Average Error: 58.6 → 0.2
Time: 10.5min
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot \frac{-2}{5} - {\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{2}{3}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot \frac{-2}{5} - {\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{2}{3}\right) - 2 \cdot \varepsilon
double code(double eps) {
	return ((double) log((((double) (1.0 - eps)) / ((double) (1.0 + eps)))));
}
↓
double code(double eps) {
	return ((double) (((double) (((double) ((((double) pow(eps, 5.0)) / ((double) pow(1.0, 5.0))) * -0.4)) - ((double) (((double) pow((eps / 1.0), 3.0)) * 0.6666666666666666)))) - ((double) (2.0 * eps))));
}

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-div58.6

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020181 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))