Kahan's exp quotient

Average Error: 39.8 → 0.3

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.02731730747026496 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\frac{3}{2}} + {1}^{\frac{3}{2}}}{x} \cdot \frac{{\left(e^{x}\right)}^{\frac{3}{2}} - {1}^{\frac{3}{2}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.0002027317307470265)) {
		VAR = (((pow(exp(x), 1.5) + pow(1.0, 1.5)) / x) * ((pow(exp(x), 1.5) - pow(1.0, 1.5)) / ((1.0 * (1.0 + exp(x))) + exp((x + 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 < -0.0002027317307470265

   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 *-commutative 0.1

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

   8. Applied sqr-pow 0.1

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

   9. Applied sqr-pow 0.1

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

   10. Applied difference-of-squares 0.1

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

   11. Applied times-frac 0.1

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

   12. Simplified 0.1

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

   13. Simplified 0.1

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

   if -0.0002027317307470265 < x

   1. Initial program 60.0

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.02731730747026496 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\frac{3}{2}} + {1}^{\frac{3}{2}}}{x} \cdot \frac{{\left(e^{x}\right)}^{\frac{3}{2}} - {1}^{\frac{3}{2}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

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))