Average Error: 58.6 → 0.2
Time: 9.3s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\varepsilon \cdot -2 - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\varepsilon \cdot -2 - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (-
  (-
   (* eps -2.0)
   (+ (* 0.6666666666666666 (pow eps 3.0)) (* 0.4 (pow eps 5.0))))
  (* 0.2857142857142857 (pow eps 7.0))))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return ((eps * -2.0) - ((0.6666666666666666 * pow(eps, 3.0)) + (0.4 * pow(eps, 5.0)))) - (0.2857142857142857 * pow(eps, 7.0));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.3
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

    \[\leadsto \color{blue}{-\left(0.4 \cdot {\varepsilon}^{5} + \left(0.2857142857142857 \cdot {\varepsilon}^{7} + \left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}}\]
  4. Using strategy rm

  5. Applied associate--l-_binary64_10390.2

    \[\leadsto \color{blue}{\left(\varepsilon \cdot -2 - \left({\varepsilon}^{3} \cdot 0.6666666666666666 + 0.4 \cdot {\varepsilon}^{5}\right)\right)} - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

  6. Simplified0.2

    \[\leadsto \left(\varepsilon \cdot -2 - \color{blue}{\left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

  7. Final simplification0.2

    \[\leadsto \left(\varepsilon \cdot -2 - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

Reproduce

herbie shell --seed 2021007 
(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))))