expax (section 3.5)

Average Error: 29.6 → 0.8

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -5.2843934468181229 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\log \left(\sqrt{e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}}\right) + \log \left(\sqrt{e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left(1 - \sqrt{1}\right) + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

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)) <= -5.284393446818123e-05)) {
		VAR = ((double) (((double) (((double) sqrt(((double) exp(((double) (a * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) log(((double) sqrt(((double) exp(((double) (((double) sqrt(((double) exp(((double) (a * x)))))) - ((double) sqrt(1.0)))))))))) + ((double) log(((double) sqrt(((double) exp(((double) (((double) sqrt(((double) exp(((double) (a * x)))))) - ((double) sqrt(1.0))))))))))))));
	} else {
		VAR = ((double) (((double) (((double) sqrt(((double) exp(((double) (a * x)))))) + ((double) sqrt(1.0)))) * ((double) (((double) (1.0 - ((double) sqrt(1.0)))) + ((double) (0.5 * ((double) (a * x))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -5.2843934468181229e-5

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

      \[\leadsto e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

   4. Applied add-sqr-sqrt 0.1

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\]

   5. Applied difference-of-squares 0.1

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

   7. Applied add-log-exp 0.1

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

   8. Applied add-log-exp 0.1

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

   9. Applied diff-log 0.1

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

   10. Simplified 0.1

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

   12. Applied add-sqr-sqrt 0.1

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

   13. Applied log-prod 0.1

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

   if -5.2843934468181229e-5 < (* a x)

   1. Initial program 44.6

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

   3. Applied add-sqr-sqrt 44.6

      \[\leadsto e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

   4. Applied add-sqr-sqrt 44.6

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\]

   5. Applied difference-of-squares 44.6

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

   7. Applied add-log-exp 44.6

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

   8. Applied add-log-exp 44.8

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

   9. Applied diff-log 44.8

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

   10. Simplified 44.8

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

   11. Taylor expanded around 0 45.9

       \[\leadsto \left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right) + 1\right)\right) - \sqrt{1}\right)}\]

   12. Simplified 4.5

       \[\leadsto \left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \color{blue}{\left(\left(1 - \sqrt{1}\right) + x \cdot \left(\frac{1}{2} \cdot a + \left(\frac{1}{8} \cdot {a}^{2}\right) \cdot x\right)\right)}\]

   13. Taylor expanded around 0 1.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -5.2843934468181229 \cdot 10^{-5}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\log \left(\sqrt{e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}}\right) + \log \left(\sqrt{e^{\sqrt{e^{a \cdot x}} - \sqrt{1}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left(1 - \sqrt{1}\right) + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(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))