Average Error: 58.5 → 0.6
Time: 8.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log 1 + 2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log 1 + 2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double f(double eps) {
        double r81394 = 1.0;
        double r81395 = eps;
        double r81396 = r81394 - r81395;
        double r81397 = r81394 + r81395;
        double r81398 = r81396 / r81397;
        double r81399 = log(r81398);
        return r81399;
}
↓
double f(double eps) {
        double r81400 = 1.0;
        double r81401 = log(r81400);
        double r81402 = 2.0;
        double r81403 = eps;
        double r81404 = 2.0;
        double r81405 = pow(r81403, r81404);
        double r81406 = pow(r81400, r81404);
        double r81407 = r81405 / r81406;
        double r81408 = r81407 + r81403;
        double r81409 = r81405 - r81408;
        double r81410 = r81402 * r81409;
        double r81411 = r81401 + r81410;
        return r81411;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))