Average Error: 29.7 → 0.6
Time: 2.8s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0839680346102975178:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0839680346102975178:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\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)) <= -0.08396803461029752)) {
		VAR = ((double) (((double) exp(((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) (x * ((double) (a + ((double) (a * ((double) (a * ((double) (x * 0.5))))))))));
	}
	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

Original29.7
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0839680346102975178

    1. Initial program 0.0

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

    if -0.0839680346102975178 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 15.1

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

    3. Simplified8.2

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

    5. Applied distribute-lft-in8.2

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

    6. Simplified8.0

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

    7. Taylor expanded around 0 9.1

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

    8. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0839680346102975178:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + a \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))