Average Error: 58.5 → 0.7
Time: 18.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{\varepsilon \cdot \varepsilon}{1 \cdot 1} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{\varepsilon \cdot \varepsilon}{1 \cdot 1} + \varepsilon\right)\right)
double f(double eps) {
        double r5884802 = 1.0;
        double r5884803 = eps;
        double r5884804 = r5884802 - r5884803;
        double r5884805 = r5884802 + r5884803;
        double r5884806 = r5884804 / r5884805;
        double r5884807 = log(r5884806);
        return r5884807;
}
↓
double f(double eps) {
        double r5884808 = 1.0;
        double r5884809 = log(r5884808);
        double r5884810 = 2.0;
        double r5884811 = eps;
        double r5884812 = r5884811 * r5884811;
        double r5884813 = r5884808 * r5884808;
        double r5884814 = r5884812 / r5884813;
        double r5884815 = r5884814 + r5884811;
        double r5884816 = r5884812 - r5884815;
        double r5884817 = r5884810 * r5884816;
        double r5884818 = r5884809 + r5884817;
        return r5884818;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019173 
(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))))