qlog (example 3.10)

Average Error: 60.8 → 0.0

Time: 24.4s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2176639 = 1.0;
        double r2176640 = x;
        double r2176641 = r2176639 - r2176640;
        double r2176642 = log(r2176641);
        double r2176643 = r2176639 + r2176640;
        double r2176644 = log(r2176643);
        double r2176645 = r2176642 / r2176644;
        return r2176645;
}

↓

double f(double x) {
        double r2176646 = x;
        double r2176647 = -r2176646;
        double r2176648 = log1p(r2176647);
        double r2176649 = log1p(r2176646);
        double r2176650 = r2176648 / r2176649;
        return r2176650;
}

Error

Bits error versus x

Target

Original	60.8
Target	0.4
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 60.8

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

2. Simplified 59.9

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

4. Applied log1p-expm1-u 59.9

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

5. Simplified 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019138 +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))))