Average Error: 58.6 → 0.7
Time: 6.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r96097 = 1.0;
        double r96098 = eps;
        double r96099 = r96097 - r96098;
        double r96100 = r96097 + r96098;
        double r96101 = r96099 / r96100;
        double r96102 = log(r96101);
        return r96102;
}
↓
double f(double eps) {
        double r96103 = 2.0;
        double r96104 = eps;
        double r96105 = 2.0;
        double r96106 = pow(r96104, r96105);
        double r96107 = 1.0;
        double r96108 = pow(r96107, r96105);
        double r96109 = r96106 / r96108;
        double r96110 = r96109 + r96104;
        double r96111 = r96106 - r96110;
        double r96112 = r96103 * r96111;
        double r96113 = log(r96107);
        double r96114 = r96112 + r96113;
        return r96114;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.3
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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019356 
(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))))