Average Error: 58.6 → 0.7
Time: 13.7s
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 r57138 = 1.0;
        double r57139 = eps;
        double r57140 = r57138 - r57139;
        double r57141 = r57138 + r57139;
        double r57142 = r57140 / r57141;
        double r57143 = log(r57142);
        return r57143;
}
↓
double f(double eps) {
        double r57144 = 2.0;
        double r57145 = eps;
        double r57146 = r57145 * r57145;
        double r57147 = 1.0;
        double r57148 = r57145 / r57147;
        double r57149 = r57148 * r57148;
        double r57150 = r57145 + r57149;
        double r57151 = r57146 - r57150;
        double r57152 = r57144 * r57151;
        double r57153 = log(r57147);
        double r57154 = r57152 + r57153;
        return r57154;
}

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

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  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 2019194 
(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))))