expax (section 3.5)

Average Error: 29.7 → 0.5

Time: 16.0min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -37.800190631006544:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{x \cdot \left(a + a\right)} + 1 \cdot \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{\frac{-1}{4} \cdot \left(a \cdot x\right)}{\frac{1}{6} - \frac{\frac{\frac{1}{2}}{a}}{x}}\\ \end{array}\]

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)) <= -37.800190631006544)) {
		VAR = (((double) (((double) pow(((double) exp(((double) (a * x)))), 3.0)) - ((double) pow(1.0, 3.0)))) / ((double) (((double) exp(((double) (x * ((double) (a + a)))))) + ((double) (1.0 * ((double) (1.0 + ((double) exp(((double) (a * x)))))))))));
	} else {
		VAR = ((double) (((double) (x * a)) + (((double) (-0.25 * ((double) (a * x)))) / ((double) (0.16666666666666666 - ((0.5 / a) / x))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.7
Target	0.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -37.800190631006544

   1. Initial program 0

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

   3. Applied flip3-- 0

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

   4. Simplified 0

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

   if -37.800190631006544 < (* a x)

   1. Initial program 44.1

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

   2. Taylor expanded around 0 14.4

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

   3. Simplified 4.7

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

   5. Applied flip-+ 4.7

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

   6. Applied associate-*r/ 4.7

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

   7. Applied associate-*r/ 4.7

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

   8. Simplified 0.7

      \[\leadsto a \cdot \left(x + \frac{\color{blue}{\left({\left(a \cdot x\right)}^{3} \cdot \frac{1}{36} + \frac{-1}{4} \cdot \left(a \cdot x\right)\right) \cdot x}}{\frac{1}{6} \cdot \left(a \cdot x\right) - \frac{1}{2}}\right)\]
   9. Using strategy rm

   10. Applied distribute-rgt-in 0.7

       \[\leadsto \color{blue}{x \cdot a + \frac{\left({\left(a \cdot x\right)}^{3} \cdot \frac{1}{36} + \frac{-1}{4} \cdot \left(a \cdot x\right)\right) \cdot x}{\frac{1}{6} \cdot \left(a \cdot x\right) - \frac{1}{2}} \cdot a}\]

   11. Simplified 0.7

       \[\leadsto x \cdot a + \color{blue}{\frac{{\left(a \cdot x\right)}^{3} \cdot \frac{1}{36} + \frac{-1}{4} \cdot \left(a \cdot x\right)}{\frac{1}{6} - \frac{\frac{\frac{1}{2}}{a}}{x}}}\]

   12. Taylor expanded around 0 0.7

       \[\leadsto x \cdot a + \frac{\color{blue}{\frac{-1}{4} \cdot \left(a \cdot x\right)}}{\frac{1}{6} - \frac{\frac{\frac{1}{2}}{a}}{x}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -37.800190631006544:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{x \cdot \left(a + a\right)} + 1 \cdot \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{\frac{-1}{4} \cdot \left(a \cdot x\right)}{\frac{1}{6} - \frac{\frac{\frac{1}{2}}{a}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))