Average Error: 61.4 → 0.4
Time: 17.8s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\log 1 + \left(\frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1} + x \cdot 1\right)}{\left(\log 1 - x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{\log 1 + \left(\frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1} + x \cdot 1\right)}{\left(\log 1 - x \cdot 1\right) - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)}}
double f(double x) {
        double r74947 = 1.0;
        double r74948 = x;
        double r74949 = r74947 - r74948;
        double r74950 = log(r74949);
        double r74951 = r74947 + r74948;
        double r74952 = log(r74951);
        double r74953 = r74950 / r74952;
        return r74953;
}
↓
double f(double x) {
        double r74954 = 1.0;
        double r74955 = 1.0;
        double r74956 = log(r74955);
        double r74957 = x;
        double r74958 = r74955 / r74957;
        double r74959 = r74957 / r74958;
        double r74960 = -0.5;
        double r74961 = r74959 * r74960;
        double r74962 = r74961 / r74955;
        double r74963 = r74957 * r74955;
        double r74964 = r74962 + r74963;
        double r74965 = r74956 + r74964;
        double r74966 = r74956 - r74963;
        double r74967 = 0.5;
        double r74968 = r74957 / r74955;
        double r74969 = r74968 * r74968;
        double r74970 = r74967 * r74969;
        double r74971 = r74966 - r74970;
        double r74972 = r74965 / r74971;
        double r74973 = r74954 / r74972;
        return r74973;
}

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. Simplified61.4

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

  3. Taylor expanded around 0 60.5

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

  4. Simplified60.5

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

  5. Taylor expanded around 0 0.4

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

  6. Simplified0.4

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

  8. Applied clear-num0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))))