v/vlib/math/log.v

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module math
2022-05-20 07:45:54 +02:00
// log_n returns log base b of x
pub fn log_n(x f64, b f64) f64 {
y := log(x)
z := log(b)
return y / z
}
// log10 returns the decimal logarithm of x.
// The special cases are the same as for log.
pub fn log10(x f64) f64 {
return log(x) * (1.0 / ln10)
}
// log2 returns the binary logarithm of x.
// The special cases are the same as for log.
pub fn log2(x f64) f64 {
frac, exp := frexp(x)
// Make sure exact powers of two give an exact answer.
// Don't depend on log(0.5)*(1/ln2)+exp being exactly exp-1.
if frac == 0.5 {
return f64(exp - 1)
}
return log(frac) * (1.0 / ln2) + f64(exp)
}
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// log1p returns log(1+x)
pub fn log1p(x f64) f64 {
y := 1.0 + x
z := y - 1.0
return log(y) - (z - x) / y // cancels errors with IEEE arithmetic
}
// log_b returns the binary exponent of x.
//
// special cases are:
// log_b(±inf) = +inf
// log_b(0) = -inf
// log_b(nan) = nan
pub fn log_b(x f64) f64 {
if x == 0 {
return inf(-1)
}
if is_inf(x, 0) {
return inf(1)
}
if is_nan(x) {
return x
}
return f64(ilog_b_(x))
}
// ilog_b returns the binary exponent of x as an integer.
//
// special cases are:
// ilog_b(±inf) = max_i32
// ilog_b(0) = min_i32
// ilog_b(nan) = max_i32
pub fn ilog_b(x f64) int {
if x == 0 {
return min_i32
}
if is_nan(x) {
return max_i32
}
if is_inf(x, 0) {
return max_i32
}
return ilog_b_(x)
}
// ilog_b returns the binary exponent of x. It assumes x is finite and
// non-zero.
fn ilog_b_(x_ f64) int {
x, exp := normalize(x_)
return int((f64_bits(x) >> shift) & mask) - bias + exp
}
// log returns the logarithm of x
//
// Method :
// 1. Argument Reduction: find k and f such that
// x = 2^k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// 2. Approximation of log(1+f).
// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
// = 2s + s*R
// We use a special Remez algorithm on [0,0.1716] to generate
// a polynomial of degree 14 to approximate R The maximum error
// of this polynomial approximation is bounded by 2**-58.45. In
// other words,
// 2 4 6 8 10 12 14
// R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
// (the values of Lg1 to Lg7 are listed in the program)
// and
// | 2 14 | -58.45
// | Lg1*s +...+Lg7*s - R(z) | <= 2
// | |
// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
// In order to guarantee error in log below 1ulp, we compute log
// by
// log(1+f) = f - s*(f - R) (if f is not too large)
// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
//
// 3. Finally, log(x) = k*ln2 + log(1+f).
// = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
// Here ln2 is split into two floating point number:
// ln2_hi + ln2_lo,
// where n*ln2_hi is always exact for |n| < 2000.
//
// Special cases:
// log(x) is NaN with signal if x < 0 (including -inf) ;
// log(+inf) is +inf; log(0) is -inf with signal;
// log(NaN) is that NaN with no signal.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
pub fn log(a f64) f64 {
ln2_hi := 6.93147180369123816490e-01 // 3fe62e42 fee00000
ln2_lo := 1.90821492927058770002e-10 // 3dea39ef 35793c76
l1 := 6.666666666666735130e-01 // 3FE55555 55555593
l2 := 3.999999999940941908e-01 // 3FD99999 9997FA04
l3 := 2.857142874366239149e-01 // 3FD24924 94229359
l4 := 2.222219843214978396e-01 // 3FCC71C5 1D8E78AF
l5 := 1.818357216161805012e-01 // 3FC74664 96CB03DE
l6 := 1.531383769920937332e-01 // 3FC39A09 D078C69F
l7 := 1.479819860511658591e-01 // 3FC2F112 DF3E5244
x := a
if is_nan(x) || is_inf(x, 1) {
return x
} else if x < 0 {
return nan()
} else if x == 0 {
return inf(-1)
}
mut f1, mut ki := frexp(x)
if f1 < sqrt2 / 2 {
f1 *= 2
ki--
}
f := f1 - 1
k := f64(ki)
// compute
s := f / (2 + f)
s2 := s * s
s4 := s2 * s2
t1 := s2 * (l1 + s4 * (l3 + s4 * (l5 + s4 * l7)))
t2 := s4 * (l2 + s4 * (l4 + s4 * l6))
r := t1 + t2
hfsq := 0.5 * f * f
return k * ln2_hi - ((hfsq - (s * (hfsq + r) + k * ln2_lo)) - f)
}