Average Error: 61.0 → 0.4
Time: 32.4s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(x \cdot x\right) \cdot \frac{-1}{2} + \left(-1 - x\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(x \cdot x\right) \cdot \frac{-1}{2} + \left(-1 - x\right)
double f(double x) {
        double r3198684 = 1.0;
        double r3198685 = x;
        double r3198686 = r3198684 - r3198685;
        double r3198687 = log(r3198686);
        double r3198688 = r3198684 + r3198685;
        double r3198689 = log(r3198688);
        double r3198690 = r3198687 / r3198689;
        return r3198690;
}
↓
double f(double x) {
        double r3198691 = x;
        double r3198692 = r3198691 * r3198691;
        double r3198693 = -0.5;
        double r3198694 = r3198692 * r3198693;
        double r3198695 = -1.0;
        double r3198696 = r3198695 - r3198691;
        double r3198697 = r3198694 + r3198696;
        return r3198697;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 
(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))))