Average Error: 29.7 → 9.0
Time: 4.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.05523141226366273 \cdot 10^{-12}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{elif}\;a \cdot x \le 7.2308568818741819 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.05523141226366273 \cdot 10^{-12}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{elif}\;a \cdot x \le 7.2308568818741819 \cdot 10^{-16}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\

\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)) <= -3.0552314122636627e-12)) {
		VAR = ((double) log(((double) exp(((double) (((double) exp(((double) (a * x)))) - 1.0))))));
	} else {
		double VAR_1;
		if ((((double) (a * x)) <= 7.230856881874182e-16)) {
			VAR_1 = ((double) fma(0.5, ((double) (((double) pow(a, 2.0)) * ((double) pow(x, 2.0)))), ((double) fma(0.16666666666666666, ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0)))), ((double) (a * x))))));
		} else {
			VAR_1 = ((double) (((double) sqrt(((double) (((double) exp(((double) (a * x)))) - 1.0)))) * ((double) sqrt(((double) (((double) exp(((double) (a * x)))) - 1.0))))));
		}
		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

Original29.7
Target0.2
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -3.0552314122636627e-12

    1. Initial program 0.6

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

    3. Applied add-log-exp0.6

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

    4. Applied add-log-exp0.7

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

    5. Applied diff-log0.7

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

    6. Simplified0.6

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

    if -3.0552314122636627e-12 < (* a x) < 7.230856881874182e-16

    1. Initial program 45.7

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    if 7.230856881874182e-16 < (* a x)

    1. Initial program 20.1

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

    3. Applied add-sqr-sqrt20.1

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.05523141226366273 \cdot 10^{-12}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{elif}\;a \cdot x \le 7.2308568818741819 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{e^{a \cdot x} - 1} \cdot \sqrt{e^{a \cdot x} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))