Average Error: 61.4 → 0.4
Time: 21.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2} + \left(1 \cdot x + \log 1\right)}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{\left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{-1}{2} + \left(1 \cdot x + \log 1\right)}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}}
double f(double x) {
        double r3845516 = 1.0;
        double r3845517 = x;
        double r3845518 = r3845516 - r3845517;
        double r3845519 = log(r3845518);
        double r3845520 = r3845516 + r3845517;
        double r3845521 = log(r3845520);
        double r3845522 = r3845519 / r3845521;
        return r3845522;
}
↓
double f(double x) {
        double r3845523 = 1.0;
        double r3845524 = x;
        double r3845525 = 1.0;
        double r3845526 = r3845524 / r3845525;
        double r3845527 = r3845526 * r3845526;
        double r3845528 = -0.5;
        double r3845529 = r3845527 * r3845528;
        double r3845530 = r3845525 * r3845524;
        double r3845531 = log(r3845525);
        double r3845532 = r3845530 + r3845531;
        double r3845533 = r3845529 + r3845532;
        double r3845534 = 0.5;
        double r3845535 = r3845534 * r3845527;
        double r3845536 = r3845530 + r3845535;
        double r3845537 = r3845531 - r3845536;
        double r3845538 = r3845533 / r3845537;
        double r3845539 = r3845523 / r3845538;
        return r3845539;
}

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 clear-num0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))))