Average Error: 58.4 → 0.3
Time: 3.7s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot -2 + \left(-0.4 \cdot {\varepsilon}^{5} - 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot -2 + \left(-0.4 \cdot {\varepsilon}^{5} - 0.6666666666666666 \cdot {\varepsilon}^{3}\right)
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (+
  (* eps -2.0)
  (- (* -0.4 (pow eps 5.0)) (* 0.6666666666666666 (pow eps 3.0)))))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return (eps * -2.0) + ((-0.4 * pow(eps, 5.0)) - (0.6666666666666666 * pow(eps, 3.0)));
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied sub-neg_binary640.3

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

  6. Applied associate-+l+_binary640.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020224 
(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))))