sqrt B

Average Error: 30.7 → 0.4

Time: 1.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x \cdot 2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 0.0) (- (* x (sqrt 2.0))) (* (sqrt x) (sqrt (* x 2.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 0.0) {
		tmp = -(x * sqrt(2.0));
	} else {
		tmp = sqrt(x) * sqrt((x * 2.0));
	}
	return tmp;
}

Fortran

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
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = -(x * sqrt(2.0d0))
    else
        tmp = sqrt(x) * sqrt((x * 2.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= 0.0) {
		tmp = -(x * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((x * 2.0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= 0.0:
		tmp = -(x * math.sqrt(2.0))
	else:
		tmp = math.sqrt(x) * math.sqrt((x * 2.0))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = -(x * sqrt(2.0));
	else
		tmp = sqrt(x) * sqrt((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, 0.0], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 0.0

   1. Initial program 30.4

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

   2. Taylor expanded in x around -inf 0.4

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot x\right)} \]

   3. Simplified 0.4

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

   if 0.0 < x

   1. Initial program 31.0

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{2 \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \sqrt{x \cdot 2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))