Average Error: 58.3 → 0.2
Time: 4.2s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) + {\varepsilon}^{5} \cdot -0.4\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) + {\varepsilon}^{5} \cdot -0.4
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (+
  (- (* eps -2.0) (* (pow eps 3.0) 0.6666666666666666))
  (* (pow eps 5.0) -0.4)))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return ((eps * -2.0) - (pow(eps, 3.0) * 0.6666666666666666)) + (pow(eps, 5.0) * -0.4);
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020232 
(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))))