Average Error: 29.3 → 0.4
Time: 44.2s
Precision: binary64
Cost: 20417
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.055700088526127754:\\ \;\;\;\;\frac{-1 + {\left(e^{a \cdot x}\right)}^{2}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -0.055700088526127754:\\
\;\;\;\;\frac{-1 + {\left(e^{a \cdot x}\right)}^{2}}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.055700088526127754)
   (/ (+ -1.0 (pow (exp (* a x)) 2.0)) (+ (exp (* a x)) 1.0))
   (+ (* a x) (* (pow (* a x) 2.0) (+ 0.5 (* (* a x) 0.16666666666666666))))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.055700088526127754) {
		tmp = (-1.0 + pow(exp(a * x), 2.0)) / (exp(a * x) + 1.0);
	} else {
		tmp = (a * x) + (pow((a * x), 2.0) * (0.5 + ((a * x) * 0.16666666666666666)));
	}
	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

Original29.3
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Alternatives

Alternative 1
Error0.4
Cost7873
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.055700088526127754:\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array}\]
Alternative 2
Error0.5
Cost7489
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.055700088526127754:\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot 0.5\\ \end{array}\]
Alternative 3
Error0.8
Cost7169
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.3304733640624707 \cdot 10^{-06}:\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
Alternative 4
Error1.2
Cost641
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.2953093660885433:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
Alternative 5
Error34.6
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -7.124667420801598 \cdot 10^{-100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 4.347966906444705 \cdot 10^{-33}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 6
Error51.2
Cost64
\[0\]

Error

Time

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -0.0557000885261277542

    1. Initial program 0.0

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

    3. Applied flip--_binary64_14170.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -0.0557000885261277542 < (*.f64 a x)

    1. Initial program 43.7

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified4.6

      \[\leadsto \color{blue}{a \cdot \left(x + 0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + 0.16666666666666666 \cdot {\left(a \cdot x\right)}^{3}}\]
    4. Using strategy rm

    5. Applied distribute-rgt-in_binary64_13924.6

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

    6. Simplified4.6

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

    7. Simplified0.6

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

    9. Applied associate-+l+_binary64_13750.6

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

    10. Simplified0.6

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

    11. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.055700088526127754:\\ \;\;\;\;\frac{-1 + {\left(e^{a \cdot x}\right)}^{2}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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))