Average Error: 58.6 → 0.2
Time: 15.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r36156 = 1.0;
        double r36157 = eps;
        double r36158 = r36156 - r36157;
        double r36159 = r36156 + r36157;
        double r36160 = r36158 / r36159;
        double r36161 = log(r36160);
        return r36161;
}
↓
double f(double eps) {
        double r36162 = eps;
        double r36163 = 3.0;
        double r36164 = pow(r36162, r36163);
        double r36165 = 1.0;
        double r36166 = pow(r36165, r36163);
        double r36167 = r36164 / r36166;
        double r36168 = -0.6666666666666666;
        double r36169 = r36167 * r36168;
        double r36170 = 0.4;
        double r36171 = 5.0;
        double r36172 = pow(r36162, r36171);
        double r36173 = pow(r36165, r36171);
        double r36174 = r36172 / r36173;
        double r36175 = r36170 * r36174;
        double r36176 = 2.0;
        double r36177 = r36176 * r36162;
        double r36178 = r36175 + r36177;
        double r36179 = r36169 - r36178;
        return r36179;
}

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

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied div-inv58.6

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

  4. Applied log-prod58.6

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

  5. Simplified58.6

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]

Reproduce

herbie shell --seed 2019235 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))