expax (section 3.5)

Average Error: 29.4 → 0.4

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 8.9719548727940208 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -5.846622341013729)) {
		temp = (((cbrt(exp((a * x))) * cbrt(exp((a * x)))) * cbrt(exp((a * x)))) - 1.0);
	} else {
		double temp_1;
		if (((a * x) <= 8.971954872794021e-08)) {
			temp_1 = fma(0.5, pow((x * a), 2.0), (x * a));
		} else {
			temp_1 = (sqrt((exp((a * x)) - 1.0)) * sqrt((exp((a * x)) - 1.0)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
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 3 regimes

2. if (* a x) < -5.846622341013729

   1. Initial program 0

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

   3. Applied add-cube-cbrt 0

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

   if -5.846622341013729 < (* a x) < 8.971954872794021e-08

   1. Initial program 44.5

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

   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 7.9

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

   5. Simplified 7.9

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

   7. Applied pow-prod-down 0.5

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

   8. Simplified 0.5

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

   if 8.971954872794021e-08 < (* a x)

   1. Initial program 7.6

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

   3. Applied add-sqr-sqrt 7.6

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.8466223410137292:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{elif}\;a \cdot x \le 8.9719548727940208 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {\left(x \cdot a\right)}^{2}, x \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))