Average Error: 61.1 → 0.4
Time: 28.8s
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 r2791554 = 1.0;
        double r2791555 = x;
        double r2791556 = r2791554 - r2791555;
        double r2791557 = log(r2791556);
        double r2791558 = r2791554 + r2791555;
        double r2791559 = log(r2791558);
        double r2791560 = r2791557 / r2791559;
        return r2791560;
}
↓
double f(double x) {
        double r2791561 = -1.0;
        double r2791562 = 0.5;
        double r2791563 = x;
        double r2791564 = r2791562 * r2791563;
        double r2791565 = r2791564 * r2791563;
        double r2791566 = r2791561 - r2791565;
        double r2791567 = r2791566 - r2791563;
        return r2791567;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.1
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 61.1

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

  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 2019149 
(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))))