Average Error: 60.8 → 0.5
Time: 17.3s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 + \left(x \cdot x\right) \cdot \frac{-1}{2}\right) - x\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\left(-1 + \left(x \cdot x\right) \cdot \frac{-1}{2}\right) - x
double f(double x) {
        double r2359042 = 1.0;
        double r2359043 = x;
        double r2359044 = r2359042 - r2359043;
        double r2359045 = log(r2359044);
        double r2359046 = r2359042 + r2359043;
        double r2359047 = log(r2359046);
        double r2359048 = r2359045 / r2359047;
        return r2359048;
}
↓
double f(double x) {
        double r2359049 = -1.0;
        double r2359050 = x;
        double r2359051 = r2359050 * r2359050;
        double r2359052 = -0.5;
        double r2359053 = r2359051 * r2359052;
        double r2359054 = r2359049 + r2359053;
        double r2359055 = r2359054 - r2359050;
        return r2359055;
}

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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 
(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))))