expax (section 3.5)

Average Error: 28.9 → 8.8

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.0066608795785513 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;a \cdot x \le 5.9195056646387966 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

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)) <= -2.0066608795785513e-08)) {
		VAR = ((double) cbrt(((double) pow(((double) cbrt(((double) pow(((double) cbrt(((double) pow(((double) cbrt(((double) pow(((double) (((double) exp(((double) (a * x)))) - 1.0)), 3.0)))), 3.0)))), 3.0)))), 3.0))));
	} else {
		double VAR_1;
		if ((((double) (a * x)) <= 5.9195056646387966e-33)) {
			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) pow(((double) exp(((double) (a * x)))), 3.0)) - ((double) pow(1.0, 3.0)))) / ((double) (((double) (((double) exp(((double) (a * x)))) * ((double) (((double) exp(((double) (a * x)))) + 1.0)))) + ((double) (1.0 * 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	8.8

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -2.0066608795785513e-8

   1. Initial program 0.2

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

   3. Applied add-cbrt-cube 0.2

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

   4. Simplified 0.2

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

   6. Applied add-cbrt-cube 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cbrt-cube 0.2

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

   10. Simplified 0.2

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

   12. Applied add-cbrt-cube 0.2

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

   13. Simplified 0.2

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

   if -2.0066608795785513e-8 < (* a x) < 5.9195056646387966e-33

   1. Initial program 44.7

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

   2. Taylor expanded around 0 12.1

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

   3. Simplified 12.1

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

   if 5.9195056646387966e-33 < (* a x)

   1. Initial program 39.1

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

   3. Applied flip3-- 39.5

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

   4. Simplified 39.5

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.0066608795785513 \cdot 10^{-8}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;a \cdot x \le 5.9195056646387966 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020164 
(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))