Average Error: 58.5 → 0.7
Time: 14.5s
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 r86148 = 1.0;
        double r86149 = eps;
        double r86150 = r86148 - r86149;
        double r86151 = r86148 + r86149;
        double r86152 = r86150 / r86151;
        double r86153 = log(r86152);
        return r86153;
}
↓
double f(double eps) {
        double r86154 = 2.0;
        double r86155 = eps;
        double r86156 = r86155 * r86155;
        double r86157 = 1.0;
        double r86158 = r86155 / r86157;
        double r86159 = r86158 * r86158;
        double r86160 = r86155 + r86159;
        double r86161 = r86156 - r86160;
        double r86162 = r86154 * r86161;
        double r86163 = log(r86157);
        double r86164 = r86162 + r86163;
        return r86164;
}

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. Simplified58.5

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019195 
(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))))