Average Error: 61.4 → 0.4
Time: 15.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\left(x \cdot 1 + \log 1\right) - \frac{\frac{1}{2} \cdot x}{1} \cdot \frac{x}{1}}{\left(\log 1 - x \cdot 1\right) - \frac{\frac{1}{2} \cdot x}{1} \cdot \frac{x}{1}}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{\left(x \cdot 1 + \log 1\right) - \frac{\frac{1}{2} \cdot x}{1} \cdot \frac{x}{1}}{\left(\log 1 - x \cdot 1\right) - \frac{\frac{1}{2} \cdot x}{1} \cdot \frac{x}{1}}}
double f(double x) {
        double r78918 = 1.0;
        double r78919 = x;
        double r78920 = r78918 - r78919;
        double r78921 = log(r78920);
        double r78922 = r78918 + r78919;
        double r78923 = log(r78922);
        double r78924 = r78921 / r78923;
        return r78924;
}
↓
double f(double x) {
        double r78925 = 1.0;
        double r78926 = x;
        double r78927 = 1.0;
        double r78928 = r78926 * r78927;
        double r78929 = log(r78927);
        double r78930 = r78928 + r78929;
        double r78931 = 0.5;
        double r78932 = r78931 * r78926;
        double r78933 = r78932 / r78927;
        double r78934 = r78926 / r78927;
        double r78935 = r78933 * r78934;
        double r78936 = r78930 - r78935;
        double r78937 = r78929 - r78928;
        double r78938 = r78937 - r78935;
        double r78939 = r78936 / r78938;
        double r78940 = r78925 / r78939;
        return r78940;
}

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

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied clear-num0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 
(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))))