Average Error: 58.5 → 0.2
Time: 5.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r94437 = 1.0;
        double r94438 = eps;
        double r94439 = r94437 - r94438;
        double r94440 = r94437 + r94438;
        double r94441 = r94439 / r94440;
        double r94442 = log(r94441);
        return r94442;
}
↓
double f(double eps) {
        double r94443 = 0.6666666666666666;
        double r94444 = eps;
        double r94445 = 3.0;
        double r94446 = pow(r94444, r94445);
        double r94447 = 1.0;
        double r94448 = pow(r94447, r94445);
        double r94449 = r94446 / r94448;
        double r94450 = r94443 * r94449;
        double r94451 = 0.4;
        double r94452 = 5.0;
        double r94453 = pow(r94444, r94452);
        double r94454 = pow(r94447, r94452);
        double r94455 = r94453 / r94454;
        double r94456 = r94451 * r94455;
        double r94457 = r94450 + r94456;
        double r94458 = 2.0;
        double r94459 = r94458 * r94444;
        double r94460 = r94457 + r94459;
        double r94461 = -r94460;
        return r94461;
}

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

Derivation

  1. Initial program 58.5

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

  6. Applied associate-+r+0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019362 
(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))))