expax (section 3.5)

Average Error: 29.1 → 9.5

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97111139828814866 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(e^{{\left(e^{a \cdot x}\right)}^{2} - 1 \cdot 1}\right)}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

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

↓

double code(double a, double x) {
	double VAR;
	if ((((double) (a * x)) <= -1.9711113982881487e-11)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))) * ((double) cbrt(((double) (((double) (((double) sqrt(((double) exp(((double) (a * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) exp(((double) (a * x)))))) - ((double) sqrt(1.0)))))))))) * ((double) (((double) cbrt(((double) log(((double) exp(((double) (((double) pow(((double) exp(((double) (a * x)))), 2.0)) - ((double) (1.0 * 1.0)))))))))) / ((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0))))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (a + ((double) (((double) (0.5 * ((double) pow(a, 2.0)))) * x)))))) + ((double) (0.16666666666666666 * ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
Target	0.2
Herbie	9.5

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.9711113982881487e-11

   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 flip-- 0.5

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

   6. Applied cbrt-div 0.5

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

   8. Applied add-log-exp 0.5

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

   9. Applied add-log-exp 0.5

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

   10. Applied diff-log 0.5

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

   11. Simplified 0.5

       \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \frac{\sqrt[3]{\log \color{blue}{\left(e^{{\left(e^{a \cdot x}\right)}^{2} - 1 \cdot 1}\right)}}}{\sqrt[3]{e^{a \cdot x} + 1}}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 0.5

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

   14. Applied add-sqr-sqrt 0.5

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

   15. Applied difference-of-squares 0.5

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

   if -1.9711113982881487e-11 < (* a x)

   1. Initial program 44.3

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

   2. Taylor expanded around 0 14.3

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

   3. Simplified 14.3

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97111139828814866 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)}\right) \cdot \frac{\sqrt[3]{\log \left(e^{{\left(e^{a \cdot x}\right)}^{2} - 1 \cdot 1}\right)}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))