expax (section 3.5)

Average Error: 29.5 → 0.8

Time: 3.6s

Precision: binary64

Math

\[e^{a \cdot x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9179393.82255176269:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]

C

double code(double a, double x) {
	return ((double) (((double) exp(((double) (a * x)))) - 1.0));
}

↓

double code(double a, double x) {
	double VAR;
	if ((((double) (a * x)) <= -9179393.822551763)) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) exp(((double) (a * x)))) - 1.0)), 3.0))));
	} else {
		VAR = ((double) (((double) (a * x)) + ((double) (((double) (0.5 * ((double) pow(((double) (a * x)), 2.0)))) + ((double) (0.16666666666666666 * ((double) pow(((double) (a * x)), 3.0))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.1
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -9179393.82255176269

   1. Initial program 0

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

   3. Applied add-cbrt-cube 0

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

   4. Simplified 0

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

   if -9179393.82255176269 < (* a x)

   1. Initial program 43.3

      \[e^{a \cdot x} - 1\]

   2. Taylor expanded around 0 14.9

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

   3. Simplified 14.9

      \[\leadsto \color{blue}{x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)}\]
   4. Using strategy rm

   5. Applied pow-prod-down 5.4

      \[\leadsto x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \color{blue}{{\left(a \cdot x\right)}^{3}}\]
   6. Using strategy rm

   7. Applied distribute-lft-in 5.4

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

   8. Simplified 5.4

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

   9. Simplified 1.1

      \[\leadsto \left(a \cdot x + \color{blue}{\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}}\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\]
   10. Using strategy rm

   11. Applied associate-+l+ 1.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -9179393.82255176269:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} \cdot {\left(a \cdot x\right)}^{2} + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))