Average Error: 29.4 → 1.0
Time: 3.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4385658.6985369977:\\ \;\;\;\;{e}^{\left(a \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -4385658.6985369977:\\
\;\;\;\;{e}^{\left(a \cdot x\right)} - 1\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\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)) <= -4385658.698536998)) {
		VAR = ((double) (((double) pow(((double) M_E), ((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) fma(0.5, ((double) (((double) fabs(((double) (x * a)))) * ((double) fabs(((double) (x * a)))))), ((double) (a * x))));
	}
	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
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -4385658.698536998

    1. Initial program 0

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

    3. Applied *-un-lft-identity0

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

    4. Applied exp-prod0

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

    5. Simplified0

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

    if -4385658.698536998 < (* a x)

    1. Initial program 43.3

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified14.3

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

    4. Taylor expanded around 0 8.5

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

    5. Simplified8.5

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

    7. Applied add-sqr-sqrt8.5

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

    8. Simplified8.5

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

    9. Simplified1.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4385658.6985369977:\\ \;\;\;\;{e}^{\left(a \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))