?

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}

Error?

Try it out?

Your Program's Arguments

Results

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

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

    neg-mul-1 [=>]0.0

    \[ \frac{{2.718281828459045}^{\left(\frac{\color{blue}{-1 \cdot {x}^{2}}}{2}\right)}}{2.5066282746310002} \]

    associate-/l* [=>]0.0

    \[ \frac{{2.718281828459045}^{\color{blue}{\left(\frac{-1}{\frac{2}{{x}^{2}}}\right)}}}{2.5066282746310002} \]

    associate-/r/ [=>]0.0

    \[ \frac{{2.718281828459045}^{\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}}}{2.5066282746310002} \]

    *-commutative [=>]0.0

    \[ \frac{{2.718281828459045}^{\color{blue}{\left({x}^{2} \cdot \frac{-1}{2}\right)}}}{2.5066282746310002} \]

    unpow2 [=>]0.0

    \[ \frac{{2.718281828459045}^{\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{2}\right)}}{2.5066282746310002} \]

    metadata-eval [=>]0.0

    \[ \frac{{2.718281828459045}^{\left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right)}}{2.5066282746310002} \]

  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

    \[ \frac{\sqrt{{7.3890560989306495}^{\left(x \cdot \left(x \cdot -0.5\right)\right)}}}{2.5066282746310002} \]

    sqr-pow [=>]0.0

    \[ \frac{\sqrt{\color{blue}{{7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)}}}}{2.5066282746310002} \]

    rem-sqrt-square [=>]0.0

    \[ \frac{\color{blue}{\left|{7.3890560989306495}^{\left(\frac{x \cdot \left(x \cdot -0.5\right)}{2}\right)}\right|}}{2.5066282746310002} \]

    associate-*r* [=>]0.0

    \[ \frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot -0.5}}{2}\right)}\right|}{2.5066282746310002} \]

    unpow2 [<=]0.0

    \[ \frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}} \cdot -0.5}{2}\right)}\right|}{2.5066282746310002} \]

    associate-/l* [=>]0.0

    \[ \frac{\left|{7.3890560989306495}^{\color{blue}{\left(\frac{{x}^{2}}{\frac{2}{-0.5}}\right)}}\right|}{2.5066282746310002} \]

    unpow2 [=>]0.0

    \[ \frac{\left|{7.3890560989306495}^{\left(\frac{\color{blue}{x \cdot x}}{\frac{2}{-0.5}}\right)}\right|}{2.5066282746310002} \]

    metadata-eval [=>]0.0

    \[ \frac{\left|{7.3890560989306495}^{\left(\frac{x \cdot x}{\color{blue}{-4}}\right)}\right|}{2.5066282746310002} \]

    sqr-pow [=>]0.0

    \[ \frac{\left|\color{blue}{{7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)}}\right|}{2.5066282746310002} \]

    fabs-sqr [=>]0.0

    \[ \frac{\color{blue}{{7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)} \cdot {7.3890560989306495}^{\left(\frac{\frac{x \cdot x}{-4}}{2}\right)}}}{2.5066282746310002} \]

    sqr-pow [<=]0.0

    \[ \frac{\color{blue}{{7.3890560989306495}^{\left(\frac{x \cdot x}{-4}\right)}}}{2.5066282746310002} \]

    unpow2 [<=]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}}}{-4}\right)}}{2.5066282746310002} \]

    metadata-eval [<=]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{{x}^{2}}{\color{blue}{\frac{2}{-0.5}}}\right)}}{2.5066282746310002} \]

    associate-/l* [<=]0.0

    \[ \frac{{7.3890560989306495}^{\color{blue}{\left(\frac{{x}^{2} \cdot -0.5}{2}\right)}}}{2.5066282746310002} \]

    unpow2 [=>]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot -0.5}{2}\right)}}{2.5066282746310002} \]

    associate-*r* [<=]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{x \cdot \left(x \cdot -0.5\right)}}{2}\right)}}{2.5066282746310002} \]

    associate-*r* [=>]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot -0.5}}{2}\right)}}{2.5066282746310002} \]

    unpow2 [<=]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{{x}^{2}} \cdot -0.5}{2}\right)}}{2.5066282746310002} \]

    *-commutative [<=]0.0

    \[ \frac{{7.3890560989306495}^{\left(\frac{\color{blue}{-0.5 \cdot {x}^{2}}}{2}\right)}}{2.5066282746310002} \]

    associate-/l* [=>]0.0

    \[ \frac{{7.3890560989306495}^{\color{blue}{\left(\frac{-0.5}{\frac{2}{{x}^{2}}}\right)}}}{2.5066282746310002} \]

    associate-/r/ [=>]0.0

    \[ \frac{{7.3890560989306495}^{\color{blue}{\left(\frac{-0.5}{2} \cdot {x}^{2}\right)}}}{2.5066282746310002} \]

  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

    \[ \frac{e^{-0.25 \cdot \left(\log 7.3890560989306495 \cdot {x}^{2}\right)}}{2.5066282746310002} \]

    associate-*r* [=>]0.0

    \[ \frac{e^{\color{blue}{\left(-0.25 \cdot \log 7.3890560989306495\right) \cdot {x}^{2}}}}{2.5066282746310002} \]

    log-pow [<=]0.0

    \[ \frac{e^{\color{blue}{\log \left({7.3890560989306495}^{-0.25}\right)} \cdot {x}^{2}}}{2.5066282746310002} \]

    unpow2 [=>]0.0

    \[ \frac{e^{\log \left({7.3890560989306495}^{-0.25}\right) \cdot \color{blue}{\left(x \cdot x\right)}}}{2.5066282746310002} \]

    exp-to-pow [=>]0.0

    \[ \frac{\color{blue}{{\left({7.3890560989306495}^{-0.25}\right)}^{\left(x \cdot x\right)}}}{2.5066282746310002} \]

  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
\[0.3989422804014327 \]

Error

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))