# ?

Average Error: 0.0 → 0.0
Time: 7.8s
Precision: binary64
Cost: 13248

# ?

$-10000000000 \leq x \land x \leq 10000000000$
$\frac{{2.718281828459045}^{\left(\frac{-{x}^{2}}{2}\right)}}{2.5066282746310002}$
$\frac{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}{2.5066282746310002}$
(FPCore (x)
:precision binary64
(/ (pow 2.718281828459045 (/ (- (pow x 2.0)) 2.0)) 2.5066282746310002))
(FPCore (x)
:precision binary64
(/ (pow (pow 7.3890560989306495 -0.25) (* x x)) 2.5066282746310002))
double code(double x) {
return pow(2.718281828459045, (-pow(x, 2.0) / 2.0)) / 2.5066282746310002;
}

double code(double x) {
return pow(pow(7.3890560989306495, -0.25), (x * x)) / 2.5066282746310002;
}

real(8) function code(x)
real(8), intent (in) :: x
code = (2.718281828459045d0 ** (-(x ** 2.0d0) / 2.0d0)) / 2.5066282746310002d0
end function

real(8) function code(x)
real(8), intent (in) :: x
code = ((7.3890560989306495d0 ** (-0.25d0)) ** (x * x)) / 2.5066282746310002d0
end function

public static double code(double x) {
return Math.pow(2.718281828459045, (-Math.pow(x, 2.0) / 2.0)) / 2.5066282746310002;
}

public static double code(double x) {
return Math.pow(Math.pow(7.3890560989306495, -0.25), (x * x)) / 2.5066282746310002;
}

def code(x):
return math.pow(2.718281828459045, (-math.pow(x, 2.0) / 2.0)) / 2.5066282746310002

def code(x):
return math.pow(math.pow(7.3890560989306495, -0.25), (x * x)) / 2.5066282746310002

function code(x)
return Float64((2.718281828459045 ^ Float64(Float64(-(x ^ 2.0)) / 2.0)) / 2.5066282746310002)
end

function code(x)
return Float64(((7.3890560989306495 ^ -0.25) ^ Float64(x * x)) / 2.5066282746310002)
end

function tmp = code(x)
tmp = (2.718281828459045 ^ (-(x ^ 2.0) / 2.0)) / 2.5066282746310002;
end

function tmp = code(x)
tmp = ((7.3890560989306495 ^ -0.25) ^ (x * x)) / 2.5066282746310002;
end

code[x_] := N[(N[Power[2.718281828459045, N[((-N[Power[x, 2.0], $MachinePrecision]) / 2.0),$MachinePrecision]], $MachinePrecision] / 2.5066282746310002),$MachinePrecision]

code[x_] := N[(N[Power[N[Power[7.3890560989306495, -0.25], $MachinePrecision], N[(x * x),$MachinePrecision]], $MachinePrecision] / 2.5066282746310002),$MachinePrecision]

\frac{{2.718281828459045}^{\left(\frac{-{x}^{2}}{2}\right)}}{2.5066282746310002}

\frac{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}{2.5066282746310002}


# Try it out?

Results

 In Out
Enter valid numbers for all inputs

# Derivation?

1. Initial program 0.0

$\frac{{2.718281828459045}^{\left(\frac{-{x}^{2}}{2}\right)}}{2.5066282746310002}$
2. Simplified0.0

$\leadsto \color{blue}{\frac{{2.718281828459045}^{\left(\left(x \cdot x\right) \cdot -0.5\right)}}{2.5066282746310002}}$
Proof
[Start]0.0 $\frac{{2.718281828459045}^{\left(\frac{-{x}^{2}}{2}\right)}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\left(\frac{\color{blue}{-1 \cdot {x}^{2}}}{2}\right)}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\color{blue}{\left(\frac{-1}{\frac{2}{{x}^{2}}}\right)}}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)}}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{2}\right)}}{2.5066282746310002}$ $\frac{{2.718281828459045}^{\left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right)}}{2.5066282746310002}$
3. Applied egg-rr0.0

$\leadsto \frac{\color{blue}{\sqrt{{7.3890560989306495}^{\left(x \cdot \left(x \cdot -0.5\right)\right)}}}}{2.5066282746310002}$
4. Simplified0.0

