Average Error: 58.9 → 0.5
Time: 29.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double f(double eps) {
        double r98561 = 1.0;
        double r98562 = eps;
        double r98563 = r98561 - r98562;
        double r98564 = r98561 + r98562;
        double r98565 = r98563 / r98564;
        double r98566 = log(r98565);
        return r98566;
}
↓
double f(double eps) {
        double r98567 = 2.0;
        double r98568 = eps;
        double r98569 = 2.0;
        double r98570 = pow(r98568, r98569);
        double r98571 = r98567 * r98570;
        double r98572 = 1.0;
        double r98573 = log(r98572);
        double r98574 = pow(r98572, r98569);
        double r98575 = r98570 / r98574;
        double r98576 = r98575 + r98568;
        double r98577 = r98567 * r98576;
        double r98578 = r98573 - r98577;
        double r98579 = r98571 + r98578;
        return r98579;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.9
Target0.1
Herbie0.5
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 
(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))))