expax (section 3.5)

Average Error: 29.5 → 9.3

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.8243751882626226 \cdot 10^{-16}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{{\left({\left(e^{a \cdot x} \cdot e^{a \cdot x}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}\right)}^{3}}{{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 + {\left(e^{a \cdot x}\right)}^{2}\right) + {\left(e^{a \cdot x}\right)}^{2} \cdot {\left(e^{a \cdot x}\right)}^{2}\right)}^{3}}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -1.8243751882626226e-16)) {
		VAR = cbrt(((pow((pow((exp((a * x)) * exp((a * x))), 3.0) - pow((1.0 * 1.0), 3.0)), 3.0) / pow((((1.0 * 1.0) * ((1.0 * 1.0) + pow(exp((a * x)), 2.0))) + (pow(exp((a * x)), 2.0) * pow(exp((a * x)), 2.0))), 3.0)) / pow((exp((a * x)) + 1.0), 3.0)));
	} 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	29.5
Target	0.1
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) < -1.8243751882626226e-16

   1. Initial program 0.9

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

   3. Applied add-cbrt-cube 0.9

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

   4. Simplified 0.9

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

   6. Applied flip-- 0.9

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

   7. Applied cube-div 0.9

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

   9. Applied flip3-- 0.9

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

   10. Applied cube-div 0.9

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

   11. Simplified 0.9

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

   if -1.8243751882626226e-16 < (* a x)

   1. Initial program 44.7

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

   2. Taylor expanded around 0 13.8

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

   3. Simplified 13.8

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.8243751882626226 \cdot 10^{-16}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{{\left({\left(e^{a \cdot x} \cdot e^{a \cdot x}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}\right)}^{3}}{{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 + {\left(e^{a \cdot x}\right)}^{2}\right) + {\left(e^{a \cdot x}\right)}^{2} \cdot {\left(e^{a \cdot x}\right)}^{2}\right)}^{3}}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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))