math.sqrt on complex, real part

Percentage Accurate: 41.3% → 90.1%

Time: 7.6s

Alternatives: 6

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im_m \cdot im_m} \leq 0:\\ \;\;\;\;0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (* 0.5 (/ im_m (sqrt (- re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re + math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0)
		tmp = 0.5 * (im_m / sqrt(-re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{if}\;re \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -6 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.5 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{elif}\;re \leq -5 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ re im_m))))))
   (if (<= re -7.5e+109)
     t_0
     (if (<= re -6e+93)
       t_1
       (if (<= re -3.8e-12)
         t_0
         (if (<= re -8.5e-117)
           (* 0.5 (sqrt (* im_m 2.0)))
           (if (<= re -5e-140)
             t_0
             (if (<= re 1.5e+102) t_1 (* 0.5 (* 2.0 (sqrt re)))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / sqrt(-re));
	double t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -7.5e+109) {
		tmp = t_0;
	} else if (re <= -6e+93) {
		tmp = t_1;
	} else if (re <= -3.8e-12) {
		tmp = t_0;
	} else if (re <= -8.5e-117) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else if (re <= -5e-140) {
		tmp = t_0;
	} else if (re <= 1.5e+102) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    if (re <= (-7.5d+109)) then
        tmp = t_0
    else if (re <= (-6d+93)) then
        tmp = t_1
    else if (re <= (-3.8d-12)) then
        tmp = t_0
    else if (re <= (-8.5d-117)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else if (re <= (-5d-140)) then
        tmp = t_0
    else if (re <= 1.5d+102) then
        tmp = t_1
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -7.5e+109) {
		tmp = t_0;
	} else if (re <= -6e+93) {
		tmp = t_1;
	} else if (re <= -3.8e-12) {
		tmp = t_0;
	} else if (re <= -8.5e-117) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else if (re <= -5e-140) {
		tmp = t_0;
	} else if (re <= 1.5e+102) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	t_1 = 0.5 * math.sqrt((2.0 * (re + im_m)))
	tmp = 0
	if re <= -7.5e+109:
		tmp = t_0
	elif re <= -6e+93:
		tmp = t_1
	elif re <= -3.8e-12:
		tmp = t_0
	elif re <= -8.5e-117:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	elif re <= -5e-140:
		tmp = t_0
	elif re <= 1.5e+102:
		tmp = t_1
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))))
	tmp = 0.0
	if (re <= -7.5e+109)
		tmp = t_0;
	elseif (re <= -6e+93)
		tmp = t_1;
	elseif (re <= -3.8e-12)
		tmp = t_0;
	elseif (re <= -8.5e-117)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	elseif (re <= -5e-140)
		tmp = t_0;
	elseif (re <= 1.5e+102)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = 0.5 * (im_m / sqrt(-re));
	t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	tmp = 0.0;
	if (re <= -7.5e+109)
		tmp = t_0;
	elseif (re <= -6e+93)
		tmp = t_1;
	elseif (re <= -3.8e-12)
		tmp = t_0;
	elseif (re <= -8.5e-117)
		tmp = 0.5 * sqrt((im_m * 2.0));
	elseif (re <= -5e-140)
		tmp = t_0;
	elseif (re <= 1.5e+102)
		tmp = t_1;
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.5e+109], t$95$0, If[LessEqual[re, -6e+93], t$95$1, If[LessEqual[re, -3.8e-12], t$95$0, If[LessEqual[re, -8.5e-117], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5e-140], t$95$0, If[LessEqual[re, 1.5e+102], t$95$1, N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{if}\;re \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.85 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{elif}\;re \leq -5 \cdot 10^{-140}:\\ \;\;\;\;0.5 \cdot \left(im_m \cdot {\left(\frac{-1}{re}\right)}^{0.5}\right)\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re)))))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ re im_m))))))
   (if (<= re -7.5e+109)
     t_0
     (if (<= re -1.85e+93)
       t_1
       (if (<= re -2e-11)
