Average Error: 58.7 → 0.2
Time: 12.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r34039 = 1.0;
        double r34040 = eps;
        double r34041 = r34039 - r34040;
        double r34042 = r34039 + r34040;
        double r34043 = r34041 / r34042;
        double r34044 = log(r34043);
        return r34044;
}
↓
double f(double eps) {
        double r34045 = 2.0;
        double r34046 = eps;
        double r34047 = r34045 * r34046;
        double r34048 = 0.6666666666666666;
        double r34049 = 3.0;
        double r34050 = pow(r34046, r34049);
        double r34051 = r34048 * r34050;
        double r34052 = 0.4;
        double r34053 = 5.0;
        double r34054 = pow(r34046, r34053);
        double r34055 = r34052 * r34054;
        double r34056 = r34051 + r34055;
        double r34057 = r34047 + r34056;
        double r34058 = -r34057;
        return r34058;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied add-exp-log58.7

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

  4. Applied add-exp-log58.7

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

  5. Applied div-exp58.7

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

  6. Applied rem-log-exp58.6

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

  7. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}}\]

  9. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  10. Final simplification0.2

    \[\leadsto -\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(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))))