Average Error: 58.7 → 0.6
Time: 14.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1
double f(double eps) {
        double r3943506 = 1.0;
        double r3943507 = eps;
        double r3943508 = r3943506 - r3943507;
        double r3943509 = r3943506 + r3943507;
        double r3943510 = r3943508 / r3943509;
        double r3943511 = log(r3943510);
        return r3943511;
}
↓
double f(double eps) {
        double r3943512 = 2.0;
        double r3943513 = eps;
        double r3943514 = r3943513 * r3943513;
        double r3943515 = 1.0;
        double r3943516 = r3943513 / r3943515;
        double r3943517 = r3943516 * r3943516;
        double r3943518 = r3943513 + r3943517;
        double r3943519 = r3943514 - r3943518;
        double r3943520 = r3943512 * r3943519;
        double r3943521 = log(r3943515);
        double r3943522 = r3943520 + r3943521;
        return r3943522;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019169 
(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))))