Average Error: 61.4 → 0.4
Time: 20.1s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\frac{1}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}{\frac{1}{\log 1 - \left(x \cdot 1 + \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\frac{1}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}{\frac{1}{\log 1 - \left(x \cdot 1 + \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}}
double f(double x) {
        double r4050434 = 1.0;
        double r4050435 = x;
        double r4050436 = r4050434 - r4050435;
        double r4050437 = log(r4050436);
        double r4050438 = r4050434 + r4050435;
        double r4050439 = log(r4050438);
        double r4050440 = r4050437 / r4050439;
        return r4050440;
}
↓
double f(double x) {
        double r4050441 = 1.0;
        double r4050442 = 1.0;
        double r4050443 = log(r4050442);
        double r4050444 = x;
        double r4050445 = r4050444 * r4050442;
        double r4050446 = r4050443 + r4050445;
        double r4050447 = 0.5;
        double r4050448 = r4050444 / r4050442;
        double r4050449 = r4050448 * r4050448;
        double r4050450 = r4050447 * r4050449;
        double r4050451 = r4050446 - r4050450;
        double r4050452 = r4050441 / r4050451;
        double r4050453 = r4050445 + r4050450;
        double r4050454 = r4050443 - r4050453;
        double r4050455 = r4050441 / r4050454;
        double r4050456 = r4050452 / r4050455;
        return r4050456;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.4

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

  2. Taylor expanded around 0 60.5

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(\log 1 + 1 \cdot x\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

  3. Simplified60.5

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}\]

  4. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}\]

  5. Simplified0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \left(\frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + x \cdot 1\right)}}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}\]
  6. Using strategy rm

  7. Applied clear-num0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}{\log 1 - \left(\frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + x \cdot 1\right)}}}\]
  8. Using strategy rm

  9. Applied div-inv0.6

    \[\leadsto \frac{1}{\color{blue}{\left(\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right) \cdot \frac{1}{\log 1 - \left(\frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + x \cdot 1\right)}}}\]

  10. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}{\frac{1}{\log 1 - \left(\frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right) + x \cdot 1\right)}}}\]

  11. Final simplification0.4

    \[\leadsto \frac{\frac{1}{\left(\log 1 + x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}{\frac{1}{\log 1 - \left(x \cdot 1 + \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))