Average Error: 61.5 → 0.4
Time: 8.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{1 \cdot x + \log 1}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{1 \cdot x + \log 1}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)} - \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}
double f(double x) {
        double r90781 = 1.0;
        double r90782 = x;
        double r90783 = r90781 - r90782;
        double r90784 = log(r90783);
        double r90785 = r90781 + r90782;
        double r90786 = log(r90785);
        double r90787 = r90784 / r90786;
        return r90787;
}
↓
double f(double x) {
        double r90788 = 1.0;
        double r90789 = 1.0;
        double r90790 = x;
        double r90791 = r90789 * r90790;
        double r90792 = log(r90789);
        double r90793 = r90791 + r90792;
        double r90794 = 0.5;
        double r90795 = 2.0;
        double r90796 = pow(r90790, r90795);
        double r90797 = pow(r90789, r90795);
        double r90798 = r90796 / r90797;
        double r90799 = r90794 * r90798;
        double r90800 = r90791 + r90799;
        double r90801 = r90792 - r90800;
        double r90802 = r90793 / r90801;
        double r90803 = r90799 / r90801;
        double r90804 = r90802 - r90803;
        double r90805 = r90788 / r90804;
        return r90805;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.5
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.5

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

  2. Taylor expanded around 0 60.6

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

  3. Taylor expanded around 0 0.4

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

  5. Applied clear-num0.4

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

  7. Applied div-sub0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))