math.sqrt on complex, real part

Percentage Accurate: 41.2% → 83.5%

Time: 8.8s

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: 41.2% 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: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.35e+154)
   (* 0.5 (sqrt (* 2.0 (* -0.5 (* im (/ im re))))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.35e+154) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.35e+154) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.35e+154:
		tmp = 0.5 * math.sqrt((2.0 * (-0.5 * (im * (im / re)))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.35e+154)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(im * Float64(im / re))))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.35e+154)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.35e+154], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.34999999999999992e154

   1. Initial program 2.8%

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

      1. hypot-udef 29.6%

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

      2. add-exp-log 16.9%

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

   3. Applied egg-rr 16.9%

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

   4. Taylor expanded in re around -inf 51.3%

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

      1. unpow2 51.3%

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

      2. associate-*r/ 61.5%

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

   6. Simplified 61.5%

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

   if -2.34999999999999992e154 < re

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. +-commutative 45.4%

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

      3. sqr-neg 45.4%

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

      4. distribute-rgt-in 45.4%

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

      5. cancel-sign-sub 45.4%

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

      6. distribute-rgt-out-- 45.4%

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

      7. sub-neg 45.4%

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

      8. remove-double-neg 45.4%

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

      9. hypot-def 85.4%

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

   3. Simplified 85.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.4e+113)
   (* 0.5 (sqrt (* 2.0 (* -0.5 (* im (/ im re))))))
   (if (<= re 2.3e-18)
     (* 0.5 (sqrt (* 2.0 (+ re (+ im (* 0.5 (/ (* re re) im)))))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.4e+113) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (re <= 2.3e-18) {
		tmp = 0.5 * sqrt((2.0 * (re + (im + (0.5 * ((re * re) / im))))));
	} else {
		tmp = 0.5 * (2.0 * 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 (re <= (-1.4d+113)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((-0.5d0) * (im * (im / re)))))
    else if (re <= 2.3d-18) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + (im + (0.5d0 * ((re * re) / im))))))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.4e+113) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (re <= 2.3e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + (im + (0.5 * ((re * re) / im))))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.4e+113:
		tmp = 0.5 * math.sqrt((2.0 * (-0.5 * (im * (im / re)))))
	elif re <= 2.3e-18:
		tmp = 0.5 * math.sqrt((2.0 * (re + (im + (0.5 * ((re * re) / im))))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.4e+113)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(im * Float64(im / re))))));
	elseif (re <= 2.3e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + Float64(im + Float64(0.5 * Float64(Float64(re * re) / im)))))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.4e+113)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	elseif (re <= 2.3e-18)
		tmp = 0.5 * sqrt((2.0 * (re + (im + (0.5 * ((re * re) / im))))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.4e+113], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[(im + N[(0.5 * N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.39999999999999999e113

   1. Initial program 5.1%

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

      1. hypot-udef 32.0%

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

      2. add-exp-log 18.5%

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

   3. Applied egg-rr 18.5%

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

   4. Taylor expanded in re around -inf 48.4%

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

      1. unpow2 48.4%

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

      2. associate-*r/ 57.4%

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

   6. Simplified 57.4%

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

   if -1.39999999999999999e113 < re < 2.3000000000000001e-18

   1. Initial program 44.1%

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

   2. Taylor expanded in re around 0 38.5%

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

      1. unpow2 38.5%

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

   4. Simplified 38.5%

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

   if 2.3000000000000001e-18 < re

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. +-commutative 52.1%

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

      3. sqr-neg 52.1%

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

      4. distribute-rgt-in 52.1%

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

      5. cancel-sign-sub 52.1%

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

      6. distribute-rgt-out-- 52.1%

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

      7. sub-neg 52.1%

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

      8. remove-double-neg 52.1%

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

      9. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. rem-square-sqrt 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + 0.5 \cdot \frac{re \cdot re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 3: 50.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+117)
   (* 0.5 (sqrt (* 2.0 (* -0.5 (* im (/ im re))))))
   (if (<= re 2.8e-20) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+117) {
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (re <= 2.8e-20) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 (re <= (-3.6d+117)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((-0.5d0) * (im * (im / re)))))
    else if (re <= 2.8d-20) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+117) {
		tmp = 0.5 * Math.sqrt((2.0 * (-0.5 * (im * (im / re)))));
	} else if (re <= 2.8e-20) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e+117:
		tmp = 0.5 * math.sqrt((2.0 * (-0.5 * (im * (im / re)))))
	elif re <= 2.8e-20:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+117)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(-0.5 * Float64(im * Float64(im / re))))));
	elseif (re <= 2.8e-20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e+117)
		tmp = 0.5 * sqrt((2.0 * (-0.5 * (im * (im / re)))));
	elseif (re <= 2.8e-20)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+117], N[(0.5 * N[Sqrt[N[(2.0 * N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-20], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.60000000000000013e117

   1. Initial program 5.1%

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

      1. hypot-udef 32.0%

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

      2. add-exp-log 18.5%

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

   3. Applied egg-rr 18.5%

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

   4. Taylor expanded in re around -inf 48.4%

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

      1. unpow2 48.4%

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

      2. associate-*r/ 57.4%

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

   6. Simplified 57.4%

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

   if -3.60000000000000013e117 < re < 2.8000000000000003e-20

   1. Initial program 44.1%

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

      1. sqr-neg 44.1%

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

      2. +-commutative 44.1%

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

      3. sqr-neg 44.1%

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

      4. distribute-rgt-in 44.1%

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

      5. cancel-sign-sub 44.1%

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

      6. distribute-rgt-out-- 44.1%

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

      7. sub-neg 44.1%

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

      8. remove-double-neg 44.1%

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

      9. hypot-def 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in re around 0 37.6%

