Average Error: 61.0 → 0.4
Time: 24.9s
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 r4517179 = 1.0;
        double r4517180 = x;
        double r4517181 = r4517179 - r4517180;
        double r4517182 = log(r4517181);
        double r4517183 = r4517179 + r4517180;
        double r4517184 = log(r4517183);
        double r4517185 = r4517182 / r4517184;
        return r4517185;
}
↓
double f(double x) {
        double r4517186 = -0.5;
        double r4517187 = x;
        double r4517188 = r4517187 * r4517187;
        double r4517189 = r4517186 * r4517188;
        double r4517190 = -1.0;
        double r4517191 = r4517189 + r4517190;
        double r4517192 = r4517191 - r4517187;
        return r4517192;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
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 61.0

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

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 
(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))))