$\leadsto \frac{\color{blue}{{7.3890560989306495}^{\left(-0.25 \cdot \left(x \cdot x\right)\right)}}}{2.5066282746310002}$
Proof
[Start]0.0 $\frac{\sqrt{{7.3890560989306495}^{\left(x \cdot \left(x \cdot -0.5\right)\right)}}}{2.5066282746310002}$ $\frac{\sqrt{\color{blue}{{7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)}}}}{2.5066282746310002}$ $\frac{\color{blue}{\left|{7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)}\right|}}{2.5066282746310002}$ $\frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot -0.5}}{2}\right)}\right|}{2.5066282746310002}$ $\frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}} \cdot -0.5}{2}\right)}\right|}{2.5066282746310002}$ $\frac{\left|{7.3890560989306495}^{\color{blue}{\left(\frac{{x}^{2}}{\frac{2}{-0.5}}\right)}}\right|}{2.5066282746310002}$ $\frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{x \cdot x}}{\frac{2}{-0.5}}\right)}\right|}{2.5066282746310002}$ $\frac{\left|{7.3890560989306495}^{\left(\frac{x \cdot x}{\color{blue}{-4}}\right)}\right|}{2.5066282746310002}$ $\frac{\left|\color{blue}{{7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)}}\right|}{2.5066282746310002}$ $\frac{\color{blue}{{7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)}}}{2.5066282746310002}$ $\frac{\color{blue}{{7.3890560989306495}^{\left(\frac{x \cdot x}{-4}\right)}}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}}}{-4}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{{x}^{2}}{\color{blue}{\frac{2}{-0.5}}}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\color{blue}{\left(\frac{{x}^{2} \cdot -0.5}{2}\right)}}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot -0.5}{2}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{x \cdot \left(x \cdot -0.5\right)}}{2}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot -0.5}}{2}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}} \cdot -0.5}{2}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\left(\frac{\color{blue}{-0.5 \cdot {x}^{2}}}{2}\right)}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\color{blue}{\left(\frac{-0.5}{\frac{2}{{x}^{2}}}\right)}}}{2.5066282746310002}$ $\frac{{7.3890560989306495}^{\color{blue}{\left(\frac{-0.5}{2} \cdot {x}^{2}\right)}}}{2.5066282746310002}$
5. Taylor expanded in x around inf 0.0

$\leadsto \frac{\color{blue}{e^{-0.25 \cdot \left(\log 7.3890560989306495 \cdot {x}^{2}\right)}}}{2.5066282746310002}$
6. Simplified0.0

$\leadsto \frac{\color{blue}{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}}{2.5066282746310002}$
Proof
[Start]0.0 $\frac{e^{-0.25 \cdot \left(\log 7.3890560989306495 \cdot {x}^{2}\right)}}{2.5066282746310002}$ $\frac{e^{\color{blue}{\left(-0.25 \cdot \log 7.3890560989306495\right) \cdot {x}^{2}}}}{2.5066282746310002}$ $\frac{e^{\color{blue}{\log \left({7.3890560989306495}^{-0.25}\right)} \cdot {x}^{2}}}{2.5066282746310002}$ $\frac{e^{\log \left({7.3890560989306495}^{-0.25}\right) \cdot \color{blue}{\left(x \cdot x\right)}}}{2.5066282746310002}$ $\frac{\color{blue}{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}}{2.5066282746310002}$
7. Final simplification0.0

$\leadsto \frac{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}{2.5066282746310002}$

# Alternatives

Alternative 1
Error0.0
Cost6912
$\frac{{2.718281828459045}^{\left(\left(x \cdot x\right) \cdot -0.5\right)}}{2.5066282746310002}$
Alternative 2
Error0.0
Cost6912
$\frac{{7.3890560989306495}^{\left(-0.25 \cdot \left(x \cdot x\right)\right)}}{2.5066282746310002}$
Alternative 3
Error0.0
Cost6912
$\frac{{54.59815003314423}^{\left(\frac{x}{\frac{-8}{x}}\right)}}{2.5066282746310002}$
Alternative 4
Error2.5
Cost64
$0.3989422804014327$

# Reproduce?

herbie shell --seed 1
(FPCore (x)
:name "2.718281828459045^(-(x^2)/2)/2.5066282746310002"
:precision binary64
:pre (and (<= -10000000000.0 x) (<= x 10000000000.0))
(/ (pow 2.718281828459045 (/ (- (pow x 2.0)) 2.0)) 2.5066282746310002))