Average Error: 58.6 → 0.6
Time: 13.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1
double f(double eps) {
        double r2449087 = 1.0;
        double r2449088 = eps;
        double r2449089 = r2449087 - r2449088;
        double r2449090 = r2449087 + r2449088;
        double r2449091 = r2449089 / r2449090;
        double r2449092 = log(r2449091);
        return r2449092;
}
↓
double f(double eps) {
        double r2449093 = 2.0;
        double r2449094 = eps;
        double r2449095 = r2449094 * r2449094;
        double r2449096 = 1.0;
        double r2449097 = r2449096 * r2449096;
        double r2449098 = r2449095 / r2449097;
        double r2449099 = r2449094 + r2449098;
        double r2449100 = r2449095 - r2449099;
        double r2449101 = r2449093 * r2449100;
        double r2449102 = log(r2449096);
        double r2449103 = r2449101 + r2449102;
        return r2449103;
}

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

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

  4. Final simplification0.6

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

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))))