Average Error: 29.5 → 4.7
Time: 3.3s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -5.519470368856438 \cdot 10^{-07}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -5.519470368856438 \cdot 10^{-07}:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.519470368856438e-07)
   (/ (+ (pow (exp (* a x)) 2.0) -1.0) (+ (exp (* a x)) 1.0))
   (*
    x
    (+
     a
     (* x (+ (* 0.5 (* a a)) (* x (* 0.16666666666666666 (pow a 3.0)))))))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.519470368856438e-07) {
		tmp = (pow(exp(a * x), 2.0) + -1.0) / (exp(a * x) + 1.0);
	} else {
		tmp = x * (a + (x * ((0.5 * (a * a)) + (x * (0.16666666666666666 * pow(a, 3.0))))));
	}
	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

Original29.5
Target0.1
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -5.5194703688564379e-7

    1. Initial program 0.2

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

    3. Applied flip--_binary640.2

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.2

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

    if -5.5194703688564379e-7 < (*.f64 a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 13.4

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

    3. Simplified7.1

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5.519470368856438 \cdot 10^{-07}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))