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/*
* // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
use crate::common::f_fmla;
use crate::dekker::Dekker;
use crate::log2p1::{log_fast, log_p_1a};
use crate::log10p1_tables::{LOG10P1_EXACT_INT_S_TABLE, LOG10P1_EXACT_INT_TABLE};
const INVLOG10H: f64 = f64::from_bits(0x3fdbcb7b1526e50e);
const INVLOG10L: f64 = f64::from_bits(0x3c695355baaafad3);
/* deal with |x| < 2^-900, then log10p1(x) ~ x/log(10) */
fn log10p1_accurate_tiny(x: f64) -> f64 {
/* first scale x to avoid truncation of l in the underflow region */
let sx = x * f64::from_bits(0x4690000000000000);
let mut px = Dekker::f64_mult(sx, Dekker::new(INVLOG10L, INVLOG10H));
let res = px.to_f64() * f64::from_bits(0x3950000000000000); // expected result
px.lo += f_fmla(-res, f64::from_bits(0x4690000000000000), px.hi);
// the correction to apply to res is l*2^-106
/* For RNDN, we have underflow for |x| <= 0x1.26bb1bbb55515p-1021,
and for rounding away, for |x| < 0x1.26bb1bbb55515p-1021. */
f_fmla(px.lo, f64::from_bits(0x3950000000000000), res)
}
fn log10p1_accurate_small(x: f64) -> f64 {
/* the following is a degree-17 polynomial approximating log10p1(x) for
|x| <= 2^-5 with relative error < 2^-105.067, cf log10p1_accurate.sollya */
static P_ACC: [u64; 25] = [
0x3fdbcb7b1526e50e,
0x3c695355baaafad4,
0xbfcbcb7b1526e50e,
0xbc595355baaae078,
0x3fc287a7636f435f,
0xbc59c871838f83ac,
0xbfbbcb7b1526e50e,
0xbc495355e23285f2,
0x3fb63c62775250d8,
0x3c4442abd5831422,
0xbfb287a7636f435f,
0x3c49d116f225c4e4,
0x3fafc3fa615105c7,
0x3c24e1d7b4790510,
0xbfabcb7b1526e512,
0x3c49f884199ab0ce,
0x3fa8b4df2f3f0486,
0xbfa63c6277522391,
0x3fa436e526a79e5c,
0xbfa287a764c5a762,
0x3fa11ac1e784daec,
0xbf9fc3eedc920817,
0x3f9da5cac3522edb,
0xbf9be5ca1f9a97cd,
0x3f9a44b64ca06e9b,
];
/* for degree 11 or more, ulp(c[d]*x^d) < 2^-105.7*|log10p1(x)|
where c[d] is the degree-d coefficient of Pacc, thus we can compute
with a double only, and even with degree 10 (this does not increase
the number of exceptional cases) */
let mut h = f_fmla(f64::from_bits(P_ACC[24]), x, f64::from_bits(P_ACC[23])); // degree 16
for i in (10..=15).rev() {
h = f_fmla(h, x, f64::from_bits(P_ACC[(i + 7) as usize])); // degree i
}
// degree 9
let px = Dekker::from_exact_mult(x, h);
let hl = Dekker::from_exact_add(f64::from_bits(P_ACC[9 + 7]), px.hi);
h = hl.hi;
let mut l = px.lo + hl.lo;
for i in (1..=8).rev() {
let mut p = Dekker::f64_mult(x, Dekker::new(l, h));
l = p.lo;
p = Dekker::from_exact_add(f64::from_bits(P_ACC[(2 * i - 2) as usize]), p.hi);
h = p.hi;
l += p.lo + f64::from_bits(P_ACC[(2 * i - 1) as usize]);
}
let pz = Dekker::f64_mult(x, Dekker::new(l, h));
pz.to_f64()
}
#[inline]
fn log10p1_fast(x: f64, e: i32) -> (Dekker, f64) {
if e < -5
/* e <= -6 thus |x| < 2^-5 */
{
if e <= -968 {
/* then |x| might be as small as 2^-968, thus h=x/log(10) might in the
binade [2^-970,2^-969), with ulp(h) = 2^-1022, and if |l| < ulp(h),
then l.ulp() might be smaller than 2^-1074. We defer that case to
the accurate path. */
let ax = x.abs();
let result = if ax < f64::from_bits(0x07b0000000000000) {
log10p1_accurate_tiny(x)
} else {
log10p1_accurate_small(x)
};
return (Dekker::new(0.0, result), 0.0);
}
let mut p = log_p_1a(x);
let p_lo = p.lo;
p = Dekker::from_exact_add(x, p.hi);
p.lo += p_lo;
/* from analyze_x_plus_p1a(rel=true,Xmax=2^-5.) in the accompanying file
log1p.sage, the relative error is bounded by 2^-61.14 with respect to
h. We use the fact that we don't need the return value err to be
positive, since we add/subtract it in the rounding test.