         t_0
         (if (<= re -1.55e-118)
           (* 0.5 (sqrt (* im_m 2.0)))
           (if (<= re -5e-140)
             (* 0.5 (* im_m (pow (/ -1.0 re) 0.5)))
             (if (<= re 1.06e+103) t_1 (* 0.5 (* 2.0 (sqrt re)))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / sqrt(-re));
	double t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -7.5e+109) {
		tmp = t_0;
	} else if (re <= -1.85e+93) {
		tmp = t_1;
	} else if (re <= -2e-11) {
		tmp = t_0;
	} else if (re <= -1.55e-118) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else if (re <= -5e-140) {
		tmp = 0.5 * (im_m * pow((-1.0 / re), 0.5));
	} else if (re <= 1.06e+103) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    t_1 = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    if (re <= (-7.5d+109)) then
        tmp = t_0
    else if (re <= (-1.85d+93)) then
        tmp = t_1
    else if (re <= (-2d-11)) then
        tmp = t_0
    else if (re <= (-1.55d-118)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else if (re <= (-5d-140)) then
        tmp = 0.5d0 * (im_m * (((-1.0d0) / re) ** 0.5d0))
    else if (re <= 1.06d+103) then
        tmp = t_1
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double t_1 = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	double tmp;
	if (re <= -7.5e+109) {
		tmp = t_0;
	} else if (re <= -1.85e+93) {
		tmp = t_1;
	} else if (re <= -2e-11) {
		tmp = t_0;
	} else if (re <= -1.55e-118) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else if (re <= -5e-140) {
		tmp = 0.5 * (im_m * Math.pow((-1.0 / re), 0.5));
	} else if (re <= 1.06e+103) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	t_1 = 0.5 * math.sqrt((2.0 * (re + im_m)))
	tmp = 0
	if re <= -7.5e+109:
		tmp = t_0
	elif re <= -1.85e+93:
		tmp = t_1
	elif re <= -2e-11:
		tmp = t_0
	elif re <= -1.55e-118:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	elif re <= -5e-140:
		tmp = 0.5 * (im_m * math.pow((-1.0 / re), 0.5))
	elif re <= 1.06e+103:
		tmp = t_1
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))))
	tmp = 0.0
	if (re <= -7.5e+109)
		tmp = t_0;
	elseif (re <= -1.85e+93)
		tmp = t_1;
	elseif (re <= -2e-11)
		tmp = t_0;
	elseif (re <= -1.55e-118)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	elseif (re <= -5e-140)
		tmp = Float64(0.5 * Float64(im_m * (Float64(-1.0 / re) ^ 0.5)));
	elseif (re <= 1.06e+103)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = 0.5 * (im_m / sqrt(-re));
	t_1 = 0.5 * sqrt((2.0 * (re + im_m)));
	tmp = 0.0;
	if (re <= -7.5e+109)
		tmp = t_0;
	elseif (re <= -1.85e+93)
		tmp = t_1;
	elseif (re <= -2e-11)
		tmp = t_0;
	elseif (re <= -1.55e-118)
		tmp = 0.5 * sqrt((im_m * 2.0));
	elseif (re <= -5e-140)
		tmp = 0.5 * (im_m * ((-1.0 / re) ^ 0.5));
	elseif (re <= 1.06e+103)
		tmp = t_1;
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.5e+109], t$95$0, If[LessEqual[re, -1.85e+93], t$95$1, If[LessEqual[re, -2e-11], t$95$0, If[LessEqual[re, -1.55e-118], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5e-140], N[(0.5 * N[(im$95$m * N[Power[N[(-1.0 / re), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e+103], t$95$1, N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{im_m}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{if}\;re \leq -3.6 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{-44} \lor \neg \left(re \leq 1.3 \cdot 10^{+18}\right) \land re \leq 1.22 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ im_m (sqrt (- re))))) (t_1 (* 0.5 (sqrt (* im_m 2.0)))))
   (if (<= re -3.6e+110)
     t_0
     (if (<= re -1e+68)
       t_1
       (if (<= re -8.2e-12)
         t_0
         (if (or (<= re 1.85e-44) (and (not (<= re 1.3e+18)) (<= re 1.22e+51)))
           t_1
           (* 0.5 (* 2.0 (sqrt re)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / sqrt(-re));
	double t_1 = 0.5 * sqrt((im_m * 2.0));
	double tmp;
	if (re <= -3.6e+110) {
		tmp = t_0;
	} else if (re <= -1e+68) {
		tmp = t_1;
	} else if (re <= -8.2e-12) {
