Average Error: 60.8 → 0.4
Time: 12.2s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 - \left(\frac{1}{2} \cdot x\right) \cdot x\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(-1 - \left(\frac{1}{2} \cdot x\right) \cdot x\right) - x
double f(double x) {
        double r1975591 = 1.0;
        double r1975592 = x;
        double r1975593 = r1975591 - r1975592;
        double r1975594 = log(r1975593);
        double r1975595 = r1975591 + r1975592;
        double r1975596 = log(r1975595);
        double r1975597 = r1975594 / r1975596;
        return r1975597;
}
↓
double f(double x) {
        double r1975598 = -1.0;
        double r1975599 = 0.5;
        double r1975600 = x;
        double r1975601 = r1975599 * r1975600;
        double r1975602 = r1975601 * r1975600;
        double r1975603 = r1975598 - r1975602;
        double r1975604 = r1975603 - r1975600;
        return r1975604;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.8
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 60.8

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

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(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))))