Kahan's exp quotient

Average Error: 39.8 → 0.3

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0502760427764326 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x \cdot 3}} + {\left(\sqrt{1}\right)}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}} \cdot \frac{\sqrt{e^{x \cdot 3}} - {\left(\sqrt{1}\right)}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.00010502760427764326)) {
		VAR = (((sqrt(exp((x * 3.0))) + pow(sqrt(1.0), 3.0)) / ((1.0 * (1.0 + exp(x))) + exp((x + x)))) * ((sqrt(exp((x * 3.0))) - pow(sqrt(1.0), 3.0)) / x));
	} else {
		VAR = ((0.16666666666666666 * pow(x, 2.0)) + ((0.5 * x) + 1.0));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	39.8
Target	40.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -0.00010502760427764326

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}{x}\]

   4. Applied associate-/l/ 0.1

      \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{x \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)}}\]

   5. Simplified 0.1

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

   7. Applied pow-exp 0.0

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

   9. Applied add-sqr-sqrt 0.0

      \[\leadsto \frac{e^{x \cdot 3} - {\color{blue}{\left(\sqrt{1} \cdot \sqrt{1}\right)}}^{3}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\]

   10. Applied unpow-prod-down 0.0

       \[\leadsto \frac{e^{x \cdot 3} - \color{blue}{{\left(\sqrt{1}\right)}^{3} \cdot {\left(\sqrt{1}\right)}^{3}}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\]

   11. Applied add-sqr-sqrt 0.1

       \[\leadsto \frac{\color{blue}{\sqrt{e^{x \cdot 3}} \cdot \sqrt{e^{x \cdot 3}}} - {\left(\sqrt{1}\right)}^{3} \cdot {\left(\sqrt{1}\right)}^{3}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\]

   12. Applied difference-of-squares 0.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt{e^{x \cdot 3}} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{x \cdot 3}} - {\left(\sqrt{1}\right)}^{3}\right)}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\]

   13. Applied times-frac 0.1

       \[\leadsto \color{blue}{\frac{\sqrt{e^{x \cdot 3}} + {\left(\sqrt{1}\right)}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}} \cdot \frac{\sqrt{e^{x \cdot 3}} - {\left(\sqrt{1}\right)}^{3}}{x}}\]

   if -0.00010502760427764326 < x

   1. Initial program 60.1

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

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0502760427764326 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x \cdot 3}} + {\left(\sqrt{1}\right)}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}} \cdot \frac{\sqrt{e^{x \cdot 3}} - {\left(\sqrt{1}\right)}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))