expax (section 3.5)

Average Error: 29.4 → 9.4

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.252602566255028 \cdot 10^{-12}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}^{3}}}}{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -1.252602566255028e-12)) {
		VAR = ((cbrt((exp((a * x)) - 1.0)) * (cbrt(((exp((a * x)) * exp((a * x))) - (1.0 * 1.0))) / cbrt((exp((a * x)) + 1.0)))) * (cbrt(cbrt(pow((pow(exp((a * x)), 3.0) - pow(1.0, 3.0)), 3.0))) / cbrt(((exp((a * x)) * exp((a * x))) + ((1.0 * 1.0) + (exp((a * x)) * 1.0))))));
	} else {
		VAR = fma(0.5, (pow(a, 2.0) * pow(x, 2.0)), fma(0.16666666666666666, (pow(a, 3.0) * pow(x, 3.0)), (a * x)));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
Target	0.2
Herbie	9.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.252602566255028e-12

   1. Initial program 0.6

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

   3. Applied add-cube-cbrt 0.6

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

   5. Applied flip3-- 0.6

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

   6. Applied cbrt-div 0.6

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

   8. Applied flip-- 0.6

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

   9. Applied cbrt-div 0.6

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

   11. Applied add-cbrt-cube 0.6

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

   12. Simplified 0.6

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

   if -1.252602566255028e-12 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 14.0

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

   3. Simplified 14.0

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.252602566255028 \cdot 10^{-12}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{{\left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}^{3}}}}{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))