Average Error: 31.0 → 0.4
Time: 1.7s
Precision: binary64
\[\sqrt{2 \cdot \left(x \cdot x\right)} \]
↓
\[\sqrt{\sqrt{2}} \cdot \left|x \cdot {2}^{0.25}\right| \]
(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))
↓
(FPCore (x)
 :precision binary64
 (* (sqrt (sqrt 2.0)) (fabs (* x (pow 2.0 0.25)))))
double code(double x) {
	return sqrt((2.0 * (x * x)));
}
↓
double code(double x) {
	return sqrt(sqrt(2.0)) * fabs((x * pow(2.0, 0.25)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x * x)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(sqrt(2.0d0)) * abs((x * (2.0d0 ** 0.25d0)))
end function

public static double code(double x) {
	return Math.sqrt((2.0 * (x * x)));
}
↓
public static double code(double x) {
	return Math.sqrt(Math.sqrt(2.0)) * Math.abs((x * Math.pow(2.0, 0.25)));
}

def code(x):
	return math.sqrt((2.0 * (x * x)))
↓
def code(x):
	return math.sqrt(math.sqrt(2.0)) * math.fabs((x * math.pow(2.0, 0.25)))

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end
↓
function code(x)
	return Float64(sqrt(sqrt(2.0)) * abs(Float64(x * (2.0 ^ 0.25))))
end

function tmp = code(x)
	tmp = sqrt((2.0 * (x * x)));
end
↓
function tmp = code(x)
	tmp = sqrt(sqrt(2.0)) * abs((x * (2.0 ^ 0.25)));
end

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[(N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision] * N[Abs[N[(x * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\sqrt{2 \cdot \left(x \cdot x\right)}
↓
\sqrt{\sqrt{2}} \cdot \left|x \cdot {2}^{0.25}\right|

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

    \[\sqrt{2 \cdot \left(x \cdot x\right)} \]

  2. Applied egg-rr31.1

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

  3. Applied egg-rr0.4

    \[\leadsto \sqrt{\sqrt{2}} \cdot \color{blue}{\left|x \cdot {2}^{0.25}\right|} \]

  4. Final simplification0.4

    \[\leadsto \sqrt{\sqrt{2}} \cdot \left|x \cdot {2}^{0.25}\right| \]

Reproduce

herbie shell --seed 2022148 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))