf(1)
}
if x < underflow {
return 0.0
}
if -near_zero < x && x < near_zero {
return 1.0 + x
}
// reduce; computed as r = hi - lo for extra precision.
mut k := 0
if x < 0 {
k = int(log2e * x - 0.5)
}
if x > 0 {
k = int(log2e * x + 0.5)
}
hi := x - f64(k) * math.ln2hi
lo := f64(k) * math.ln2lo
// compute
return expmulti(hi, lo, k)
}
// exp2 returns 2**x, the base-2 exponential of x.
//
// special cases are the same as exp.
pub fn exp2(x f64) f64 {
overflow := 1.0239999999999999e+03
underflow := -1.0740e+03
if is_nan(x) || is_inf(x, 1) {
return x
}
if is_inf(x, -1) {
return 0
}
if x > overflow {
return inf(1)
}
if x < underflow {
return 0
}
// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
// computed as r = hi - lo for extra precision.
mut k := 0
if x > 0 {
k = int(x + 0.5)
}
if x < 0 {
k = int(x - 0.5)
}
mut t := x - f64(k)
hi := t * math.ln2hi
lo := -t * math.ln2lo
// compute
return expmulti(hi, lo, k)
}
pub fn ldexp(frac f64, exp int) f64 {
return scalbn(frac, exp)
}
// frexp breaks f into a normalized fraction
// and an integral power of two.
// It returns frac and exp satisfying f == frac × 2**exp,
// with the absolute value of frac in the interval [½, 1).
//
// special cases are:
// frexp(±0) = ±0, 0
// frexp(±inf) = ±inf, 0
// frexp(nan) = nan, 0
// pub fn frexp(f f64) (f64, int) {
// mut y := f64_bits(x)
// ee := int((y >> 52) & 0x7ff)
// // special cases
// if ee == 0 {
// if x != 0.0 {
// x1p64 := f64_from_bits(0x43f0000000000000)
// z,e_ := frexp(x * x1p64)
// return z,e_ - 64
// }
// return x,0
// } else if ee == 0x7ff {
// return x,0
// }
// e_ := ee - 0x3fe
// y &= 0x800fffffffffffff
// y |= 0x3fe0000000000000
// return f64_from_bits(y),e_
pub fn frexp(x f64) (f64, int) {
mut y := f64_bits(x)
ee := int((y >> 52) & 0x7ff)
if ee == 0 {
if x != 0.0 {
x1p64 := f64_from_bits(u64(0x43f0000000000000))
z, e_ := frexp(x * x1p64)
return z, e_ - 64
}
return x, 0
} else if ee == 0x7ff {
return x, 0
}
e_ := ee - 0x3fe
y &= u64(0x800fffffffffffff)
y |= u64(0x3fe0000000000000)
return f64_from_bits(y), e_
}
// special cases are:
// expm1(+inf) = +inf
// expm1(-inf) = -1
// expm1(nan) = nan
pub fn expm1(x f64) f64 {
if is_inf(x, 1) || is_nan(x) {
return x
}
if is_inf(x, -1) {
return f64(-1)
}
// FIXME: this should be improved
if abs(x) < ln2 { // Compute the taylor series S = x + (1/2!) x^2 + (1/3!) x^3 + ...
mut i := 1.0
mut sum := x
mut term := x / 1.0
i++
term *= x / f64(i)
sum += term
for abs(term) > abs(sum) * internal.f64_epsilon {
i++
term *= x / f64(i)
sum += term
}
return sum
} else {
return exp(x) - 1
}
}
// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
fn expmulti(hi f64, lo f64, k int) f64 {
exp_p1 := 1.66666666666666657415e-01 // 0x3FC55555; 0x55555555
exp_p2 := -2.77777777770155933842e-03 // 0xBF66C16C; 0x16BEBD93
exp_p3 := 6.61375632143793436117e-05 // 0x3F11566A; 0xAF25DE2C
exp_p4 := -1.65339022054652515390e-06 // 0xBEBBBD41; 0xC5D26BF1
exp_p5 := 4.13813679705723846039e-08 // 0x3E663769; 0x72BEA4D0
r := hi - lo
t := r * r
c := r - t * (exp_p1 + t * (exp_p2 + t * (exp_p3 + t * (exp_p4 + t * exp_p5))))
y := 1 - ((lo - (r * c) / (2 - c)) - hi)
// TODO(rsc): make sure ldexp can handle boundary k
return ldexp(y, k)
}