Average Error: 58.8 → 0.6
Time: 18.0s
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 r118657 = 1.0;
        double r118658 = eps;
        double r118659 = r118657 - r118658;
        double r118660 = r118657 + r118658;
        double r118661 = r118659 / r118660;
        double r118662 = log(r118661);
        return r118662;
}
↓
double f(double eps) {
        double r118663 = 2.0;
        double r118664 = eps;
        double r118665 = r118664 * r118664;
        double r118666 = 1.0;
        double r118667 = r118664 / r118666;
        double r118668 = r118667 * r118667;
        double r118669 = r118664 + r118668;
        double r118670 = r118665 - r118669;
        double r118671 = r118663 * r118670;
        double r118672 = log(r118666);
        double r118673 = r118671 + r118672;
        return r118673;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

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

  2. Simplified58.8

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. 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 2019196 
(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))))