We also get that the ratio |l/h| is bounded by 2^-50.96. */
/* now we multiply h+l by 1/log(2) */
p = Dekker::quick_mult(p, Dekker::new(INVLOG10L, INVLOG10H));
/* the d_mul() call decomposes into:
a_mul (h_out, l1, h, INVLOG10H)
l2 = __builtin_fma (h, INVLOG10L, l1)
l_out = __builtin_fma (l, INVLOG10H, l2)
we have |l1| <= ulp(h_out)
since |INVLOG10L/INVLOG10H| < 2^-55, then |h*INVLOG10L| <= 2^-55*|h_out|
and since |x| < 2^53*ulp(x): |h*INVLOG10L| <= ulp(h_out)/4
thus |l2| <= 5/4*ulp(h_out).
Now since |l/h| < 2^-50.96, |l*INVLOG10H| < 2^-50.96*|h*INVLOG10H|
< 2^-50.96*(1+2^-52)*|h_out| < 2^-50.95*|h_out| < 4.15*ulp(h_out),
thus |l_out| < o(4.15*ulp(h_out)+5/4*ulp(h_out)) < 5.5*ulp(h_out).
The rounding errors are bounded by ulp(l2)+ulp(l_out)
<= ulp(5/4*ulp(h_out)) + ulp(5.5*ulp(h_out))
<= 2^-52*(5/4*ulp(h_out)+5.5*ulp(h_out)) [using ulp(x) <= 2^-52*|x|]
<= 2^-49.2*ulp(h_out)
We also have to take into account the ignored term l*INVLOG10L:
|l*INVLOG10L| < 2^-50.96*|h|*2^-55.97*|INVLOG10H|
< 2^-106.93*(1+2^-52)*|h_out|
< 2^-106.92*|h_out|
< 2^-51.92*ulp(h_out) [using |x| < 2^53*ulp(x)]
and the approximation error in INVLOG10H+INVLOG10L:
|INVLOG10H + INVLOG10L - 1/log(10)| < 2^-109.84/log(10)
The total error of d_mul() is thus bounded by:
(2^-49.2+2^-51.92)*ulp(h_out) < 2^-48.99*ulp(h_out) < 2^-100.99*|h_out|,
using again ulp(x) <= 2^-52*|x|.
The relative error is thus bounded by
(1+2^-61.14)*(1+2^-100.99)*(1+2^-109.84)-1 < 2^-61.13 */
return (p, f64::from_bits(0x3c1d400000000000) * p.hi); /* 2^-61.13 < 0x1.d4p-62 */
}
/* (xh,xl) <- 1+x */
let zx = if x > 1.0 {
if x < f64::from_bits(0x7fefffffffffffff) {
Dekker::from_exact_add(x, 1.0)
} else {
// avoid spurious overflow for RNDU
Dekker::new(1.0, x)
}
} else {
Dekker::from_exact_add(1.0, x)
};
let mut v_u = zx.hi.to_bits();
let e = ((v_u >> 52) as i32).wrapping_sub(0x3ff);
v_u = (0x3ffu64 << 52) | (v_u & 0xfffffffffffff);
let mut p = log_fast(e, v_u);
/* log(xh+xl) = log(xh) + log(1+xl/xh) */
let c = if zx.hi <= f64::from_bits(0x7fd0000000000000) || zx.lo.abs() >= 4.0 {
zx.lo / zx.hi
} else {
0.
