Average Error: 60.9 → 0.0
Time: 18.6s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\mathsf{log1p}\left(-x \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\mathsf{log1p}\left(-x \cdot x\right)}{\mathsf{log1p}\left(x\right)} - \frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(x\right)}
double f(double x) {
        double r1456078 = 1.0;
        double r1456079 = x;
        double r1456080 = r1456078 - r1456079;
        double r1456081 = log(r1456080);
        double r1456082 = r1456078 + r1456079;
        double r1456083 = log(r1456082);
        double r1456084 = r1456081 / r1456083;
        return r1456084;
}
↓
double f(double x) {
        double r1456085 = x;
        double r1456086 = r1456085 * r1456085;
        double r1456087 = -r1456086;
        double r1456088 = log1p(r1456087);
        double r1456089 = log1p(r1456085);
        double r1456090 = r1456088 / r1456089;
        double r1456091 = r1456089 / r1456089;
        double r1456092 = r1456090 - r1456091;
        return r1456092;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.9
Target0.4
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.9

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

  2. Simplified59.9

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

  4. Applied flip--59.9

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

  5. Applied log-div59.9

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

  6. Applied div-sub59.9

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

  7. Simplified59.9

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

  8. Simplified0.6

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

  10. Applied sub-neg0.6

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

  11. Applied log1p-def0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +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))))