Average Error: 60.8 → 0.5
Time: 15.9s
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 r3000084 = 1.0;
        double r3000085 = x;
        double r3000086 = r3000084 - r3000085;
        double r3000087 = log(r3000086);
        double r3000088 = r3000084 + r3000085;
        double r3000089 = log(r3000088);
        double r3000090 = r3000087 / r3000089;
        return r3000090;
}
↓
double f(double x) {
        double r3000091 = x;
        double r3000092 = r3000091 * r3000091;
        double r3000093 = -0.5;
        double r3000094 = r3000092 * r3000093;
        double r3000095 = -1.0;
        double r3000096 = r3000095 - r3000091;
        double r3000097 = r3000094 + r3000096;
        return r3000097;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.8
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.8

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019121 
(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))))