Average Error: 29.0 → 3.3
Time: 2.6s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.283265312368245 \cdot 10^{-05}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{elif}\;a \cdot x \leq -1.0908319111938347 \cdot 10^{-40} \lor \neg \left(a \cdot x \leq -4.132494059883518 \cdot 10^{-139}\right):\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -1.283265312368245 \cdot 10^{-05}:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{elif}\;a \cdot x \leq -1.0908319111938347 \cdot 10^{-40} \lor \neg \left(a \cdot x \leq -4.132494059883518 \cdot 10^{-139}\right):\\
\;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \left(a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\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)) <= -1.283265312368245e-05)) {
		VAR = ((double) (((double) exp(((double) (a * x)))) - 1.0));
	} else {
		double VAR_1;
		if (((((double) (a * x)) <= -1.0908319111938347e-40) || !(((double) (a * x)) <= -4.132494059883518e-139))) {
			VAR_1 = ((double) (((double) (a * x)) + ((double) (a * ((double) (((double) (a * 0.5)) * ((double) (x * x))))))));
		} else {
			VAR_1 = ((double) (x * ((double) (a + ((double) (((double) (a * ((double) (a * ((double) (x * 0.5)))))) + ((double) (x * ((double) (x * ((double) (0.16666666666666666 * ((double) pow(a, 3.0))))))))))))));
		}
		VAR = VAR_1;
	}
	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

Original29.0
Target0.2
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -1.28326531236824496e-5

    1. Initial program 0.1

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

    if -1.28326531236824496e-5 < (* a x) < -1.0908319111938347e-40 or -4.13249405988351797e-139 < (* a x)

    1. Initial program 41.4

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

    2. Taylor expanded around 0 11.7

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified6.5

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]

    4. Taylor expanded around 0 6.6

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x}\]

    5. Simplified3.7

      \[\leadsto \color{blue}{a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)}\]

    if -1.0908319111938347e-40 < (* a x) < -4.13249405988351797e-139

    1. Initial program 61.5

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

    2. Taylor expanded around 0 27.4

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified13.5

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot 0.5\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied distribute-lft-in13.5

      \[\leadsto x \cdot \left(a + \color{blue}{\left(x \cdot \left(a \cdot \left(a \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\right)\]

    6. Simplified13.5

      \[\leadsto x \cdot \left(a + \left(\color{blue}{a \cdot \left(a \cdot \left(0.5 \cdot x\right)\right)} + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.283265312368245 \cdot 10^{-05}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{elif}\;a \cdot x \leq -1.0908319111938347 \cdot 10^{-40} \lor \neg \left(a \cdot x \leq -4.132494059883518 \cdot 10^{-139}\right):\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(a \cdot \left(a \cdot \left(x \cdot 0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))