expax (section 3.5)

Average Error: 28.9 → 0.4

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.369328679075431043:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(x \cdot a\right) \cdot \left(x \cdot a\right), \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -0.36932867907543104)) {
		VAR = (exp((a * x)) - 1.0);
	} else {
		VAR = fma(0.5, ((x * a) * (x * a)), fma(0.16666666666666666, pow((x * a), 3.0), (a * x)));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.36932867907543104

   1. Initial program 0.0

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

   if -0.36932867907543104 < (* a x)

   1. Initial program 43.8

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

   2. Taylor expanded around 0 14.6

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

   3. Simplified 14.6

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

   5. Applied pow-prod-down 8.4

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

   6. Simplified 8.4

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

   8. Applied unpow2 8.4

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

   9. Applied unpow2 8.4

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

   10. Applied unswap-sqr 0.7

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

   11. Simplified 0.7

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

   12. Simplified 0.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.369328679075431043:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(x \cdot a\right) \cdot \left(x \cdot a\right), \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))