sqrt C

Average Error: 30.2 → 0.4

Time: 1.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.25} \cdot \left(x \cdot {2}^{0.25}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 0.0) (- (* x (sqrt 2.0))) (* (pow 2.0 0.25) (* x (pow 2.0 0.25)))))

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 = pow(2.0, 0.25) * (x * pow(2.0, 0.25));
	}
	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 = (2.0d0 ** 0.25d0) * (x * (2.0d0 ** 0.25d0))
    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.pow(2.0, 0.25) * (x * Math.pow(2.0, 0.25));
	}
	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.pow(2.0, 0.25) * (x * math.pow(2.0, 0.25))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(-Float64(x * sqrt(2.0)));
	else
		tmp = Float64((2.0 ^ 0.25) * Float64(x * (2.0 ^ 0.25)));
	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 = (2.0 ^ 0.25) * (x * (2.0 ^ 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, 0.0], (-N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 0.0

   1. Initial program 30.1

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

   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 30.3

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

   2. Applied egg-rr 32.6

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

   3. Applied egg-rr 30.5

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{x \cdot \left(2 \cdot x\right)} \cdot \left(\sqrt[3]{x \cdot \sqrt{2}} \cdot \left(x \cdot \sqrt{2}\right)\right)}} \]

   4. Applied egg-rr 0.4

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;-x \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.25} \cdot \left(x \cdot {2}^{0.25}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (x)
  :name "sqrt C"
  :precision binary64
  (sqrt (* 2.0 (* x x))))