expax (section 3.5)

Average Error: 30.1 → 10.2

Time: 21.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -7.654795418996283:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\frac{\left(-{\left(1 \cdot 1\right)}^{3}\right) + {\left(e^{a \cdot x}\right)}^{6}}{\mathsf{fma}\left(1 \cdot 1, \mathsf{fma}\left(1, 1, {\left(e^{a \cdot x}\right)}^{2}\right), {\left(e^{a \cdot x}\right)}^{4}\right)}\right)}^{3}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -7.654795418996283)) {
		temp = cbrt((pow(((-pow((1.0 * 1.0), 3.0) + pow(exp((a * x)), 6.0)) / fma((1.0 * 1.0), fma(1.0, 1.0, pow(exp((a * x)), 2.0)), pow(exp((a * x)), 4.0))), 3.0) / pow((exp((a * x)) + 1.0), 3.0)));
	} else {
		temp = fma(0.5, (pow(a, 2.0) * pow(x, 2.0)), fma(0.16666666666666663, (pow(a, 3.0) * pow(x, 3.0)), (1.0 * (a * x))));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	30.1
Target	0.2
Herbie	10.2

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -7.654795418996283

   1. Initial program 0

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

   3. Applied add-cbrt-cube 0.0

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

   4. Simplified 0.0

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

   6. Applied flip-- 0.0

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

   7. Applied cube-div 0.0

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

   9. Applied flip3-- 0.0

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

   10. Simplified 0.0

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

   11. Simplified 0.0

       \[\leadsto \sqrt[3]{\frac{{\left(\frac{\left(-{\left(1 \cdot 1\right)}^{3}\right) + {\left(e^{a \cdot x}\right)}^{6}}{\color{blue}{\mathsf{fma}\left(1 \cdot 1, \mathsf{fma}\left(1, 1, {\left(e^{a \cdot x}\right)}^{2}\right), {\left(e^{a \cdot x}\right)}^{4}\right)}}\right)}^{3}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}\]

   if -7.654795418996283 < (* a x)

   1. Initial program 44.6

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

   3. Applied add-cbrt-cube 44.6

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

   4. Simplified 44.6

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

   6. Applied flip-- 44.6

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

   7. Applied cube-div 44.7

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

   8. Taylor expanded around 0 15.1

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

   9. Simplified 15.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -7.654795418996283:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\frac{\left(-{\left(1 \cdot 1\right)}^{3}\right) + {\left(e^{a \cdot x}\right)}^{6}}{\mathsf{fma}\left(1 \cdot 1, \mathsf{fma}\left(1, 1, {\left(e^{a \cdot x}\right)}^{2}\right), {\left(e^{a \cdot x}\right)}^{4}\right)}\right)}^{3}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(0.16666666666666663, {a}^{3} \cdot {x}^{3}, 1 \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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))