Average Error: 61.0 → 0.0
Time: 31.0s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}
double f(double x) {
        double r9639308 = 1.0;
        double r9639309 = x;
        double r9639310 = r9639308 - r9639309;
        double r9639311 = log(r9639310);
        double r9639312 = r9639308 + r9639309;
        double r9639313 = log(r9639312);
        double r9639314 = r9639311 / r9639313;
        return r9639314;
}
↓
double f(double x) {
        double r9639315 = x;
        double r9639316 = -r9639315;
        double r9639317 = log1p(r9639316);
        double r9639318 = log1p(r9639315);
        double r9639319 = r9639317 / r9639318;
        return r9639319;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Simplified60.0

    \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\log_* (1 + x)}}\]
  3. Using strategy rm

  4. Applied sub-neg60.0

    \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]

  5. Applied log1p-def0.0

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

  6. Final simplification0.0

    \[\leadsto \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(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))))