Average Error: 58.7 → 0.5
Time: 12.0s
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 r39537 = 1.0;
        double r39538 = eps;
        double r39539 = r39537 - r39538;
        double r39540 = r39537 + r39538;
        double r39541 = r39539 / r39540;
        double r39542 = log(r39541);
        return r39542;
}
↓
double f(double eps) {
        double r39543 = 2.0;
        double r39544 = eps;
        double r39545 = 2.0;
        double r39546 = pow(r39544, r39545);
        double r39547 = 1.0;
        double r39548 = pow(r39547, r39545);
        double r39549 = r39546 / r39548;
        double r39550 = r39549 + r39544;
        double r39551 = r39546 - r39550;
        double r39552 = r39543 * r39551;
        double r39553 = log(r39547);
        double r39554 = r39552 + r39553;
        return r39554;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019298 
(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))))