Average Error: 58.7 → 0.1
Time: 4.4s
Precision: binary64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[{\varepsilon}^{7} \cdot -0.2857142857142857 - \left(\varepsilon \cdot 2 + \mathsf{fma}\left(0.4, {\varepsilon}^{5}, 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{7} \cdot -0.2857142857142857 - \left(\varepsilon \cdot 2 + \mathsf{fma}\left(0.4, {\varepsilon}^{5}, 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)

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

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.1
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

  1. Initial program 58.7

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

  2. Simplified0.0

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

  3. Taylor expanded in eps around 0 0.1

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

  4. Applied fma-def_binary640.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022081 
(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))))