Average Error: 29.8 → 0.7
Time: 5.7s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.495199940381929 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -1.495199940381929 \cdot 10^{-08}:\\
\;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1.495199940381929e-08)
   (cbrt (pow (- (exp (* a x)) 1.0) 3.0))
   (* a x)))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.495199940381929e-08) {
		tmp = cbrt(pow((exp(a * x) - 1.0), 3.0));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -1.4951999403819288e-8

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube_binary64_18190.2

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]

    4. Simplified0.2

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{a \cdot x} - 1\right)}^{3}}}\]

    if -1.4951999403819288e-8 < (*.f64 a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{a \cdot x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.495199940381929 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2021027 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :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))