Average Error: 61.4 → 0.4
Time: 25.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right)}{\left(1 \cdot x + \log 1\right) + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right)}{\left(1 \cdot x + \log 1\right) + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2}}}\right)
double f(double x) {
        double r4242590 = 1.0;
        double r4242591 = x;
        double r4242592 = r4242590 - r4242591;
        double r4242593 = log(r4242592);
        double r4242594 = r4242590 + r4242591;
        double r4242595 = log(r4242594);
        double r4242596 = r4242593 / r4242595;
        return r4242596;
}
↓
double f(double x) {
        double r4242597 = 1.0;
        double r4242598 = log(r4242597);
        double r4242599 = x;
        double r4242600 = r4242597 * r4242599;
        double r4242601 = r4242599 / r4242597;
        double r4242602 = r4242601 * r4242601;
        double r4242603 = 0.5;
        double r4242604 = r4242602 * r4242603;
        double r4242605 = r4242600 + r4242604;
        double r4242606 = r4242598 - r4242605;
        double r4242607 = r4242600 + r4242598;
        double r4242608 = -0.5;
        double r4242609 = r4242602 * r4242608;
        double r4242610 = r4242607 + r4242609;
        double r4242611 = r4242606 / r4242610;
        double r4242612 = exp(r4242611);
        double r4242613 = log(r4242612);
        return r4242613;
}

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. 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

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied add-log-exp0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(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))))