Average Error: 58.6 → 0.6
Time: 13.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double f(double eps) {
        double r53954 = 1.0;
        double r53955 = eps;
        double r53956 = r53954 - r53955;
        double r53957 = r53954 + r53955;
        double r53958 = r53956 / r53957;
        double r53959 = log(r53958);
        return r53959;
}
↓
double f(double eps) {
        double r53960 = 2.0;
        double r53961 = eps;
        double r53962 = 2.0;
        double r53963 = pow(r53961, r53962);
        double r53964 = r53960 * r53963;
        double r53965 = 1.0;
        double r53966 = log(r53965);
        double r53967 = pow(r53965, r53962);
        double r53968 = r53963 / r53967;
        double r53969 = r53968 + r53961;
        double r53970 = r53960 * r53969;
        double r53971 = r53966 - r53970;
        double r53972 = r53964 + r53971;
        return r53972;
}

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019212 
(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))))