expax (section 3.5)

Average Error: 28.8 → 9.7

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.6776443470702676 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(e^{a \cdot x} - 1\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)\\ \end{array}\]

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)) <= -0.00026776443470702676)) {
		VAR = ((double) cbrt(((double) cbrt(((double) pow(((double) pow(((double) (((double) exp(((double) (a * x)))) - 1.0)), 3.0)), 3.0))))));
	} else {
		VAR = ((double) (((double) (0.5 * ((double) (((double) pow(a, 2.0)) * ((double) pow(x, 2.0)))))) + ((double) (((double) (0.16666666666666666 * ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0)))))) + ((double) (a * x))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.8
Target	0.2
Herbie	9.7

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.00026776443470702676

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube 0.0

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

   4. Simplified 0.0

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

   6. Applied add-cbrt-cube 0.0

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

   7. Simplified 0.0

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

   if -0.00026776443470702676 < (* a x)

   1. Initial program 43.5

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

   3. Applied add-cbrt-cube 43.6

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

   4. Simplified 43.6

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

   6. Applied add-cbrt-cube 43.6

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

   7. Simplified 43.6

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

   8. 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. Recombined 2 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.6776443470702676 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(e^{a \cdot x} - 1\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))