expax (section 3.5)

Average Error: 30.0 → 9.8

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.1104584441958173 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \frac{{\left(e^{a \cdot x}\right)}^{2} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \frac{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

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)) <= -3.110458444195817e-05)) {
		VAR = ((double) (((double) cbrt(((double) (((double) (((double) exp(((double) (a * x)))) - 1.0)) * ((double) (((double) (((double) pow(((double) exp(((double) (a * x)))), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (((double) exp(((double) (a * x)))) + 1.0)))))))) * ((double) (((double) cbrt(((double) (((double) (((double) exp(((double) (a * x)))) * ((double) exp(((double) (a * x)))))) - ((double) (1.0 * 1.0)))))) / ((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0))))))));
	} else {
		VAR = ((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))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	30.0
Target	0.1
Herbie	9.8

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -3.1104584441958173e-5

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube 0.1

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

   4. Simplified 0.1

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

   6. Applied add-cube-cbrt 0.1

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

   7. Applied unpow-prod-down 0.1

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

   8. Applied cbrt-prod 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

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

   12. Applied flip-- 0.1

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

   13. Applied cbrt-div 0.1

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

   15. Applied flip-- 0.1

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

   16. Simplified 0.1

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

   if -3.1104584441958173e-5 < (* a x)

   1. Initial program 45.2

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

   2. Taylor expanded around 0 14.8

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

   3. Simplified 14.8

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.1104584441958173 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \frac{{\left(e^{a \cdot x}\right)}^{2} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \frac{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(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))