}; // avoid spurious underflow
/* Since |xl| < ulp(xh), we have |xl| < 2^-52 |xh|,
thus |c| < 2^-52, and since |log(1+x)-x| < x^2 for |x| < 0.5,
we have |log(1+c)-c)| < c^2 < 2^-104. */
p.lo += c;
/* Since |l_in| < 2^-18.69 (from the analysis of cr_log_fast, see file
../log/log.c), and |c| < 2^-52, we have |l| < 2^-18.68, thus the
rounding error in *l += c is bounded by ulp(2^-18.68) = 2^-71.
The total absolute error is thus bounded by:
0x1.b6p-69 + 2^-104 + 2^-71 < 2^-68.02. */
/* now multiply h+l by 1/log(2) */
p = Dekker::quick_mult(p, Dekker::new(INVLOG10L, INVLOG10H));
/* the d_mul() call decomposes into:
a_mul (h_out, l1, h, INVLOG10H)
l2 = __builtin_fma (h, INVLOG10L, l1)
l_out = __builtin_fma (l, INVLOG10H, l2)
We have three errors:
* the rounding error in l2 = __builtin_fma (h, INVLOG10L, l1)
* the rounding error in l_out = __builtin_fma (l, INVLOG10H, l2)
* the ignored term l * INVLOG10L
We have |h| < 745 thus |h*INVLOG10H| < 324 thus |h_out| <= 324
and |l1| <= ulp(h_out) <= 2^-44.
Then |h*INVLOG10L+l1| <= 745*INVLOG2L+2^-44 < 2^-43.6
thus |l2| < 2^-43.6*(1+2^-52) < 2^-43.5
and the first rounding error is bounded by ulp(2^-43.5) = 2^-96.
Now |l*INVLOG10H+l2| < 2^-18.68*INVLOG10H+2^-43.5 < 2^-19.8
thus |l_out| < 2^-19.8*(1+2^-52) < 2^-19.7
and the second rounding error is bounded by ulp(2^-19.7) = 2^-72.
The ignored term is bounded by |l*INVLOG10L| < 2^-18.68*INVLOG10L
< 2^-75.0.
Thus the absolute error from d_mul() is bounded by:
2^-96 + 2^-72 + 2^-75.0 < 2^-71.83.
Adding to the maximal absolute error of 2^-68.02 before d_mul(),
we get 2^-68.02 + 2^-71.83 < 2^-67.92.
*/
(p, f64::from_bits(0x3bb0a00000000000)) /* 2^-67.92 < 0x1.0ap-68 */
}
/// Computes log(x+1)
///
/// Max ULP 0.504
#[inline]
pub fn f_log10p1(x: f64) -> f64 {
let x_u = x.to_bits();
let e = (((x_u >> 52) & 0x7ff) as i32).wrapping_sub(0x3ff);
if e == 0x400 || x == 0. || x <= -1.0 {
/* case NaN/Inf, +/-0 or x <= -1 */
if e == 0x400 && x.to_bits() != 0xfffu64 << 52 {
/* NaN or + Inf*/
return x + x;
}
if x <= -1.0
/* we use the fact that NaN < -1 is false */
{
/* log2p(x<-1) is NaN, log2p(-1) is -Inf and raises DivByZero */
return if x < -1.0 {
f64::NAN
} else {
// x=-1
f64::NEG_INFINITY
};
}
return x + x; /* +/-0 */
}
/* check x=10^n-1 for 1 <= n <= 15, where log10p1(x) is exact,
and we shouldn't raise the inexact flag */
if 3 <= e && e <= 49 && x == f64::from_bits(LOG10P1_EXACT_INT_TABLE[e as usize]) {
return LOG10P1_EXACT_INT_S_TABLE[e as usize] as f64;
}
/* now x = m*2^e with 1 <= m < 2 (m = v.f) and -1074 <= e <= 1023 */
let (p, err) = log10p1_fast(x, e);
p.hi + (p.lo - err)
}