Average Error: 58.6 → 0.7
Time: 29.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\varepsilon \cdot \varepsilon - \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right) - \varepsilon\right) \cdot 2 + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\varepsilon \cdot \varepsilon - \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right) - \varepsilon\right) \cdot 2 + \log 1
double f(double eps) {
        double r2934325 = 1.0;
        double r2934326 = eps;
        double r2934327 = r2934325 - r2934326;
        double r2934328 = r2934325 + r2934326;
        double r2934329 = r2934327 / r2934328;
        double r2934330 = log(r2934329);
        return r2934330;
}
↓
double f(double eps) {
        double r2934331 = eps;
        double r2934332 = r2934331 * r2934331;
        double r2934333 = 1.0;
        double r2934334 = r2934331 / r2934333;
        double r2934335 = r2934334 * r2934334;
        double r2934336 = r2934332 - r2934335;
        double r2934337 = r2934336 - r2934331;
        double r2934338 = 2.0;
        double r2934339 = r2934337 * r2934338;
        double r2934340 = log(r2934333);
        double r2934341 = r2934339 + r2934340;
        return r2934341;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.3
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019168 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))