Average Error: 58.6 → 0.2
Time: 4.9s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[-\mathsf{fma}\left(0.2857142857142857, {\varepsilon}^{7}, \mathsf{fma}\left(\varepsilon, 2, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, 0.4 \cdot {\varepsilon}^{5}\right)\right)\right) \]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(0.2857142857142857, {\varepsilon}^{7}, \mathsf{fma}\left(\varepsilon, 2, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, 0.4 \cdot {\varepsilon}^{5}\right)\right)\right)

(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))
↓
(FPCore (eps)
 :precision binary64
 (-
  (fma
   0.2857142857142857
   (pow eps 7.0)
   (fma
    eps
    2.0
    (fma (pow eps 3.0) 0.6666666666666666 (* 0.4 (pow eps 5.0)))))))
double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}
↓
double code(double eps) {
	return -fma(0.2857142857142857, pow(eps, 7.0), fma(eps, 2.0, fma(pow(eps, 3.0), 0.6666666666666666, (0.4 * pow(eps, 5.0)))));
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
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. Simplified50.5

    \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]

  3. Taylor expanded in eps around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022129 
(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))))