Average Error: 61.2 → 0.4
Time: 20.5s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - 1 \cdot x}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1} - \frac{\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - 1 \cdot x}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1} - \frac{\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1}
double f(double x) {
        double r3656845 = 1.0;
        double r3656846 = x;
        double r3656847 = r3656845 - r3656846;
        double r3656848 = log(r3656847);
        double r3656849 = r3656845 + r3656846;
        double r3656850 = log(r3656849);
        double r3656851 = r3656848 / r3656850;
        return r3656851;
}
↓
double f(double x) {
        double r3656852 = 1.0;
        double r3656853 = log(r3656852);
        double r3656854 = x;
        double r3656855 = r3656852 * r3656854;
        double r3656856 = r3656853 - r3656855;
        double r3656857 = r3656854 / r3656852;
        double r3656858 = r3656857 * r3656857;
        double r3656859 = 0.5;
        double r3656860 = r3656858 * r3656859;
        double r3656861 = r3656855 - r3656860;
        double r3656862 = r3656861 + r3656853;
        double r3656863 = r3656856 / r3656862;
        double r3656864 = r3656860 / r3656862;
        double r3656865 = r3656863 - r3656864;
        return r3656865;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
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 61.2

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

  2. Taylor expanded around 0 60.4

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

  3. Simplified60.4

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied div-sub0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 
(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))))