Average Error: 61.3 → 0.5
Time: 21.2s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\left(\log 1 - 1 \cdot x\right) - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}{\left(\log 1 + 1 \cdot x\right) - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\left(\log 1 - 1 \cdot x\right) - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}{\left(\log 1 + 1 \cdot x\right) - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}}
double f(double x) {
        double r4252254 = 1.0;
        double r4252255 = x;
        double r4252256 = r4252254 - r4252255;
        double r4252257 = log(r4252256);
        double r4252258 = r4252254 + r4252255;
        double r4252259 = log(r4252258);
        double r4252260 = r4252257 / r4252259;
        return r4252260;
}
↓
double f(double x) {
        double r4252261 = 1.0;
        double r4252262 = log(r4252261);
        double r4252263 = x;
        double r4252264 = r4252261 * r4252263;
        double r4252265 = r4252262 - r4252264;
        double r4252266 = r4252263 / r4252261;
        double r4252267 = r4252266 * r4252266;
        double r4252268 = 0.5;
        double r4252269 = r4252267 * r4252268;
        double r4252270 = r4252265 - r4252269;
        double r4252271 = r4252262 + r4252264;
        double r4252272 = r4252271 - r4252269;
        double r4252273 = r4252270 / r4252272;
        return r4252273;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.3

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

  2. Taylor expanded around 0 60.5

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

  3. Simplified60.5

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))