Average Error: 58.4 → 0.7
Time: 7.1s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double code(double eps) {
	return ((double) log((((double) (1.0 - eps)) / ((double) (1.0 + eps)))));
}
↓
double code(double eps) {
	return ((double) (((double) log(1.0)) + ((double) (2.0 * ((double) (((double) (eps * eps)) - ((double) ((((double) pow(eps, 2.0)) / ((double) pow(1.0, 2.0))) + eps))))))));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020182 
(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))))