expax (section 3.5)

Average Error: 28.9 → 8.9

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.57004817984858424 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + {1}^{\frac{3}{2}}\right) \cdot \frac{\frac{{\left({\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3}\right)}^{3} - {\left({\left({1}^{\frac{3}{2}}\right)}^{3}\right)}^{3}}{\left({\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{6} + {\left({1}^{\frac{3}{2}}\right)}^{6}\right) + {\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} \cdot {\left({1}^{\frac{3}{2}}\right)}^{3}}}{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right) + \sqrt{{\left(e^{a \cdot x}\right)}^{3}} \cdot {1}^{\frac{3}{2}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -4.570048179848584e-12)) {
		VAR = (((sqrt(pow(exp((a * x)), 3.0)) + pow(1.0, 1.5)) * (((pow(pow(sqrt(pow(exp((a * x)), 3.0)), 3.0), 3.0) - pow(pow(pow(1.0, 1.5), 3.0), 3.0)) / ((pow(sqrt(pow(exp((a * x)), 3.0)), 6.0) + pow(pow(1.0, 1.5), 6.0)) + (pow(sqrt(pow(exp((a * x)), 3.0)), 3.0) * pow(pow(1.0, 1.5), 3.0)))) / ((pow(exp((a * x)), 3.0) + pow(1.0, 3.0)) + (sqrt(pow(exp((a * x)), 3.0)) * pow(1.0, 1.5))))) / ((exp((a * x)) * (exp((a * x)) + 1.0)) + (1.0 * 1.0)));
	} else {
		VAR = ((x * (a + ((0.5 * pow(a, 2.0)) * x))) + (0.16666666666666666 * (pow(a, 3.0) * pow(x, 3.0))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	8.9

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -4.570048179848584e-12

   1. Initial program 0.5

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

   3. Applied flip3-- 0.5

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

   4. Simplified 0.5

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

   6. Applied sqr-pow 0.5

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

   7. Applied add-sqr-sqrt 0.5

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

   8. Applied difference-of-squares 0.5

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

   9. Simplified 0.5

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

   10. Simplified 0.5

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

   12. Applied flip3-- 0.5

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

   13. Simplified 0.5

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

   15. Applied flip3-- 0.5

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

   16. Simplified 0.5

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

   if -4.570048179848584e-12 < (* a x)

   1. Initial program 43.6

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

   2. Taylor expanded around 0 13.2

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

   3. Simplified 13.2

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4.57004817984858424 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + {1}^{\frac{3}{2}}\right) \cdot \frac{\frac{{\left({\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3}\right)}^{3} - {\left({\left({1}^{\frac{3}{2}}\right)}^{3}\right)}^{3}}{\left({\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{6} + {\left({1}^{\frac{3}{2}}\right)}^{6}\right) + {\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} \cdot {\left({1}^{\frac{3}{2}}\right)}^{3}}}{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right) + \sqrt{{\left(e^{a \cdot x}\right)}^{3}} \cdot {1}^{\frac{3}{2}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(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))