Kahan's exp quotient

Average Error: 39.5 → 0.4

Time: 2.9s

Precision: 64

Math

\[\frac{e^{x} - 1}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.7128650325012662 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}} \cdot \left({\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}\right)}{\left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\left(\frac{1}{4} \cdot x + 1\right) - \frac{5}{96} \cdot {x}^{2}\right) + \frac{25}{9216} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}\\ \end{array}\]

C

double code(double x) {
	return ((double) (((double) (((double) exp(x)) - 1.0)) / x));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -0.00017128650325012662)) {
		VAR = ((double) (((double) (((double) exp(x)) / x)) - ((double) (1.0 / x))));
	} else {
		VAR = ((double) (((double) (((double) sqrt(((double) (((double) pow(((double) (0.16666666666666666 * ((double) pow(x, 2.0)))), 3.0)) + ((double) pow(((double) (((double) (0.5 * x)) + 1.0)), 3.0)))))) * ((double) (((double) pow(((double) (0.052083333333333336 * ((double) pow(x, 2.0)))), 3.0)) + ((double) pow(((double) (((double) (0.25 * x)) + 1.0)), 3.0)))))) / ((double) (((double) (((double) (((double) (((double) (0.25 * x)) + 1.0)) * ((double) (((double) (((double) (0.25 * x)) + 1.0)) - ((double) (0.052083333333333336 * ((double) pow(x, 2.0)))))))) + ((double) (0.002712673611111111 * ((double) (((double) pow(x, 2.0)) * ((double) pow(x, 2.0)))))))) * ((double) sqrt(((double) (((double) (((double) (0.16666666666666666 * ((double) pow(x, 2.0)))) * ((double) (0.16666666666666666 * ((double) pow(x, 2.0)))))) + ((double) (((double) (((double) (((double) (0.5 * x)) + 1.0)) * ((double) (((double) (0.5 * x)) + 1.0)))) - ((double) (((double) (0.16666666666666666 * ((double) pow(x, 2.0)))) * ((double) (((double) (0.5 * x)) + 1.0))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	39.5
Target	39.8
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.00017128650325012662

   1. Initial program 0.0

      \[\frac{e^{x} - 1}{x}\]
   2. Using strategy rm

   3. Applied div-sub 0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{x} - \frac{1}{x}}\]

   if -0.00017128650325012662 < x

   1. Initial program 59.9

      \[\frac{e^{x} - 1}{x}\]

   2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.6

      \[\leadsto \color{blue}{\sqrt{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \sqrt{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\]

   5. Taylor expanded around 0 0.5

      \[\leadsto \sqrt{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \color{blue}{\left(\frac{5}{96} \cdot {x}^{2} + \left(\frac{1}{4} \cdot x + 1\right)\right)}\]
   6. Using strategy rm

   7. Applied flip3-+ 0.5

      \[\leadsto \sqrt{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \color{blue}{\frac{{\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}}{\left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{5}{96} \cdot {x}^{2}\right) + \left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\frac{1}{4} \cdot x + 1\right) - \left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{1}{4} \cdot x + 1\right)\right)}}\]

   8. Applied flip3-+ 0.5

      \[\leadsto \sqrt{\color{blue}{\frac{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}} \cdot \frac{{\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}}{\left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{5}{96} \cdot {x}^{2}\right) + \left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\frac{1}{4} \cdot x + 1\right) - \left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{1}{4} \cdot x + 1\right)\right)}\]

   9. Applied sqrt-div 0.5

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}}}{\sqrt{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}} \cdot \frac{{\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}}{\left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{5}{96} \cdot {x}^{2}\right) + \left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\frac{1}{4} \cdot x + 1\right) - \left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{1}{4} \cdot x + 1\right)\right)}\]

   10. Applied frac-times 0.5

       \[\leadsto \color{blue}{\frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}} \cdot \left({\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}\right)}{\sqrt{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)} \cdot \left(\left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{5}{96} \cdot {x}^{2}\right) + \left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\frac{1}{4} \cdot x + 1\right) - \left(\frac{5}{96} \cdot {x}^{2}\right) \cdot \left(\frac{1}{4} \cdot x + 1\right)\right)\right)}}\]

   11. Simplified 0.5

       \[\leadsto \frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}} \cdot \left({\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}\right)}{\color{blue}{\left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\left(\frac{1}{4} \cdot x + 1\right) - \frac{5}{96} \cdot {x}^{2}\right) + \frac{25}{9216} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.7128650325012662 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left(\frac{1}{6} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{2} \cdot x + 1\right)}^{3}} \cdot \left({\left(\frac{5}{96} \cdot {x}^{2}\right)}^{3} + {\left(\frac{1}{4} \cdot x + 1\right)}^{3}\right)}{\left(\left(\frac{1}{4} \cdot x + 1\right) \cdot \left(\left(\frac{1}{4} \cdot x + 1\right) - \frac{5}{96} \cdot {x}^{2}\right) + \frac{25}{9216} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \left(\left(\frac{1}{2} \cdot x + 1\right) \cdot \left(\frac{1}{2} \cdot x + 1\right) - \left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot x + 1\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))