Average Error: 61.1 → 0.4
Time: 31.8s
Precision: 64
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(x \cdot x\right) \cdot \frac{-1}{2} + \left(-1 - x\right)\]
double f(double x) {
        double r3169020 = 1.0;
        double r3169021 = x;
        double r3169022 = r3169020 - r3169021;
        double r3169023 = log(r3169022);
        double r3169024 = r3169020 + r3169021;
        double r3169025 = log(r3169024);
        double r3169026 = r3169023 / r3169025;
        return r3169026;
}
↓
double f(double x) {
        double r3169027 = x;
        double r3169028 = r3169027 * r3169027;
        double r3169029 = -0.5;
        double r3169030 = r3169028 * r3169029;
        double r3169031 = -1.0;
        double r3169032 = r3169031 - r3169027;
        double r3169033 = r3169030 + r3169032;
        return r3169033;
}


\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(x \cdot x\right) \cdot \frac{-1}{2} + \left(-1 - x\right)

Error

Bits error versus x

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019102 
(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))))