Average Error: 58.7 → 0.2
Time: 3.9s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon - 0.6666666666666666 \cdot {\varepsilon}^{3}\right) - 0.4 \cdot {\varepsilon}^{5}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon - 0.6666666666666666 \cdot {\varepsilon}^{3}\right) - 0.4 \cdot {\varepsilon}^{5}
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (-
  (- (* -2.0 eps) (* 0.6666666666666666 (pow eps 3.0)))
  (* 0.4 (pow eps 5.0))))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return ((-2.0 * eps) - (0.6666666666666666 * pow(eps, 3.0))) - (0.4 * pow(eps, 5.0));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(0.4 \cdot {\varepsilon}^{5} + \left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon - 0.6666666666666666 \cdot {\varepsilon}^{3}\right) - 0.4 \cdot {\varepsilon}^{5}}\]

  4. Final simplification0.2

    \[\leadsto \left(-2 \cdot \varepsilon - 0.6666666666666666 \cdot {\varepsilon}^{3}\right) - 0.4 \cdot {\varepsilon}^{5}\]

Reproduce

herbie shell --seed 2021173 
(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))))