Average Error: 58.6 → 0.6
Time: 14.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1
double f(double eps) {
        double r3807778 = 1.0;
        double r3807779 = eps;
        double r3807780 = r3807778 - r3807779;
        double r3807781 = r3807778 + r3807779;
        double r3807782 = r3807780 / r3807781;
        double r3807783 = log(r3807782);
        return r3807783;
}
↓
double f(double eps) {
        double r3807784 = 2.0;
        double r3807785 = eps;
        double r3807786 = r3807785 * r3807785;
        double r3807787 = 1.0;
        double r3807788 = r3807785 / r3807787;
        double r3807789 = r3807788 * r3807788;
        double r3807790 = r3807785 + r3807789;
        double r3807791 = r3807786 - r3807790;
        double r3807792 = r3807784 * r3807791;
        double r3807793 = log(r3807787);
        double r3807794 = r3807792 + r3807793;
        return r3807794;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(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))))