expax (section 3.5)

Average Error: 29.3 → 9.0

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\right)} \cdot \sqrt[3]{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}{\sqrt[3]{\left(e^{a \cdot x} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right)} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -1.9961401647367767e-21)) {
		temp = ((cbrt((((exp((a * x)) * exp((a * x))) - (1.0 * 1.0)) * ((pow(sqrt(exp((a * x))), 3.0) + pow(sqrt(1.0), 3.0)) * (sqrt(exp((a * x))) - sqrt(1.0))))) * cbrt((pow(exp((a * x)), 3.0) - pow(1.0, 3.0)))) / (cbrt(((exp((a * x)) + 1.0) * ((sqrt(exp((a * x))) * sqrt(exp((a * x)))) + ((sqrt(1.0) * sqrt(1.0)) - (sqrt(exp((a * x))) * sqrt(1.0)))))) * cbrt(((exp((a * x)) * exp((a * x))) + ((1.0 * 1.0) + (exp((a * x)) * 1.0))))));
	} else {
		temp = ((0.5 * (pow(a, 2.0) * pow(x, 2.0))) + ((0.16666666666666666 * (pow(a, 3.0) * pow(x, 3.0))) + (a * x)));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	9.0

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.9961401647367767e-21

   1. Initial program 2.0

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

   3. Applied add-cbrt-cube 2.0

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

   4. Simplified 2.0

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

   6. Applied add-cube-cbrt 2.0

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

   7. Simplified 2.0

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

   9. Applied add-sqr-sqrt 2.0

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

   10. Applied add-sqr-sqrt 2.0

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

   11. Applied difference-of-squares 2.0

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

   13. Applied flip3-- 2.0

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

   14. Applied cbrt-div 2.0

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

   15. Applied flip3-+ 2.0

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

   16. Applied associate-*l/ 2.0

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

   17. Applied flip-- 2.0

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

   18. Applied frac-times 2.0

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

   19. Applied cbrt-div 2.0

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

   20. Applied frac-times 2.0

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

   21. Applied cube-div 2.0

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

   22. Applied cbrt-div 2.0

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

   23. Simplified 2.0

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

   24. Simplified 2.0

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

   if -1.9961401647367767e-21 < (* a x)

   1. Initial program 44.5

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

   3. Applied add-cbrt-cube 44.6

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

   4. Simplified 44.6

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

   6. Applied add-cube-cbrt 44.6

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

   7. Simplified 44.6

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

   8. Taylor expanded around 0 13.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\right)} \cdot \sqrt[3]{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}}{\sqrt[3]{\left(e^{a \cdot x} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{a \cdot x}} \cdot \sqrt{1}\right)\right)} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))