Average Error: 58.6 → 0.2
Time: 4.2s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[-2 \cdot \varepsilon - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right) \]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-2 \cdot \varepsilon - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)

(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

Original58.6
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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2021204 
(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))))