Average Error: 29.4 → 0.4
Time: 2.8s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00353149015182299643:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00353149015182299643:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{else}:\\
\;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\

\end{array}
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)) <= -0.0035314901518229964)) {
		VAR = ((double) (((double) exp(((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) (((double) (a * ((double) (x + ((double) (x * ((double) (a * ((double) (x * 0.5)))))))))) + ((double) (0.16666666666666666 * ((double) pow(((double) (a * x)), 3.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00353149015182299643

    1. Initial program 0.0

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

    if -0.00353149015182299643 < (* a x)

    1. Initial program 44.6

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

    2. Taylor expanded around 0 14.8

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

    3. Simplified7.9

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

    5. Applied distribute-lft-in7.9

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

    6. Simplified7.8

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

    7. Taylor expanded around inf 14.8

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

    8. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00353149015182299643:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))