expax (section 3.5)

Average Error: 29.6 → 0.4

Time: 12.3s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.3848736434257956 \cdot 10^{-5}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\\ \end{array}\]

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)) <= -1.3848736434257956e-05)) {
		VAR = ((double) log(((double) exp(((double) (((double) exp(((double) (a * x)))) - 1.0))))));
	} else {
		VAR = ((double) (((double) (a * x)) + ((double) (0.5 * ((double) pow(((double) (a * x)), 2.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -1.3848736434257956e-5

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied add-log-exp 0.1

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

   5. Applied diff-log 0.1

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

   6. Simplified 0.1

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

   if -1.3848736434257956e-5 < (* a x)

   1. Initial program 45.1

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

   3. Applied add-log-exp 45.1

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

   4. Applied add-log-exp 45.2

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

   5. Applied diff-log 45.3

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

   6. Simplified 45.2

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

   7. Taylor expanded around 0 15.4

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

   8. Simplified 8.1

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

   9. Taylor expanded around 0 8.9

      \[\leadsto \color{blue}{a \cdot x + \frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right)}\]
   10. Using strategy rm

   11. Applied pow-prod-down 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.3848736434257956 \cdot 10^{-5}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \frac{1}{2} \cdot {\left(a \cdot x\right)}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))