Average Error: 58.7 → 0.2
Time: 17.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r2508074 = 1.0;
        double r2508075 = eps;
        double r2508076 = r2508074 - r2508075;
        double r2508077 = r2508074 + r2508075;
        double r2508078 = r2508076 / r2508077;
        double r2508079 = log(r2508078);
        return r2508079;
}
↓
double f(double eps) {
        double r2508080 = -0.6666666666666666;
        double r2508081 = eps;
        double r2508082 = r2508080 * r2508081;
        double r2508083 = r2508081 * r2508081;
        double r2508084 = r2508082 * r2508083;
        double r2508085 = -2.0;
        double r2508086 = r2508085 * r2508081;
        double r2508087 = 5.0;
        double r2508088 = pow(r2508081, r2508087);
        double r2508089 = 0.4;
        double r2508090 = r2508088 * r2508089;
        double r2508091 = r2508086 - r2508090;
        double r2508092 = r2508084 + r2508091;
        return r2508092;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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