Average Error: 61.2 → 0.4
Time: 15.6s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-\left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(1 + x\right)\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
-\left(\left(x \cdot x\right) \cdot \frac{1}{2} + \left(1 + x\right)\right)
double f(double x) {
        double r916782 = 1.0;
        double r916783 = x;
        double r916784 = r916782 - r916783;
        double r916785 = log(r916784);
        double r916786 = r916782 + r916783;
        double r916787 = log(r916786);
        double r916788 = r916785 / r916787;
        return r916788;
}
↓
double f(double x) {
        double r916789 = x;
        double r916790 = r916789 * r916789;
        double r916791 = 0.5;
        double r916792 = r916790 * r916791;
        double r916793 = 1.0;
        double r916794 = r916793 + r916789;
        double r916795 = r916792 + r916794;
        double r916796 = -r916795;
        return r916796;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
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 61.2

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

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019154 
(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))))