expax (section 3.5)

Average Error: 29.4 → 9.8

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.78394354237175143 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \frac{\log \left(e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right)}{2} \cdot \left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

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)) <= -4.7839435423717514e-18)) {
		VAR = ((double) (((double) log(((double) sqrt(((double) exp(((double) (((double) exp(((double) (a * x)))) - 1.0)))))))) + ((double) (((double) (((double) log(((double) exp(((double) (((double) sqrt(((double) exp(((double) (a * x)))))) - ((double) sqrt(1.0)))))))) / 2.0)) * ((double) (((double) sqrt(((double) exp(((double) (a * x)))))) + ((double) sqrt(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	29.4
Target	0.2
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) < -4.78394354237175143e-18

   1. Initial program 1.1

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

   3. Applied add-log-exp 1.1

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

   4. Applied add-log-exp 1.1

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

   5. Applied diff-log 1.1

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

   6. Simplified 1.1

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

   8. Applied add-sqr-sqrt 1.1

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

   9. Applied log-prod 1.1

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

   11. Applied add-sqr-sqrt 1.1

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

   12. Applied add-sqr-sqrt 1.1

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

   13. Applied difference-of-squares 1.1

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

   14. Applied exp-prod 1.1

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

   15. Applied sqrt-pow1 1.1

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

   16. Applied log-pow 1.1

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

   17. Simplified 1.1

       \[\leadsto \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \frac{\sqrt{e^{a \cdot x}} - \sqrt{1}}{2} \cdot \color{blue}{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right)}\]
   18. Using strategy rm

   19. Applied add-log-exp 1.1

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

   20. Applied add-log-exp 1.1

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

   21. Applied diff-log 1.1

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

   22. Simplified 1.1

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

   if -4.78394354237175143e-18 < (* a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 14.5

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

   3. Simplified 14.5

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4.78394354237175143 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \frac{\log \left(e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right)}{2} \cdot \left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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))