expax (section 3.5)

Average Error: 29.2 → 9.4

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{\left(\log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right) + \log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right)\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -3.569795479894664e-07)) {
		temp = ((cbrt(((log(sqrt(sqrt(exp((exp((a * x)) - 1.0))))) + log(sqrt(sqrt(exp((exp((a * x)) - 1.0)))))) + log(sqrt(exp((exp((a * x)) - 1.0)))))) * cbrt((exp((a * x)) - 1.0))) * cbrt((exp((a * x)) - 1.0)));
	} else {
		double temp_1;
		if (((a * x) <= 2.9747332644534994e-19)) {
			temp_1 = fma(0.5, (pow(a, 2.0) * pow(x, 2.0)), fma(0.16666666666666666, (pow(a, 3.0) * pow(x, 3.0)), (a * x)));
		} else {
			temp_1 = ((cbrt((log(sqrt(exp((exp((a * x)) - 1.0)))) + ((cbrt(log(sqrt(exp((exp((a * x)) - 1.0))))) * cbrt(log(sqrt(exp((exp((a * x)) - 1.0)))))) * cbrt(log(sqrt(exp((exp((a * x)) - 1.0)))))))) * cbrt((exp((a * x)) - 1.0))) * cbrt((exp((a * x)) - 1.0)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
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 3 regimes

2. if (* a x) < -3.569795479894664e-07

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.2

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

   5. Applied add-log-exp 0.2

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

   6. Applied add-log-exp 0.2

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

   7. Applied diff-log 0.2

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

   8. Simplified 0.2

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

   10. Applied add-sqr-sqrt 0.2

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

   11. Applied log-prod 0.2

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

   13. Applied add-sqr-sqrt 0.2

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

   14. Applied sqrt-prod 0.2

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

   15. Applied log-prod 0.2

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

   if -3.569795479894664e-07 < (* a x) < 2.9747332644534994e-19

   1. Initial program 45.1

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

   2. Taylor expanded around 0 13.5

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

   3. Simplified 13.5

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

   if 2.9747332644534994e-19 < (* a x)

   1. Initial program 25.4

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

   3. Applied add-cube-cbrt 25.5

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

   5. Applied add-log-exp 25.5

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

   6. Applied add-log-exp 31.7

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

   7. Applied diff-log 31.8

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

   8. Simplified 31.8

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

   10. Applied add-sqr-sqrt 31.9

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

   11. Applied log-prod 31.9

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

   13. Applied add-cube-cbrt 31.9

       \[\leadsto \left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{\left(\log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right) + \log \left(\sqrt{\sqrt{e^{e^{a \cdot x} - 1}}}\right)\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \left(\sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)} \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))