qlog (example 3.10)

Average Error: 60.9 → 0.0

Time: 17.4s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r5357226 = 1.0;
        double r5357227 = x;
        double r5357228 = r5357226 - r5357227;
        double r5357229 = log(r5357228);
        double r5357230 = r5357226 + r5357227;
        double r5357231 = log(r5357230);
        double r5357232 = r5357229 / r5357231;
        return r5357232;
}

↓

double f(double x) {
        double r5357233 = x;
        double r5357234 = -r5357233;
        double r5357235 = log1p(r5357234);
        double r5357236 = log1p(r5357233);
        double r5357237 = r5357235 / r5357236;
        return r5357237;
}

Error

Bits error versus x

Target

Original	60.9
Target	0.3
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 60.9

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

2. Simplified 60.0

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

4. Applied sub-neg 60.0

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

5. Applied log1p-def 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019120 +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))))