expax (section 3.5)

Average Error: 29.5 → 9.9

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.9941401337104723 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \mathbf{elif}\;a \cdot x \le 1.941994952414224 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \end{array}\]

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)) <= -4.994140133710472e-15)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))) * ((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))))) * ((double) (((double) (((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))))) * ((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))))))) * ((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0))))))))));
	} else {
		double VAR_1;
		if ((((double) (a * x)) <= 1.941994952414224e-53)) {
			VAR_1 = ((double) (((double) (x * ((double) (a + ((double) (((double) (0.5 * ((double) pow(a, 2.0)))) * x)))))) + ((double) (0.16666666666666666 * ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))) * ((double) cbrt(((double) (((double) (((double) sqrt(((double) exp(((double) (a * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) exp(((double) (a * x)))))) - ((double) sqrt(1.0)))))))))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))) * ((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))))))) * ((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -4.994140133710472e-15

   1. Initial program 1.1

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

   3. Applied add-cube-cbrt 1.1

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

   5. Applied add-cube-cbrt 1.1

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

   if -4.994140133710472e-15 < (* a x) < 1.941994952414224e-53

   1. Initial program 44.5

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

   2. Taylor expanded around 0 11.4

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

   3. Simplified 11.4

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

   if 1.941994952414224e-53 < (* a x)

   1. Initial program 48.1

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

   3. Applied add-cube-cbrt 48.1

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

   5. Applied add-cube-cbrt 48.1

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

   6. Applied cbrt-prod 48.1

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

   8. Applied add-sqr-sqrt 48.1

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

   9. Applied add-sqr-sqrt 48.2

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

   10. Applied difference-of-squares 48.2

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4.9941401337104723 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \mathbf{elif}\;a \cdot x \le 1.941994952414224 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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))