Average Error: 61.2 → 0.4
Time: 20.9s
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 r1983421 = 1.0;
        double r1983422 = x;
        double r1983423 = r1983421 - r1983422;
        double r1983424 = log(r1983423);
        double r1983425 = r1983421 + r1983422;
        double r1983426 = log(r1983425);
        double r1983427 = r1983424 / r1983426;
        return r1983427;
}
↓
double f(double x) {
        double r1983428 = -0.5;
        double r1983429 = x;
        double r1983430 = r1983429 * r1983429;
        double r1983431 = r1983428 * r1983430;
        double r1983432 = -1.0;
        double r1983433 = r1983431 + r1983432;
        double r1983434 = r1983433 - r1983429;
        return r1983434;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.2

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019137 
(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))))