Average Error: 61.3 → 0.5
Time: 23.5s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot x + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right)}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(1 \cdot x + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right)}{\left(1 \cdot x - \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot \frac{1}{2}\right) + \log 1}
double f(double x) {
        double r4180272 = 1.0;
        double r4180273 = x;
        double r4180274 = r4180272 - r4180273;
        double r4180275 = log(r4180274);
        double r4180276 = r4180272 + r4180273;
        double r4180277 = log(r4180276);
        double r4180278 = r4180275 / r4180277;
        return r4180278;
}
↓
double f(double x) {
        double r4180279 = 1.0;
        double r4180280 = log(r4180279);
        double r4180281 = x;
        double r4180282 = r4180279 * r4180281;
        double r4180283 = r4180281 / r4180279;
        double r4180284 = r4180283 * r4180283;
        double r4180285 = 0.5;
        double r4180286 = r4180284 * r4180285;
        double r4180287 = r4180282 + r4180286;
        double r4180288 = r4180280 - r4180287;
        double r4180289 = r4180282 - r4180286;
        double r4180290 = r4180289 + r4180280;
        double r4180291 = r4180288 / r4180290;
        return r4180291;
}

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

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

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))))