Average Error: 61.0 → 0.4
Time: 24.4s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(\frac{-1}{2} \cdot \left(x \cdot x\right) + -1\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(\frac{-1}{2} \cdot \left(x \cdot x\right) + -1\right) - x
double f(double x) {
        double r3465643 = 1.0;
        double r3465644 = x;
        double r3465645 = r3465643 - r3465644;
        double r3465646 = log(r3465645);
        double r3465647 = r3465643 + r3465644;
        double r3465648 = log(r3465647);
        double r3465649 = r3465646 / r3465648;
        return r3465649;
}
↓
double f(double x) {
        double r3465650 = -0.5;
        double r3465651 = x;
        double r3465652 = r3465651 * r3465651;
        double r3465653 = r3465650 * r3465652;
        double r3465654 = -1.0;
        double r3465655 = r3465653 + r3465654;
        double r3465656 = r3465655 - r3465651;
        return r3465656;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 
(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))))