Average Error: 58.5 → 0.2
Time: 8.1s
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 r357 = 1.0;
        double r358 = eps;
        double r359 = r357 - r358;
        double r360 = r357 + r358;
        double r361 = r359 / r360;
        double r362 = log(r361);
        return r362;
}
↓
double f(double eps) {
        double r363 = 0.6666666666666666;
        double r364 = eps;
        double r365 = 3.0;
        double r366 = pow(r364, r365);
        double r367 = 1.0;
        double r368 = pow(r367, r365);
        double r369 = r366 / r368;
        double r370 = r363 * r369;
        double r371 = 0.4;
        double r372 = 5.0;
        double r373 = pow(r364, r372);
        double r374 = pow(r367, r372);
        double r375 = r373 / r374;
        double r376 = r371 * r375;
        double r377 = r370 + r376;
        double r378 = 2.0;
        double r379 = r378 * r364;
        double r380 = r377 + r379;
        double r381 = -r380;
        return r381;
}

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 2020025 
(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))))