		tmp = t_0;
	} else if ((re <= 1.85e-44) || (!(re <= 1.3e+18) && (re <= 1.22e+51))) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (im_m / sqrt(-re))
    t_1 = 0.5d0 * sqrt((im_m * 2.0d0))
    if (re <= (-3.6d+110)) then
        tmp = t_0
    else if (re <= (-1d+68)) then
        tmp = t_1
    else if (re <= (-8.2d-12)) then
        tmp = t_0
    else if ((re <= 1.85d-44) .or. (.not. (re <= 1.3d+18)) .and. (re <= 1.22d+51)) then
        tmp = t_1
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = 0.5 * (im_m / Math.sqrt(-re));
	double t_1 = 0.5 * Math.sqrt((im_m * 2.0));
	double tmp;
	if (re <= -3.6e+110) {
		tmp = t_0;
	} else if (re <= -1e+68) {
		tmp = t_1;
	} else if (re <= -8.2e-12) {
		tmp = t_0;
	} else if ((re <= 1.85e-44) || (!(re <= 1.3e+18) && (re <= 1.22e+51))) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = 0.5 * (im_m / math.sqrt(-re))
	t_1 = 0.5 * math.sqrt((im_m * 2.0))
	tmp = 0
	if re <= -3.6e+110:
		tmp = t_0
	elif re <= -1e+68:
		tmp = t_1
	elif re <= -8.2e-12:
		tmp = t_0
	elif (re <= 1.85e-44) or (not (re <= 1.3e+18) and (re <= 1.22e+51)):
		tmp = t_1
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))))
	t_1 = Float64(0.5 * sqrt(Float64(im_m * 2.0)))
	tmp = 0.0
	if (re <= -3.6e+110)
		tmp = t_0;
	elseif (re <= -1e+68)
		tmp = t_1;
	elseif (re <= -8.2e-12)
		tmp = t_0;
	elseif ((re <= 1.85e-44) || (!(re <= 1.3e+18) && (re <= 1.22e+51)))
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = 0.5 * (im_m / sqrt(-re));
	t_1 = 0.5 * sqrt((im_m * 2.0));
	tmp = 0.0;
	if (re <= -3.6e+110)
		tmp = t_0;
	elseif (re <= -1e+68)
		tmp = t_1;
	elseif (re <= -8.2e-12)
		tmp = t_0;
	elseif ((re <= 1.85e-44) || (~((re <= 1.3e+18)) && (re <= 1.22e+51)))
		tmp = t_1;
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.6e+110], t$95$0, If[LessEqual[re, -1e+68], t$95$1, If[LessEqual[re, -8.2e-12], t$95$0, If[Or[LessEqual[re, 1.85e-44], And[N[Not[LessEqual[re, 1.3e+18]], $MachinePrecision], LessEqual[re, 1.22e+51]]], t$95$1, N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.15 \cdot 10^{-44} \lor \neg \left(re \leq 1.25 \cdot 10^{+18}\right) \land re \leq 3.8 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \sqrt{im_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re 1.15e-44) (and (not (<= re 1.25e+18)) (<= re 3.8e+51)))
   (* 0.5 (sqrt (* im_m 2.0)))
   (* 0.5 (* 2.0 (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= 1.15e-44) || (!(re <= 1.25e+18) && (re <= 3.8e+51))) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re <= 1.15d-44) .or. (.not. (re <= 1.25d+18)) .and. (re <= 3.8d+51)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re <= 1.15e-44) || (!(re <= 1.25e+18) && (re <= 3.8e+51))) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= 1.15e-44) or (not (re <= 1.25e+18) and (re <= 3.8e+51)):
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= 1.15e-44) || (!(re <= 1.25e+18) && (re <= 3.8e+51)))
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re <= 1.15e-44) || (~((re <= 1.25e+18)) && (re <= 3.8e+51)))
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, 1.15e-44], And[N[Not[LessEqual[re, 1.25e+18]], $MachinePrecision], LessEqual[re, 3.8e+51]]], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \sqrt{im_m \cdot 2} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (sqrt (* im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * sqrt((im_m * 2.0));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.5d0 * sqrt((im_m * 2.0d0))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5 * Math.sqrt((im_m * 2.0));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * math.sqrt((im_m * 2.0))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * sqrt(Float64(im_m * 2.0)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5 * sqrt((im_m * 2.0));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 48.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024008 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))