expax (section 3.5)

Average Error: 29.3 → 9.0

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -1.9961401647367767e-21)) {
		temp = cbrt(pow((cbrt(((exp((a * x)) - 1.0) * log(exp((exp((a * x)) - 1.0))))) * cbrt(fma(sqrt(exp((a * x))), sqrt(exp((a * x))), -1.0))), 3.0));
	} else {
		temp = 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 temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	9.0

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.9961401647367767e-21

   1. Initial program 2.0

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

   3. Applied add-cbrt-cube 2.0

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

   4. Simplified 2.0

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

   6. Applied add-cube-cbrt 2.0

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

   7. Simplified 2.0

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

   9. Applied add-sqr-sqrt 2.0

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

   10. Applied fma-neg 2.0

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

   12. Applied add-log-exp 2.0

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

   13. Applied add-log-exp 2.0

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

   14. Applied diff-log 2.0

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

   15. Simplified 2.0

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

   if -1.9961401647367767e-21 < (* a x)

   1. Initial program 44.5

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 13.0

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.9961401647367767 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))