Average Error: 61.0 → 0.5
Time: 21.7s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(\frac{-1}{2} \cdot \left(x \cdot x\right) + -1\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(\frac{-1}{2} \cdot \left(x \cdot x\right) + -1\right) - x
double f(double x) {
        double r4040537 = 1.0;
        double r4040538 = x;
        double r4040539 = r4040537 - r4040538;
        double r4040540 = log(r4040539);
        double r4040541 = r4040537 + r4040538;
        double r4040542 = log(r4040541);
        double r4040543 = r4040540 / r4040542;
        return r4040543;
}
↓
double f(double x) {
        double r4040544 = -0.5;
        double r4040545 = x;
        double r4040546 = r4040545 * r4040545;
        double r4040547 = r4040544 * r4040546;
        double r4040548 = -1.0;
        double r4040549 = r4040547 + r4040548;
        double r4040550 = r4040549 - r4040545;
        return r4040550;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
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 61.0

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

  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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 
(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))))