Average Error: 30.2 → 0.3

Time: 16.9s

Precision: binary64

Cost: 13120

\[\sqrt{2 \cdot {x}^{2}} \]

↓

\[\left(\left|x\right| \cdot \sqrt{0.2222222222222222}\right) \cdot 3 \]

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

↓

(FPCore (x) :precision binary64 (* (* (fabs x) (sqrt 0.2222222222222222)) 3.0))

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

↓

double code(double x) {
	return (fabs(x) * sqrt(0.2222222222222222)) * 3.0;
}

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (abs(x) * sqrt(0.2222222222222222d0)) * 3.0d0
end function

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

↓

public static double code(double x) {
	return (Math.abs(x) * Math.sqrt(0.2222222222222222)) * 3.0;
}

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

↓

def code(x):
	return (math.fabs(x) * math.sqrt(0.2222222222222222)) * 3.0

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

↓

function code(x)
	return Float64(Float64(abs(x) * sqrt(0.2222222222222222)) * 3.0)
end

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

↓

function tmp = code(x)
	tmp = (abs(x) * sqrt(0.2222222222222222)) * 3.0;
end

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

↓

code[x_] := N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[0.2222222222222222], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]

\sqrt{2 \cdot {x}^{2}}

↓

\left(\left|x\right| \cdot \sqrt{0.2222222222222222}\right) \cdot 3

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 30.2
\[\sqrt{2 \cdot {x}^{2}} \]
Applied egg-rr0.5
\[\leadsto \color{blue}{\frac{\left|x\right| \cdot \sqrt{2}}{3} \cdot 3} \]
Applied egg-rr0.3
\[\leadsto \color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{2}{9}}\right)} \cdot 3 \]
Simplified0.3
\[\leadsto \color{blue}{\left(\left|x\right| \cdot \sqrt{0.2222222222222222}\right)} \cdot 3 \]
Proof

Alternatives

Alternative 1
Error	0.4
Cost	12992

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

Alternative 2
Error	0.4
Cost	12992

\[\frac{\left|x\right|}{\sqrt{0.5}} \]

Alternative 3
Error	0.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x \cdot -2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \]

Alternative 4
Error	0.4
Cost	6788

\[\begin{array}{l} t_0 := \sqrt{2} \cdot x\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	31.1
Cost	6592

\[\sqrt{2} \cdot x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))