Average Error: 30.1 → 0.1
Time: 4.6s
Precision: binary64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.4521224485863904 \cdot 10^{-05}:\\ \;\;\;\;x \cdot x + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}{{\left(e^{x}\right)}^{3} + e^{x} \cdot \left(2 \cdot \left(e^{x} + 2\right)\right)}\\ \end{array}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\begin{array}{l}
\mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.4521224485863904 \cdot 10^{-05}:\\
\;\;\;\;x \cdot x + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}{{\left(e^{x}\right)}^{3} + e^{x} \cdot \left(2 \cdot \left(e^{x} + 2\right)\right)}\\

\end{array}
double code(double x) {
	return ((double) (((double) (((double) exp(x)) - 2.0)) + ((double) exp(((double) -(x))))));
}
↓
double code(double x) {
	double VAR;
	if ((((double) (((double) (((double) exp(x)) - 2.0)) + ((double) exp(((double) -(x)))))) <= 1.4521224485863904e-05)) {
		VAR = ((double) (((double) (x * x)) + ((double) (((double) (0.002777777777777778 * ((double) pow(x, 6.0)))) + ((double) (0.08333333333333333 * ((double) pow(x, 4.0))))))));
	} else {
		VAR = ((double) (((double) (((double) (2.0 * 2.0)) + ((double) (((double) exp(x)) * ((double) (((double) (((double) pow(((double) exp(x)), 3.0)) - ((double) pow(2.0, 3.0)))) + ((double) (((double) exp(x)) + 2.0)))))))) / ((double) (((double) pow(((double) exp(x)), 3.0)) + ((double) (((double) exp(x)) * ((double) (2.0 * ((double) (((double) exp(x)) + 2.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.1
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (exp x) 2.0) (exp (neg x))) < 1.4521224485864e-5

    1. Initial program 30.6

      \[\left(e^{x} - 2\right) + e^{-x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{{x}^{2} + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{x \cdot x + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)}\]

    if 1.4521224485864e-5 < (+ (- (exp x) 2.0) (exp (neg x)))

    1. Initial program 3.1

      \[\left(e^{x} - 2\right) + e^{-x}\]
    2. Using strategy rm

    3. Applied exp-neg2.9

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

    4. Applied flip3--5.1

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

    5. Applied frac-add6.2

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

    6. Simplified5.9

      \[\leadsto \frac{\color{blue}{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}}{\left(e^{x} \cdot e^{x} + \left(2 \cdot 2 + e^{x} \cdot 2\right)\right) \cdot e^{x}}\]

    7. Simplified5.9

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 1.4521224485863904 \cdot 10^{-05}:\\ \;\;\;\;x \cdot x + \left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot 2 + e^{x} \cdot \left(\left({\left(e^{x}\right)}^{3} - {2}^{3}\right) + \left(e^{x} + 2\right)\right)}{{\left(e^{x}\right)}^{3} + e^{x} \cdot \left(2 \cdot \left(e^{x} + 2\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))