expax (section 3.5)

Average Error: 29.9 → 3.2

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.003398217270803261:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array}\]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

↓

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.003398217270803261)
   (- (exp (* a x)) 1.0)
   (+ (* a x) (* a (* (* a 0.5) (* x x))))))

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)) <= -0.003398217270803261)) {
		VAR = ((double) (((double) exp(((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) (((double) (a * x)) + ((double) (a * ((double) (((double) (a * 0.5)) * ((double) (x * x))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.9
Target	0.2
Herbie	3.2

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0033982172708032611

   1. Initial program 0.0

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

   if -0.0033982172708032611 < (* a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 14.4

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

   3. Simplified 7.5

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]

   4. Taylor expanded around 0 8.1

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x}\]

   5. Simplified 4.8

      \[\leadsto \color{blue}{a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.003398217270803261:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))