Average Error: 60.8 → 0.5
Time: 35.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(\left(x \cdot x\right) \cdot \frac{-1}{2} - x\right) + -1\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(\left(x \cdot x\right) \cdot \frac{-1}{2} - x\right) + -1
double f(double x) {
        double r5547797 = 1.0;
        double r5547798 = x;
        double r5547799 = r5547797 - r5547798;
        double r5547800 = log(r5547799);
        double r5547801 = r5547797 + r5547798;
        double r5547802 = log(r5547801);
        double r5547803 = r5547800 / r5547802;
        return r5547803;
}
↓
double f(double x) {
        double r5547804 = x;
        double r5547805 = r5547804 * r5547804;
        double r5547806 = -0.5;
        double r5547807 = r5547805 * r5547806;
        double r5547808 = r5547807 - r5547804;
        double r5547809 = -1.0;
        double r5547810 = r5547808 + r5547809;
        return r5547810;
}

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. Using strategy rm

  5. Applied sub-neg0.5

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

  6. Applied associate-+l+0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019125 
(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))))