Average Error: 61.4 → 0.5
Time: 39.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot x + \frac{x}{1} \cdot \left(\frac{1}{2} \cdot \frac{x}{1}\right)\right)}{1 \cdot x + \left(\log 1 - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(1 \cdot x + \frac{x}{1} \cdot \left(\frac{1}{2} \cdot \frac{x}{1}\right)\right)}{1 \cdot x + \left(\log 1 - \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}
double f(double x) {
        double r4364810 = 1.0;
        double r4364811 = x;
        double r4364812 = r4364810 - r4364811;
        double r4364813 = log(r4364812);
        double r4364814 = r4364810 + r4364811;
        double r4364815 = log(r4364814);
        double r4364816 = r4364813 / r4364815;
        return r4364816;
}
↓
double f(double x) {
        double r4364817 = 1.0;
        double r4364818 = log(r4364817);
        double r4364819 = x;
        double r4364820 = r4364817 * r4364819;
        double r4364821 = r4364819 / r4364817;
        double r4364822 = 0.5;
        double r4364823 = r4364822 * r4364821;
        double r4364824 = r4364821 * r4364823;
        double r4364825 = r4364820 + r4364824;
        double r4364826 = r4364818 - r4364825;
        double r4364827 = r4364821 * r4364821;
        double r4364828 = r4364822 * r4364827;
        double r4364829 = r4364818 - r4364828;
        double r4364830 = r4364820 + r4364829;
        double r4364831 = r4364826 / r4364830;
        return r4364831;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

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

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 
(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))))