Kahan's exp quotient

Average Error: 39.4 → 0.4

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.144325969769503 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left(e^{x \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left({\left(e^{x}\right)}^{6} + e^{x \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.00011443259697695033)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) exp(((double) (x * 3.0)))), 3.0)) - ((double) pow(((double) pow(1.0, 3.0)), 3.0)))) / ((double) (((double) (((double) pow(((double) exp(x)), 6.0)) + ((double) (((double) exp(((double) (x * 3.0)))) * ((double) pow(1.0, 3.0)))))) + ((double) pow(1.0, 6.0)))))) / ((double) (((double) (((double) (1.0 * ((double) (1.0 + ((double) exp(x)))))) + ((double) exp(((double) (x + x)))))) * x))));
	} else {
		VAR = ((double) (((double) (((double) (((double) pow(x, 2.0)) * ((double) (((double) (x * 0.16666666666666666)) + 0.5)))) + x)) / x));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	39.4
Target	39.9
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 < -0.00011443259697695033

   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 0.1

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

   9. Applied flip3-- 0.1

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

   10. Simplified 0.1

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

   if -0.00011443259697695033 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.5

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

   3. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.144325969769503 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left(e^{x \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left({\left(e^{x}\right)}^{6} + e^{x \cdot 3} \cdot {1}^{3}\right) + {1}^{6}}}{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))