math.sqrt on complex, real part

Percentage Accurate: 40.9% → 82.5%

Time: 8.2s

Alternatives: 7

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: 40.9% 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: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\frac{re}{-0.5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.02e+96)
   (* 0.5 (sqrt (* 2.0 (* im (/ im (/ re -0.5))))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+96) {
		tmp = 0.5 * sqrt((2.0 * (im * (im / (re / -0.5)))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.02e+96) {
		tmp = 0.5 * Math.sqrt((2.0 * (im * (im / (re / -0.5)))));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.02e+96:
		tmp = 0.5 * math.sqrt((2.0 * (im * (im / (re / -0.5)))))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.02e+96)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im * Float64(im / Float64(re / -0.5))))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.02e+96)
		tmp = 0.5 * sqrt((2.0 * (im * (im / (re / -0.5)))));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.02e+96], N[(0.5 * N[Sqrt[N[(2.0 * N[(im * N[(im / N[(re / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.02000000000000001e96

   1. Initial program 5.0%

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

   2. Taylor expanded in re around -inf 19.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-1 \cdot re + -0.5 \cdot \frac{{im}^{2}}{re}\right)} + re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + -1 \cdot re\right)} + re\right)} \]

      2. mul-1-neg 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(-0.5 \cdot \frac{{im}^{2}}{re} + \color{blue}{\left(-re\right)}\right) + re\right)} \]

      3. unsub-neg 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} - re\right)} + re\right)} \]

      4. associate-*r/ 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{-0.5 \cdot {im}^{2}}{re}} - re\right) + re\right)} \]

      5. unpow2 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{-0.5 \cdot \color{blue}{\left(im \cdot im\right)}}{re} - re\right) + re\right)} \]

      6. associate-*r* 19.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{\color{blue}{\left(-0.5 \cdot im\right) \cdot im}}{re} - re\right) + re\right)} \]

   4. Simplified 19.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\frac{\left(-0.5 \cdot im\right) \cdot im}{re} - re\right)} + re\right)} \]

   5. Taylor expanded in im around 0 62.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 62.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{-0.5 \cdot {im}^{2}}{re}}} \]

      2. unpow2 62.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{-0.5 \cdot \color{blue}{\left(im \cdot im\right)}}{re}} \]

      3. associate-*r* 62.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\left(-0.5 \cdot im\right) \cdot im}}{re}} \]

      4. *-commutative 62.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\left(im \cdot -0.5\right)} \cdot im}{re}} \]

      5. associate-*l/ 66.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{im \cdot -0.5}{re} \cdot im\right)}} \]

      6. *-commutative 66.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im \cdot \frac{im \cdot -0.5}{re}\right)}} \]

      7. associate-/l* 66.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \color{blue}{\frac{im}{\frac{re}{-0.5}}}\right)} \]

   7. Simplified 66.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im \cdot \frac{im}{\frac{re}{-0.5}}\right)}} \]

   if -1.02000000000000001e96 < re

   1. Initial program 50.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 50.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 90.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 90.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 89.3%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}} \]

      2. sqrt-unprod 90.0%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}} \]

      3. *-commutative 90.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)} \]

      4. *-commutative 90.0%

         \[\leadsto \sqrt{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot 0.5\right)}} \]

      5. swap-sqr 90.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]

      6. add-sqr-sqrt 90.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      7. *-commutative 90.0%

         \[\leadsto \sqrt{\color{blue}{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right)} \cdot \left(0.5 \cdot 0.5\right)} \]

      8. metadata-eval 90.0%

         \[\leadsto \sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 90.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. associate-*l* 90.0%

         \[\leadsto \sqrt{\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \left(2 \cdot 0.25\right)}} \]

      2. metadata-eval 90.0%

         \[\leadsto \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot \color{blue}{0.5}} \]