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

      1. *-commutative 37.6%

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

   6. Simplified 37.6%

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

   if 2.8000000000000003e-20 < re

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. +-commutative 52.1%

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

      3. sqr-neg 52.1%

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

      4. distribute-rgt-in 52.1%

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

      5. cancel-sign-sub 52.1%

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

      6. distribute-rgt-out-- 52.1%

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

      7. sub-neg 52.1%

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

      8. remove-double-neg 52.1%

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

      9. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. rem-square-sqrt 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \left(im \cdot \frac{im}{re}\right)\right)}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 4: 48.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e+154)
   (* 0.5 (sqrt (/ (* im (- im)) re)))
   (if (<= re 1.06e-17) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+154) {
		tmp = 0.5 * sqrt(((im * -im) / re));
	} else if (re <= 1.06e-17) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 (re <= (-2.3d+154)) then
        tmp = 0.5d0 * sqrt(((im * -im) / re))
    else if (re <= 1.06d-17) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+154) {
		tmp = 0.5 * Math.sqrt(((im * -im) / re));
	} else if (re <= 1.06e-17) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.3e+154:
		tmp = 0.5 * math.sqrt(((im * -im) / re))
	elif re <= 1.06e-17:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e+154)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-im)) / re)));
	elseif (re <= 1.06e-17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.3e+154)
		tmp = 0.5 * sqrt(((im * -im) / re));
	elseif (re <= 1.06e-17)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.3e+154], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.06e-17], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.3e154

   1. Initial program 2.8%

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

      1. sqr-neg 2.8%

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

      2. +-commutative 2.8%

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

      3. sqr-neg 2.8%

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

      4. distribute-rgt-in 2.8%

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

      5. cancel-sign-sub 2.8%

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

      6. distribute-rgt-out-- 2.8%

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

      7. sub-neg 2.8%

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

      8. remove-double-neg 2.8%

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

      9. hypot-def 29.6%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in re around -inf 51.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 51.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{-1 \cdot {im}^{2}}{re}}} \]

      2. neg-mul-1 51.3%

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

      3. unpow2 51.3%

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

      4. distribute-rgt-neg-in 51.3%

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

   6. Simplified 51.3%

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

   if -2.3e154 < re < 1.06000000000000006e-17

   1. Initial program 43.0%

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

      1. sqr-neg 43.0%

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

      2. +-commutative 43.0%

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

      3. sqr-neg 43.0%

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

      4. distribute-rgt-in 43.0%

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

      5. cancel-sign-sub 43.0%

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

      6. distribute-rgt-out-- 43.0%

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

      7. sub-neg 43.0%

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

      8. remove-double-neg 43.0%

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

      9. hypot-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in re around 0 37.2%

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

      1. *-commutative 37.2%

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

   6. Simplified 37.2%

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

   if 1.06000000000000006e-17 < re

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. +-commutative 52.1%

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

      3. sqr-neg 52.1%

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

      4. distribute-rgt-in 52.1%

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

      5. cancel-sign-sub 52.1%

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

      6. distribute-rgt-out-- 52.1%

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

      7. sub-neg 52.1%

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

      8. remove-double-neg 52.1%

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

      9. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. rem-square-sqrt 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 1.06 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 42.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.7 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.7e-21) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.7e-21) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 (re <= 6.7d-21) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.7e-21) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.7e-21:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.7e-21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.7e-21)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.7e-21], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.6999999999999997e-21

   1. Initial program 34.9%

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

      1. sqr-neg 34.9%

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

      2. +-commutative 34.9%

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

      3. sqr-neg 34.9%

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

      4. distribute-rgt-in 34.9%

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

      5. cancel-sign-sub 34.9%

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

      6. distribute-rgt-out-- 34.9%

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

      7. sub-neg 34.9%

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

      8. remove-double-neg 34.9%

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

      9. hypot-def 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in re around 0 32.0%

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

      1. *-commutative 32.0%

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

   6. Simplified 32.0%

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

   if 6.6999999999999997e-21 < re

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. +-commutative 52.1%

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

      3. sqr-neg 52.1%

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

      4. distribute-rgt-in 52.1%

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

      5. cancel-sign-sub 52.1%

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

      6. distribute-rgt-out-- 52.1%

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

      7. sub-neg 52.1%

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

      8. remove-double-neg 52.1%

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

      9. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.3%

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

      1. *-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. rem-square-sqrt 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.7 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 26.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. sqr-neg 38.8%

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

   2. +-commutative 38.8%

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

   3. sqr-neg 38.8%

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

   4. distribute-rgt-in 38.8%

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

   5. cancel-sign-sub 38.8%

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

   6. distribute-rgt-out-- 38.8%

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

   7. sub-neg 38.8%

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

   8. remove-double-neg 38.8%

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

   9. hypot-def 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in re around 0 27.8%

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

   1. *-commutative 27.8%

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

6. Simplified 27.8%

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

7. Final simplification 27.8%

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

Developer target: 47.9% 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 2023287 
(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)))))