Average Error: 61.3 → 0.4
Time: 36.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - 1 \cdot x}{\log 1 + \left(1 \cdot x + \frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1}\right)} - \frac{\frac{\frac{x}{\frac{1}{x}}}{1}}{\frac{\log 1 + \left(1 \cdot x + \frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1}\right)}{\frac{1}{2}}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - 1 \cdot x}{\log 1 + \left(1 \cdot x + \frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1}\right)} - \frac{\frac{\frac{x}{\frac{1}{x}}}{1}}{\frac{\log 1 + \left(1 \cdot x + \frac{\frac{x}{\frac{1}{x}} \cdot \frac{-1}{2}}{1}\right)}{\frac{1}{2}}}
double f(double x) {
        double r110959 = 1.0;
        double r110960 = x;
        double r110961 = r110959 - r110960;
        double r110962 = log(r110961);
        double r110963 = r110959 + r110960;
        double r110964 = log(r110963);
        double r110965 = r110962 / r110964;
        return r110965;
}
↓
double f(double x) {
        double r110966 = 1.0;
        double r110967 = log(r110966);
        double r110968 = x;
        double r110969 = r110966 * r110968;
        double r110970 = r110967 - r110969;
        double r110971 = r110966 / r110968;
        double r110972 = r110968 / r110971;
        double r110973 = -0.5;
        double r110974 = r110972 * r110973;
        double r110975 = r110974 / r110966;
        double r110976 = r110969 + r110975;
        double r110977 = r110967 + r110976;
        double r110978 = r110970 / r110977;
        double r110979 = r110972 / r110966;
        double r110980 = 0.5;
        double r110981 = r110977 / r110980;
        double r110982 = r110979 / r110981;
        double r110983 = r110978 - r110982;
        return r110983;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
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 61.3

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

  2. Simplified61.3

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

  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 div-sub0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(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))))