Average Error: 58.5 → 0.2
Time: 13.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r87949 = 1.0;
        double r87950 = eps;
        double r87951 = r87949 - r87950;
        double r87952 = r87949 + r87950;
        double r87953 = r87951 / r87952;
        double r87954 = log(r87953);
        return r87954;
}
↓
double f(double eps) {
        double r87955 = eps;
        double r87956 = 1.0;
        double r87957 = r87955 / r87956;
        double r87958 = 3.0;
        double r87959 = pow(r87957, r87958);
        double r87960 = -0.6666666666666666;
        double r87961 = r87959 * r87960;
        double r87962 = 0.4;
        double r87963 = 5.0;
        double r87964 = pow(r87955, r87963);
        double r87965 = pow(r87956, r87963);
        double r87966 = r87964 / r87965;
        double r87967 = r87962 * r87966;
        double r87968 = 2.0;
        double r87969 = r87968 * r87955;
        double r87970 = r87967 + r87969;
        double r87971 = r87961 - r87970;
        return r87971;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 58.5

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. 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-prod58.5

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

  5. Simplified58.5

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

  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

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(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))))