expax (section 3.5)

Average Error: 29.8 → 9.3

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.9833472177783894 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3}\right)}^{3} - {\left({\left({1}^{3}\right)}^{3}\right)}^{3}}{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{6} + {\left({1}^{3}\right)}^{6}\right) + {\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} \cdot {\left({1}^{3}\right)}^{3}}}{\left(\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -2.9833472177783894e-09)) {
		temp = (((pow(pow(exp(((a * x) * 3.0)), 3.0), 3.0) - pow(pow(pow(1.0, 3.0), 3.0), 3.0)) / ((pow(exp(((a * x) * 3.0)), 6.0) + pow(pow(1.0, 3.0), 6.0)) + (pow(exp(((a * x) * 3.0)), 3.0) * pow(pow(1.0, 3.0), 3.0)))) / (((exp((((a * x) * 3.0) + ((a * x) * 3.0))) + pow(1.0, 6.0)) + (exp(((a * x) * 3.0)) * pow(1.0, 3.0))) * ((exp((a * x)) * (exp((a * x)) + 1.0)) + (1.0 * 1.0))));
	} else {
		temp = ((x * (a + ((0.5 * pow(a, 2.0)) * x))) + (0.16666666666666666 * (pow(a, 3.0) * pow(x, 3.0))));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
Target	0.2
Herbie	9.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -2.9833472177783894e-09

   1. Initial program 0.3

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

   3. Applied flip3-- 0.3

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

   4. Simplified 0.3

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

   6. Applied pow-exp 0.3

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

   8. Applied flip3-- 0.3

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

   9. Applied associate-/l/ 0.3

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

   10. Simplified 0.3

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

   12. Applied flip3-- 0.3

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

   13. Simplified 0.3

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

   if -2.9833472177783894e-09 < (* a x)

   1. Initial program 44.8

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

   2. Taylor expanded around 0 13.9

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

   3. Simplified 13.9

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.9833472177783894 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3}\right)}^{3} - {\left({\left({1}^{3}\right)}^{3}\right)}^{3}}{\left({\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{6} + {\left({1}^{3}\right)}^{6}\right) + {\left(e^{\left(a \cdot x\right) \cdot 3}\right)}^{3} \cdot {\left({1}^{3}\right)}^{3}}}{\left(\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + {1}^{6}\right) + e^{\left(a \cdot x\right) \cdot 3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))