Average Error: 61.1 → 0.0
Time: 31.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \left(\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\sqrt[3]{\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \left(\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\right)}
double f(double x) {
        double r6621241 = 1.0;
        double r6621242 = x;
        double r6621243 = r6621241 - r6621242;
        double r6621244 = log(r6621243);
        double r6621245 = r6621241 + r6621242;
        double r6621246 = log(r6621245);
        double r6621247 = r6621244 / r6621246;
        return r6621247;
}
↓
double f(double x) {
        double r6621248 = x;
        double r6621249 = -r6621248;
        double r6621250 = log1p(r6621249);
        double r6621251 = log1p(r6621248);
        double r6621252 = r6621250 / r6621251;
        double r6621253 = r6621252 * r6621252;
        double r6621254 = r6621252 * r6621253;
        double r6621255 = cbrt(r6621254);
        return r6621255;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.1
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 61.1

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

  2. Simplified60.1

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

  4. Applied sub-neg60.1

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

  5. Applied log1p-def0.0

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

  7. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\right) \cdot \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}}}\]

  8. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \left(\frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)} \cdot \frac{\mathsf{log1p}\left(\left(-x\right)\right)}{\mathsf{log1p}\left(x\right)}\right)}\]

Reproduce

herbie shell --seed 2019124 +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))))