Average Error: 58.5 → 0.7
Time: 6.0s
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 r109991 = 1.0;
        double r109992 = eps;
        double r109993 = r109991 - r109992;
        double r109994 = r109991 + r109992;
        double r109995 = r109993 / r109994;
        double r109996 = log(r109995);
        return r109996;
}
↓
double f(double eps) {
        double r109997 = 2.0;
        double r109998 = eps;
        double r109999 = 2.0;
        double r110000 = pow(r109998, r109999);
        double r110001 = 1.0;
        double r110002 = pow(r110001, r109999);
        double r110003 = r110000 / r110002;
        double r110004 = r110003 + r109998;
        double r110005 = r110000 - r110004;
        double r110006 = r109997 * r110005;
        double r110007 = log(r110001);
        double r110008 = r110006 + r110007;
        return r110008;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.5
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.5

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

  2. 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 2020056 
(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))))