Average Error: 29.7 → 0.5
Time: 4.0s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.08386209981862915:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + 0.5 \cdot {\left(a \cdot x\right)}^{2}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -0.08386209981862915:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(e^{a \cdot x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + 0.5 \cdot {\left(a \cdot x\right)}^{2}\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.08386209981862915)
   (/
    (- (pow (exp (* a x)) 3.0) 1.0)
    (+ (* (exp (* a x)) (exp (* a x))) (+ (exp (* a x)) 1.0)))
   (+ (* a x) (* 0.5 (pow (* a x) 2.0)))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.08386209981862915) {
		tmp = (pow(exp(a * x), 3.0) - 1.0) / ((exp(a * x) * exp(a * x)) + (exp(a * x) + 1.0));
	} else {
		tmp = (a * x) + (0.5 * pow((a * x), 2.0));
	}
	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

Original29.7
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -0.0838620998186291461

    1. Initial program 0.0

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

    3. Applied flip3--_binary64_14460.0

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

    if -0.0838620998186291461 < (*.f64 a x)

    1. Initial program 44.6

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

    2. Taylor expanded around 0 8.5

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

    3. Simplified4.7

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

    5. Applied associate-*r*_binary64_13820.8

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

    7. Applied distribute-rgt-in_binary64_13920.8

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

    8. Simplified0.8

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

    9. Simplified0.8

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

    10. Taylor expanded around 0 8.5

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

    11. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.08386209981862915:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + 0.5 \cdot {\left(a \cdot x\right)}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))