expq2 (section 3.11)

Average Error: 41.1 → 0.1

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.998544414394856195:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}} \cdot \sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}}\right) \cdot \sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{elif}\;e^{x} \le 1.0000034109656726:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((double) exp(x)) <= 0.9985444143948562)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) exp(x)) / ((double) log(((double) exp(((double) (((double) exp(x)) - 1.0)))))))))) * ((double) cbrt(((double) (((double) exp(x)) / ((double) log(((double) exp(((double) (((double) exp(x)) - 1.0)))))))))))) * ((double) cbrt(((double) (((double) exp(x)) / ((double) log(((double) exp(((double) (((double) exp(x)) - 1.0))))))))))));
	} else {
		double VAR_1;
		if ((((double) exp(x)) <= 1.0000034109656726)) {
			VAR_1 = ((double) (((double) fma(0.08333333333333333, x, ((double) (1.0 / x)))) + 0.5));
		} else {
			VAR_1 = ((double) (1.0 / ((double) (1.0 - ((double) (1.0 / ((double) exp(x))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	41.1
Target	40.6
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if (exp x) < 0.9985444143948562

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied add-log-exp 0.0

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

   5. Applied diff-log 0.0

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

   6. Simplified 0.0

      \[\leadsto \frac{e^{x}}{\log \color{blue}{\left(e^{e^{x} - 1}\right)}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 0.0

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

   if 0.9985444143948562 < (exp x) < 1.0000034109656726

   1. Initial program 62.6

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}}\]

   if 1.0000034109656726 < (exp x)

   1. Initial program 36.4

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

   3. Applied clear-num 36.4

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

   4. Simplified 2.4

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.998544414394856195:\\ \;\;\;\;\left(\sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}} \cdot \sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}}\right) \cdot \sqrt[3]{\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{elif}\;e^{x} \le 1.0000034109656726:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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