expax (section 3.5)

Average Error: 29.3 → 0.4

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.55569637749741916 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, a \cdot x\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -3.555696377497419e-05)) {
		VAR = fma(sqrt(exp((a * x))), sqrt(exp((a * x))), -1.0);
	} else {
		VAR = fma(0.5, pow((x * a), 2.0), (a * x));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
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) < -3.555696377497419e-05

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

   4. Applied fma-neg 0.1

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

   if -3.555696377497419e-05 < (* a x)

   1. Initial program 44.4

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

   2. Taylor expanded around 0 14.0

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

   3. Simplified 14.0

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

   4. Taylor expanded around 0 8.0

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

   5. Simplified 8.0

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

   7. Applied pow-prod-down 0.6

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

   8. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.55569637749741916 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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))