   7. Simplified 90.0%

      \[\leadsto \color{blue}{\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.02 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\frac{re}{-0.5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(re + im\right)\\ \mathbf{if}\;im \leq -9.2 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + t_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ re im))))
   (if (<= im -9.2e-116)
     (* 0.5 (sqrt (* 2.0 (- re im))))
     (if (<= im 9.8e-186)
       (sqrt re)
       (if (<= im 2.05e-159)
         (* 0.5 (sqrt (* 2.0 im)))
         (if (<= im 1.1e-111)
           (sqrt re)
           (if (<= im 3.5e-72)
             (* 0.5 (sqrt (+ (/ (* re re) im) t_0)))
             (* 0.5 (sqrt t_0)))))))))

C

double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double tmp;
	if (im <= -9.2e-116) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 9.8e-186) {
		tmp = sqrt(re);
	} else if (im <= 2.05e-159) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if (im <= 1.1e-111) {
		tmp = sqrt(re);
	} else if (im <= 3.5e-72) {
		tmp = 0.5 * sqrt((((re * re) / im) + t_0));
	} else {
		tmp = 0.5 * sqrt(t_0);
	}
	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 = 2.0d0 * (re + im)
    if (im <= (-9.2d-116)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 9.8d-186) then
        tmp = sqrt(re)
    else if (im <= 2.05d-159) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if (im <= 1.1d-111) then
        tmp = sqrt(re)
    else if (im <= 3.5d-72) then
        tmp = 0.5d0 * sqrt((((re * re) / im) + t_0))
    else
        tmp = 0.5d0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double tmp;
	if (im <= -9.2e-116) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 9.8e-186) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.05e-159) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if (im <= 1.1e-111) {
		tmp = Math.sqrt(re);
	} else if (im <= 3.5e-72) {
		tmp = 0.5 * Math.sqrt((((re * re) / im) + t_0));
	} else {
		tmp = 0.5 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 * (re + im)
	tmp = 0
	if im <= -9.2e-116:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 9.8e-186:
		tmp = math.sqrt(re)
	elif im <= 2.05e-159:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif im <= 1.1e-111:
		tmp = math.sqrt(re)
	elif im <= 3.5e-72:
		tmp = 0.5 * math.sqrt((((re * re) / im) + t_0))
	else:
		tmp = 0.5 * math.sqrt(t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 * Float64(re + im))
	tmp = 0.0
	if (im <= -9.2e-116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 9.8e-186)
		tmp = sqrt(re);
	elseif (im <= 2.05e-159)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif (im <= 1.1e-111)
		tmp = sqrt(re);
	elseif (im <= 3.5e-72)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(re * re) / im) + t_0)));
	else
		tmp = Float64(0.5 * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 * (re + im);
	tmp = 0.0;
	if (im <= -9.2e-116)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 9.8e-186)
		tmp = sqrt(re);
	elseif (im <= 2.05e-159)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif (im <= 1.1e-111)
		tmp = sqrt(re);
	elseif (im <= 3.5e-72)
		tmp = 0.5 * sqrt((((re * re) / im) + t_0));
	else
		tmp = 0.5 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9.2e-116], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.8e-186], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.05e-159], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e-111], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 3.5e-72], N[(0.5 * N[Sqrt[N[(N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -9.20000000000000006e-116

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 80.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around -inf 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\left(-im\right)}\right)} \]

      2. sub-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   6. Simplified 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   if -9.20000000000000006e-116 < im < 9.7999999999999992e-186 or 2.05000000000000007e-159 < im < 1.1e-111

   1. Initial program 35.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 35.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 71.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around 0 43.9%

      \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 43.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

      2. unpow2 43.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

      3. rem-square-sqrt 44.8%

         \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

      4. metadata-eval 44.8%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      5. *-lft-identity 44.8%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 9.7999999999999992e-186 < im < 2.05000000000000007e-159

   1. Initial program 26.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 26.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 76.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 65.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 65.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 65.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 1.1e-111 < im < 3.5e-72

   1. Initial program 78.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 78.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 78.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 64.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{re}^{2}}{im} + \left(2 \cdot im + 2 \cdot re\right)}} \]
   5. Step-by-step derivation

      1. unpow2 64.0%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{re \cdot re}}{im} + \left(2 \cdot im + 2 \cdot re\right)} \]

