Average Error: 60.9 → 0.5
Time: 25.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 - \left(x \cdot x\right) \cdot \frac{1}{2}\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(-1 - \left(x \cdot x\right) \cdot \frac{1}{2}\right) - x
double f(double x) {
        double r2156609 = 1.0;
        double r2156610 = x;
        double r2156611 = r2156609 - r2156610;
        double r2156612 = log(r2156611);
        double r2156613 = r2156609 + r2156610;
        double r2156614 = log(r2156613);
        double r2156615 = r2156612 / r2156614;
        return r2156615;
}
↓
double f(double x) {
        double r2156616 = -1.0;
        double r2156617 = x;
        double r2156618 = r2156617 * r2156617;
        double r2156619 = 0.5;
        double r2156620 = r2156618 * r2156619;
        double r2156621 = r2156616 - r2156620;
        double r2156622 = r2156621 - r2156617;
        return r2156622;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.9
Target0.3
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

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