Average Error: 58.5 → 0.2
Time: 11.8s
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 r2234673 = 1.0;
        double r2234674 = eps;
        double r2234675 = r2234673 - r2234674;
        double r2234676 = r2234673 + r2234674;
        double r2234677 = r2234675 / r2234676;
        double r2234678 = log(r2234677);
        return r2234678;
}
↓
double f(double eps) {
        double r2234679 = -0.6666666666666666;
        double r2234680 = eps;
        double r2234681 = r2234679 * r2234680;
        double r2234682 = r2234680 * r2234680;
        double r2234683 = r2234681 * r2234682;
        double r2234684 = -2.0;
        double r2234685 = r2234684 * r2234680;
        double r2234686 = 5.0;
        double r2234687 = pow(r2234680, r2234686);
        double r2234688 = 0.4;
        double r2234689 = r2234687 * r2234688;
        double r2234690 = r2234685 - r2234689;
        double r2234691 = r2234683 + r2234690;
        return r2234691;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.2

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

  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 2019158 
(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))))