Average Error: 61.1 → 0.5
Time: 42.8s
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 r8432424 = 1.0;
        double r8432425 = x;
        double r8432426 = r8432424 - r8432425;
        double r8432427 = log(r8432426);
        double r8432428 = r8432424 + r8432425;
        double r8432429 = log(r8432428);
        double r8432430 = r8432427 / r8432429;
        return r8432430;
}
↓
double f(double x) {
        double r8432431 = x;
        double r8432432 = r8432431 * r8432431;
        double r8432433 = -0.5;
        double r8432434 = r8432432 * r8432433;
        double r8432435 = -1.0;
        double r8432436 = r8432435 - r8432431;
        double r8432437 = r8432434 + r8432436;
        return r8432437;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.1
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 61.1

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

  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 2019124 
(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))))