expax (section 3.5)

Average Error: 29.5 → 9.2

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.43868542208870495 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}\right)}{\left(\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666652 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

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)) <= -1.438685422088705e-13)) {
		VAR = ((double) (((double) log(((double) exp(((double) (((double) pow(((double) exp(((double) (((double) (a * x)) * 3.0)))), 3.0)) - ((double) pow(((double) pow(1.0, 3.0)), 3.0)))))))) / ((double) (((double) (((double) (((double) exp(((double) (((double) (((double) (a * x)) * 3.0)) + ((double) (((double) (a * x)) * 3.0)))))) + ((double) pow(1.0, 6.0)))) + ((double) (((double) exp(((double) (((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))))))));
	} else {
		VAR = ((double) (((double) (0.5 * ((double) (((double) pow(a, 2.0)) * ((double) pow(x, 2.0)))))) + ((double) (((double) (0.16666666666666652 * ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0)))))) + ((double) (1.0 * ((double) (a * x))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	9.2

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.43868542208870495e-13

   1. Initial program 0.7

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

   3. Applied flip3-- 0.7

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

   4. Simplified 0.7

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

   6. Applied pow-exp 0.7

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

   8. Applied flip3-- 0.7

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

   9. Applied associate-/l/ 0.7

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

   10. Simplified 0.7

       \[\leadsto \frac{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\color{blue}{\left(\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}}\]
   11. Using strategy rm

   12. Applied add-log-exp 0.7

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

   13. Applied add-log-exp 0.7

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

   14. Applied diff-log 0.7

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

   15. Simplified 0.7

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

   if -1.43868542208870495e-13 < (* a x)

   1. Initial program 44.4

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

   3. Applied flip3-- 44.5

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

   4. Simplified 44.5

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

   5. Taylor expanded around 0 13.6

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666652 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.43868542208870495 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}\right)}{\left(\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666652 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))