Kahan's exp quotient

Average Error: 39.2 → 0.3

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.45555334023096547 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{\frac{x}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \left(x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right) + 1\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -0.00014555533402309655)) {
		temp = (1.0 / (x / (exp(x) - 1.0)));
	} else {
		temp = ((cbrt(((0.16666666666666666 * pow(x, 2.0)) + ((0.5 * x) + 1.0))) * ((x * (0.16666666666666666 + (x * 0.027777777777777776))) + 1.0)) * (1.0 + (x * (0.16666666666666666 + (x * 0.027777777777777776)))));
	}
	return temp;
}

Error

Bits error versus x

Target

Original	39.2
Target	39.6
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 < -0.00014555533402309655

   1. Initial program 0.0

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

   3. Applied clear-num 0.0

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

   if -0.00014555533402309655 < x

   1. Initial program 59.8

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

   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-cube-cbrt 0.5

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

   5. Taylor expanded around 0 0.5

      \[\leadsto \left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \color{blue}{\left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

   6. Simplified 0.5

      \[\leadsto \left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right) + 1\right)}\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

   7. Taylor expanded around 0 0.5

      \[\leadsto \left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \left(x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right) + 1\right)\right) \cdot \color{blue}{\left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)}\]

   8. Simplified 0.5

      \[\leadsto \left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \left(x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right) + 1\right)\right) \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.45555334023096547 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{\frac{x}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)} \cdot \left(x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right) + 1\right)\right) \cdot \left(1 + x \cdot \left(\frac{1}{6} + x \cdot \frac{1}{36}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))