Average Error: 58.5 → 58.5
Time: 4.0s
Precision: binary64
Cost: 6848
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}
↓
double code(double eps) {
	return log((1.0 - eps) / (1.0 + eps));
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.5
Target0.2
Herbie58.5
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Final simplification58.5

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

Reproduce

herbie shell --seed 2021023 
(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))))