expq2 (section 3.11)

Average Error: 41.1 → 0.3

Time: 2.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.3403842967041447e+217

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0 \lor \neg \left(e^{x} \le 1.000000720604215\right):\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if (((exp(x) <= 0.0) || !(exp(x) <= 1.000000720604215))) {
		temp = (1.0 / (1.0 - (1.0 / exp(x))));
	} else {
		temp = (0.5 + ((0.08333333333333333 * x) + (1.0 / x)));
	}
	return temp;
}

Error

Bits error versus x

Target

Original	41.1
Target	40.8
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (exp x) < 0.0 or 1.000000720604215 < (exp x)

   1. Initial program 1.3

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

   3. Applied clear-num 1.3

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

   4. Simplified 0.2

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

   if 0.0 < (exp x) < 1.000000720604215

   1. Initial program 61.9

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

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0 \lor \neg \left(e^{x} \le 1.000000720604215\right):\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))