      2. distribute-lft-out 64.0%

         \[\leadsto 0.5 \cdot \sqrt{\frac{re \cdot re}{im} + \color{blue}{2 \cdot \left(im + re\right)}} \]

   6. Simplified 64.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{re \cdot re}{im} + 2 \cdot \left(im + re\right)}} \]

   if 3.5e-72 < im

   1. Initial program 41.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 41.8%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 84.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 65.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im + 2 \cdot re}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 65.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(im + re\right)}} \]

      2. +-commutative 65.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]

      3. *-commutative 65.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(re + im\right) \cdot 2}} \]

      4. +-commutative 65.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right)} \cdot 2} \]

   6. Simplified 65.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.2 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 10^{-185} \lor \neg \left(im \leq 3.4 \cdot 10^{-159}\right) \land im \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.5e-116)
   (* 0.5 (sqrt (* im -2.0)))
   (if (or (<= im 1e-185) (and (not (<= im 3.4e-159)) (<= im 1.65e-111)))
     (sqrt re)
     (* 0.5 (sqrt (* 2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.5e-116) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if ((im <= 1e-185) || (!(im <= 3.4e-159) && (im <= 1.65e-111))) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.5d-116)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if ((im <= 1d-185) .or. (.not. (im <= 3.4d-159)) .and. (im <= 1.65d-111)) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.5e-116) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if ((im <= 1e-185) || (!(im <= 3.4e-159) && (im <= 1.65e-111))) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.5e-116:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif (im <= 1e-185) or (not (im <= 3.4e-159) and (im <= 1.65e-111)):
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.5e-116)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif ((im <= 1e-185) || (!(im <= 3.4e-159) && (im <= 1.65e-111)))
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.5e-116)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif ((im <= 1e-185) || (~((im <= 3.4e-159)) && (im <= 1.65e-111)))
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.5e-116], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 1e-185], And[N[Not[LessEqual[im, 3.4e-159]], $MachinePrecision], LessEqual[im, 1.65e-111]]], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.50000000000000013e-116

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 80.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around -inf 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 69.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot -2}} \]

   6. Simplified 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot -2}} \]

   if -1.50000000000000013e-116 < im < 9.9999999999999999e-186 or 3.39999999999999984e-159 < im < 1.65e-111

   1. Initial program 35.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 35.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 71.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around 0 43.9%

      \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 43.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

      2. unpow2 43.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

      3. rem-square-sqrt 44.8%

         \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

      4. metadata-eval 44.8%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      5. *-lft-identity 44.8%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 9.9999999999999999e-186 < im < 3.39999999999999984e-159 or 1.65e-111 < im

   1. Initial program 43.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 43.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 83.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 63.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 63.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 10^{-185} \lor \neg \left(im \leq 3.4 \cdot 10^{-159}\right) \land im \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 4: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{-185} \lor \neg \left(im \leq 2.2 \cdot 10^{-157}\right) \land im \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -8.2e-116)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (or (<= im 6.6e-185) (and (not (<= im 2.2e-157)) (<= im 1.4e-111)))
     (sqrt re)
     (* 0.5 (sqrt (* 2.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -8.2e-116) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if ((im <= 6.6e-185) || (!(im <= 2.2e-157) && (im <= 1.4e-111))) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-8.2d-116)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if ((im <= 6.6d-185) .or. (.not. (im <= 2.2d-157)) .and. (im <= 1.4d-111)) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -8.2e-116) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if ((im <= 6.6e-185) || (!(im <= 2.2e-157) && (im <= 1.4e-111))) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -8.2e-116:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif (im <= 6.6e-185) or (not (im <= 2.2e-157) and (im <= 1.4e-111)):
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -8.2e-116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif ((im <= 6.6e-185) || (!(im <= 2.2e-157) && (im <= 1.4e-111)))
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -8.2e-116)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif ((im <= 6.6e-185) || (~((im <= 2.2e-157)) && (im <= 1.4e-111)))
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -8.2e-116], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 6.6e-185], And[N[Not[LessEqual[im, 2.2e-157]], $MachinePrecision], LessEqual[im, 1.4e-111]]], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.1999999999999998e-116

