Kahan's exp quotient

Average Error: 40.0 → 0.3

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.6139978353163021 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{\left(e^{2}\right)}^{x} - 1 \cdot 1}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.0001613997835316302)) {
		VAR = ((double) (((double) (((double) log(((double) exp(((double) (((double) pow(((double) exp(2.0)), x)) - ((double) (1.0 * 1.0)))))))) / ((double) (((double) exp(x)) + 1.0)))) / x));
	} else {
		VAR = ((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	40.0
Target	40.5
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 < -1.6139978353163021e-4

   1. Initial program 0.0

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

   3. Applied flip-- 0.0

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 0.0

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

   6. Applied exp-prod 0.0

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

   7. Applied *-un-lft-identity 0.0

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

   8. Applied exp-prod 0.0

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

   9. Applied pow-prod-down 0.0

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

   10. Simplified 0.0

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

   12. Applied add-log-exp 0.0

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

   13. Applied add-log-exp 0.0

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

   14. Applied diff-log 0.1

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

   15. Simplified 0.0

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

   if -1.6139978353163021e-4 < x

   1. Initial program 60.1

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

   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. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6139978353163021 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{\left(e^{2}\right)}^{x} - 1 \cdot 1}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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))