Average Error: 60.9 → 0.4
Time: 39.5s
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 r8241286 = 1.0;
        double r8241287 = x;
        double r8241288 = r8241286 - r8241287;
        double r8241289 = log(r8241288);
        double r8241290 = r8241286 + r8241287;
        double r8241291 = log(r8241290);
        double r8241292 = r8241289 / r8241291;
        return r8241292;
}
↓
double f(double x) {
        double r8241293 = x;
        double r8241294 = r8241293 * r8241293;
        double r8241295 = -0.5;
        double r8241296 = r8241294 * r8241295;
        double r8241297 = -1.0;
        double r8241298 = r8241297 - r8241293;
        double r8241299 = r8241296 + r8241298;
        return r8241299;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.9
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 60.9

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

  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 2019107 
(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))))