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 80.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around -inf 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\left(-im\right)}\right)} \]

      2. sub-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   6. Simplified 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   if -8.1999999999999998e-116 < im < 6.5999999999999995e-185 or 2.2000000000000001e-157 < im < 1.39999999999999998e-111

   1. Initial program 35.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 35.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 71.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around 0 43.9%

      \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 43.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

      2. unpow2 43.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

      3. rem-square-sqrt 44.8%

         \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

      4. metadata-eval 44.8%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      5. *-lft-identity 44.8%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 6.5999999999999995e-185 < im < 2.2000000000000001e-157 or 1.39999999999999998e-111 < im

   1. Initial program 43.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 43.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 83.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 63.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 63.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{-185} \lor \neg \left(im \leq 2.2 \cdot 10^{-157}\right) \land im \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 5: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 5.1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.05e-116)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 5.1e-185) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.05e-116) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 5.1e-185) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.05d-116)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 5.1d-185) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.05e-116) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 5.1e-185) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.05e-116:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 5.1e-185:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.05e-116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 5.1e-185)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.05e-116)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 5.1e-185)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.05e-116], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.1e-185], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.05e-116

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 80.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around -inf 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + -1 \cdot im\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\left(-im\right)}\right)} \]

      2. sub-neg 70.0%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   6. Simplified 70.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re - im\right)}} \]

   if -1.05e-116 < im < 5.1000000000000003e-185

   1. Initial program 31.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 31.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 70.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around 0 40.7%

      \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 40.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

      2. unpow2 40.7%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

      3. rem-square-sqrt 41.4%

         \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

      4. metadata-eval 41.4%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      5. *-lft-identity 41.4%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 41.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 5.1000000000000003e-185 < im

   1. Initial program 44.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 83.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in re around 0 61.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im + 2 \cdot re}} \]
   5. Step-by-step derivation

      1. distribute-lft-out 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(im + re\right)}} \]

      2. +-commutative 61.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]

      3. *-commutative 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(re + im\right) \cdot 2}} \]

      4. +-commutative 61.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right)} \cdot 2} \]

   6. Simplified 61.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 5.1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6: 43.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.4e-116) (* 0.5 (sqrt (* im -2.0))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.4e-116) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-3.4d-116)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.4e-116) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.4e-116:
		tmp = 0.5 * math.sqrt((im * -2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.4e-116)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.4e-116)
		tmp = 0.5 * sqrt((im * -2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.4e-116], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.39999999999999992e-116

   1. Initial program 45.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 80.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around -inf 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 69.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot -2}} \]

   6. Simplified 69.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot -2}} \]

   if -3.39999999999999992e-116 < im

   1. Initial program 40.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 40.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

      2. hypot-def 78.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   4. Taylor expanded in im around 0 30.8%

      \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 30.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

      2. unpow2 30.8%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

      3. rem-square-sqrt 31.4%

         \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

      4. metadata-eval 31.4%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      5. *-lft-identity 31.4%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   6. Simplified 31.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 7: 26.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

double code(double re, double im) {
	return sqrt(re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

def code(re, im):
	return math.sqrt(re)

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt(re);
end

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 42.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Step-by-step derivation

   1. +-commutative 42.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]

   2. hypot-def 79.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

3. Simplified 79.1%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

4. Taylor expanded in im around 0 24.9%

   \[\leadsto \color{blue}{0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 24.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \sqrt{re}} \]

   2. unpow2 24.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}\right) \cdot \sqrt{re} \]

   3. rem-square-sqrt 25.4%

      \[\leadsto \left(0.5 \cdot \color{blue}{2}\right) \cdot \sqrt{re} \]

   4. metadata-eval 25.4%

      \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   5. *-lft-identity 25.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]

6. Simplified 25.4%

   \[\leadsto \color{blue}{\sqrt{re}} \]

7. Final simplification 25.4%

   \[\leadsto \sqrt{re} \]

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 2023193 
(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)))))