Average Error: 61.2 → 0.3
Time: 20.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-\left(\frac{1}{2} \cdot \left(x \cdot x\right) + x\right)\right) + -1\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(-\left(\frac{1}{2} \cdot \left(x \cdot x\right) + x\right)\right) + -1
double f(double x) {
        double r1252794 = 1.0;
        double r1252795 = x;
        double r1252796 = r1252794 - r1252795;
        double r1252797 = log(r1252796);
        double r1252798 = r1252794 + r1252795;
        double r1252799 = log(r1252798);
        double r1252800 = r1252797 / r1252799;
        return r1252800;
}
↓
double f(double x) {
        double r1252801 = 0.5;
        double r1252802 = x;
        double r1252803 = r1252802 * r1252802;
        double r1252804 = r1252801 * r1252803;
        double r1252805 = r1252804 + r1252802;
        double r1252806 = -r1252805;
        double r1252807 = -1.0;
        double r1252808 = r1252806 + r1252807;
        return r1252808;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
Target0.2
Herbie0.3
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.2

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 
(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))))