sqrt D

Average Error: 31.6 → 0.4

Time: 1.7s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 0.0

   1. Initial program 31.0

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

   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 32.3

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

   2. Taylor expanded in x around 0 0.4

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

4. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022156 
(FPCore (x)
  :name "sqrt D"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))