expax (section 3.5)

Average Error: 28.9 → 8.9

Time: 4.9s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double a, double x) {
	double VAR;
	if (((a * x) <= -4.570048179848584e-12)) {
		VAR = (((cbrt((exp((a * x)) - 1.0)) * cbrt((sqrt(exp((a * x))) + sqrt(1.0)))) * cbrt((sqrt(exp((a * x))) - sqrt(1.0)))) * cbrt(log(exp((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	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 add-cube-cbrt 0.5

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

   5. Applied add-sqr-sqrt 0.5

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

   6. Applied add-sqr-sqrt 0.5

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

   7. Applied difference-of-squares 0.5

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

   8. Applied cbrt-prod 0.5

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

   9. Applied associate-*r* 0.5

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

   11. Applied add-log-exp 0.5

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

   12. Applied add-log-exp 0.5

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

   13. Applied diff-log 0.5

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

   14. Simplified 0.5

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

   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

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

4. Final simplification 8.9

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

Reproduce

herbie shell --seed 2020091 +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))