expax (section 3.5)

Average Error: 29.4 → 0.8

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4385658.6985369977:\\ \;\;\;\;{e}^{\left(a \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{1}\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot {\left(x \cdot a\right)}^{3}\\ \end{array}\]

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)) <= -4385658.698536998)) {
		VAR = ((double) (((double) pow(((double) M_E), ((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) (((double) (x * ((double) (a + ((double) (((double) (0.5 * ((double) pow(a, 1.0)))) * ((double) (a * x)))))))) + ((double) (0.16666666666666666 * ((double) pow(((double) (x * a)), 3.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
Target	0.2
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) < -4385658.698536998

   1. Initial program 0

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

   3. Applied *-un-lft-identity 0

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

   4. Applied exp-prod 0

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

   5. Simplified 0

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

   if -4385658.698536998 < (* a x)

   1. Initial program 43.3

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

   2. Taylor expanded around 0 14.3

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

   3. Simplified 14.3

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

   5. Applied pow1 14.3

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

   6. Applied pow-pow 14.3

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

   7. Applied pow1 14.3

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

   8. Applied pow-pow 14.3

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

   9. Applied pow-prod-down 4.9

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

   10. Simplified 4.9

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

   12. Applied sqr-pow 4.9

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

   13. Applied associate-*r* 4.9

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

   14. Applied associate-*l* 1.2

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

   15. Simplified 1.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4385658.6985369977:\\ \;\;\;\;{e}^{\left(a \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{1}\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot {\left(x \cdot a\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))