Average Error: 61.0 → 0.4
Time: 15.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 + \left(x \cdot x\right) \cdot \frac{-1}{2}\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(-1 + \left(x \cdot x\right) \cdot \frac{-1}{2}\right) - x
double f(double x) {
        double r3447467 = 1.0;
        double r3447468 = x;
        double r3447469 = r3447467 - r3447468;
        double r3447470 = log(r3447469);
        double r3447471 = r3447467 + r3447468;
        double r3447472 = log(r3447471);
        double r3447473 = r3447470 / r3447472;
        return r3447473;
}
↓
double f(double x) {
        double r3447474 = -1.0;
        double r3447475 = x;
        double r3447476 = r3447475 * r3447475;
        double r3447477 = -0.5;
        double r3447478 = r3447476 * r3447477;
        double r3447479 = r3447474 + r3447478;
        double r3447480 = r3447479 - r3447475;
        return r3447480;
}

Error

Bits error versus x

Try it out

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(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))))