qlog (example 3.10)

Average Error: 61.2 → 0.0

Time: 33.2s

Precision: 64

\[-1 \lt x \land x \lt 1\]

Math

\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

↓

\[\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\]

C

double f(double x) {
        double r9008343 = 1.0;
        double r9008344 = x;
        double r9008345 = r9008343 - r9008344;
        double r9008346 = log(r9008345);
        double r9008347 = r9008343 + r9008344;
        double r9008348 = log(r9008347);
        double r9008349 = r9008346 / r9008348;
        return r9008349;
}

↓

double f(double x) {
        double r9008350 = x;
        double r9008351 = -r9008350;
        double r9008352 = log1p(r9008351);
        double r9008353 = log1p(r9008350);
        double r9008354 = r9008352 / r9008353;
        return r9008354;
}

Error

Bits error versus x

Target

Original	61.2
Target	0.2
Herbie	0.0

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

1. Initial program 61.2

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

2. Simplified 60.2

   \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\mathsf{log1p}\left(x\right)}}\]
3. Using strategy rm

4. Applied log1p-expm1-u 60.2

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\log \left(1 - x\right)\right)\right)\right)\right)}}{\mathsf{log1p}\left(x\right)}\]

5. Simplified 0.0

   \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\left(-x\right)}\right)}{\mathsf{log1p}\left(x\right)}\]

6. Final simplification 0.0

   \[\leadsto \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))