Average Error: 58.6 → 0.2
Time: 9.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-\left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right) - \frac{{\left(\frac{\varepsilon}{1}\right)}^{3}}{\frac{3}{2}}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-\left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right) - \frac{{\left(\frac{\varepsilon}{1}\right)}^{3}}{\frac{3}{2}}
double f(double eps) {
        double r337516 = 1.0;
        double r337517 = eps;
        double r337518 = r337516 - r337517;
        double r337519 = r337516 + r337517;
        double r337520 = r337518 / r337519;
        double r337521 = log(r337520);
        return r337521;
}
↓
double f(double eps) {
        double r337522 = 2.0;
        double r337523 = 5.0;
        double r337524 = r337522 / r337523;
        double r337525 = eps;
        double r337526 = pow(r337525, r337523);
        double r337527 = 1.0;
        double r337528 = pow(r337527, r337523);
        double r337529 = r337526 / r337528;
        double r337530 = r337524 * r337529;
        double r337531 = 2.0;
        double r337532 = r337531 * r337525;
        double r337533 = r337530 + r337532;
        double r337534 = -r337533;
        double r337535 = r337525 / r337527;
        double r337536 = 3.0;
        double r337537 = pow(r337535, r337536);
        double r337538 = r337536 / r337522;
        double r337539 = r337537 / r337538;
        double r337540 = r337534 - r337539;
        return r337540;
}

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

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 350497007 
(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))))