Average Error: 58.4 → 0.2
Time: 4.4s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot -0.4 + \left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot -0.6666666666666666 - \varepsilon \cdot 2\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot -0.4 + \left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot -0.6666666666666666 - \varepsilon \cdot 2\right)
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (+
  (* (/ (pow eps 5.0) (pow 1.0 5.0)) -0.4)
  (- (* (pow (/ eps 1.0) 3.0) -0.6666666666666666) (* eps 2.0))))
double code(double eps) {
	return ((double) log((((double) (1.0 - eps)) / ((double) (1.0 + eps)))));
}
↓
double code(double eps) {
	return ((double) (((double) ((((double) pow(eps, 5.0)) / ((double) pow(1.0, 5.0))) * -0.4)) + ((double) (((double) (((double) pow((eps / 1.0), 3.0)) * -0.6666666666666666)) - ((double) (eps * 2.0))))));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.4

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div_binary6458.4

    \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)}\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(0.6666666666666666 \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(0.4 \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{\frac{{\varepsilon}^{5}}{{1}^{5}} \cdot -0.4 + \left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot -0.6666666666666666 - \varepsilon \cdot 2\right)}\]

  6. Final simplification0.2

    \[\leadsto \frac{{\varepsilon}^{5}}{{1}^{5}} \cdot -0.4 + \left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot -0.6666666666666666 - \varepsilon \cdot 2\right)\]

Reproduce

herbie shell --seed 2020210 
(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))))