math.sqrt on complex, real part

Percentage Accurate: 42.0% → 82.5%

Time: 5.8s

Alternatives: 4

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: 42.0% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e+160)
   (* 0.5 (sqrt (* (/ im -1.0) (/ im re))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.4e+160) {
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.4e+160) {
		tmp = 0.5 * Math.sqrt(((im / -1.0) * (im / re)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.4e+160:
		tmp = 0.5 * math.sqrt(((im / -1.0) * (im / re)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.4e+160)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * Float64(im / re))));
	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 <= -4.4e+160)
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.4e+160], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(im / re), $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 < -4.39999999999999984e160

   1. Initial program 2.5%

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

      1. sqr-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. sqr-neg 2.5%

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

      4. +-commutative 2.5%

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

      5. distribute-rgt-in 2.5%

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

      6. cancel-sign-sub 2.5%

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

      7. distribute-rgt-out-- 2.5%

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

      8. sub-neg 2.5%

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

      9. remove-double-neg 2.5%

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

      10. +-commutative 2.5%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in re around -inf 70.1%

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

      1. mul-1-neg 70.1%

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

      2. distribute-neg-frac2 70.1%

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

   7. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. neg-mul-1 70.1%

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

      3. times-frac 74.4%

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

   9. Applied egg-rr 74.4%

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

   if -4.39999999999999984e160 < re

   1. Initial program 50.8%

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

      1. sqr-neg 50.8%

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

      2. +-commutative 50.8%

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

      3. sqr-neg 50.8%

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

      4. +-commutative 50.8%

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

      5. distribute-rgt-in 50.8%

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

      6. cancel-sign-sub 50.8%

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

      7. distribute-rgt-out-- 50.8%

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

      8. sub-neg 50.8%

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

      9. remove-double-neg 50.8%

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

      10. +-commutative 50.8%

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

   3. Simplified 89.6%

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

      1. *-commutative 89.6%

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

      2. hypot-define 50.8%

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

      3. +-commutative 50.8%

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

      4. *-commutative 50.8%

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

      5. add-sqr-sqrt 50.4%

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

      6. sqrt-unprod 50.8%

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

      7. *-commutative 50.8%

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

      8. *-commutative 50.8%

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

      9. swap-sqr 50.8%

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

   6. Applied egg-rr 89.6%

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

      1. *-commutative 89.6%

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

      2. associate-*r* 89.6%

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

      3. metadata-eval 89.6%

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

   8. Simplified 89.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e+160)
   (* 0.5 (sqrt (* (/ im -1.0) (/ im re))))
   (if (<= re 4.5e-82) (sqrt (* 0.5 im)) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+160) {
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 4.5e-82) {
		tmp = sqrt((0.5 * im));
	} 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 (re <= (-3.4d+160)) then
        tmp = 0.5d0 * sqrt(((im / (-1.0d0)) * (im / re)))
    else if (re <= 4.5d-82) then
        tmp = sqrt((0.5d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+160) {
		tmp = 0.5 * Math.sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 4.5e-82) {
		tmp = Math.sqrt((0.5 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e+160:
		tmp = 0.5 * math.sqrt(((im / -1.0) * (im / re)))
	elif re <= 4.5e-82:
		tmp = math.sqrt((0.5 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e+160)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * Float64(im / re))));
	elseif (re <= 4.5e-82)
		tmp = sqrt(Float64(0.5 * im));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e+160)
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	elseif (re <= 4.5e-82)
		tmp = sqrt((0.5 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e+160], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e-82], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.4000000000000003e160

   1. Initial program 2.5%

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

      1. sqr-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. sqr-neg 2.5%

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

      4. +-commutative 2.5%

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

      5. distribute-rgt-in 2.5%

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

      6. cancel-sign-sub 2.5%

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

      7. distribute-rgt-out-- 2.5%

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

      8. sub-neg 2.5%

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

      9. remove-double-neg 2.5%

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

      10. +-commutative 2.5%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in re around -inf 70.1%

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

      1. mul-1-neg 70.1%

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

      2. distribute-neg-frac2 70.1%

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

   7. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. neg-mul-1 70.1%

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

      3. times-frac 74.4%

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

   9. Applied egg-rr 74.4%

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

   if -3.4000000000000003e160 < re < 4.4999999999999998e-82

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. +-commutative 52.6%

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

      5. distribute-rgt-in 52.6%

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

      6. cancel-sign-sub 52.6%

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

      7. distribute-rgt-out-- 52.6%

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

      8. sub-neg 52.6%

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

      9. remove-double-neg 52.6%

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

      10. +-commutative 52.6%

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

   3. Simplified 83.1%

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

      1. *-commutative 83.1%

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

      2. hypot-define 52.6%

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

      3. +-commutative 52.6%

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

      4. *-commutative 52.6%

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

      5. add-sqr-sqrt 52.3%

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

      6. sqrt-unprod 52.6%

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

      7. *-commutative 52.6%

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

      8. *-commutative 52.6%

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

      9. swap-sqr 52.6%

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

   6. Applied egg-rr 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-*r* 83.1%

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

      3. metadata-eval 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in re around 0 36.7%

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

   if 4.4999999999999998e-82 < re

   1. Initial program 47.8%

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

      1. sqr-neg 47.8%

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

      2. +-commutative 47.8%

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

      3. sqr-neg 47.8%

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

      4. +-commutative 47.8%

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

      5. distribute-rgt-in 47.8%

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

      6. cancel-sign-sub 47.8%

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

      7. distribute-rgt-out-- 47.8%

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

      8. sub-neg 47.8%

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

      9. remove-double-neg 47.8%

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

      10. +-commutative 47.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. rem-square-sqrt 83.7%

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

      4. associate-*r* 83.7%

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

      5. metadata-eval 83.7%

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

   7. Simplified 83.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+160}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.8e-79) (sqrt (* 0.5 im)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.8e-79) {
		tmp = sqrt((0.5 * im));
	} 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 (re <= 9.8d-79) then
        tmp = sqrt((0.5d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.8e-79) {
		tmp = Math.sqrt((0.5 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.8e-79:
		tmp = math.sqrt((0.5 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.8e-79)
		tmp = sqrt(Float64(0.5 * im));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.8e-79)
		tmp = sqrt((0.5 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.8e-79], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.8000000000000001e-79

   1. Initial program 45.7%

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

      1. sqr-neg 45.7%

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

      2. +-commutative 45.7%

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

      3. sqr-neg 45.7%

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

      4. +-commutative 45.7%

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

      5. distribute-rgt-in 45.7%

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

      6. cancel-sign-sub 45.7%

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

      7. distribute-rgt-out-- 45.7%

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

      8. sub-neg 45.7%

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

      9. remove-double-neg 45.7%

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

      10. +-commutative 45.7%

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

   3. Simplified 72.2%

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

      1. *-commutative 72.2%

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

      2. hypot-define 45.7%

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

      3. +-commutative 45.7%

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

      4. *-commutative 45.7%

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

      5. add-sqr-sqrt 45.4%

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

      6. sqrt-unprod 45.7%

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

      7. *-commutative 45.7%

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

      8. *-commutative 45.7%

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

      9. swap-sqr 45.7%

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

   6. Applied egg-rr 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*r* 72.2%

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

      3. metadata-eval 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in re around 0 32.0%

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

   if 9.8000000000000001e-79 < re

   1. Initial program 47.8%

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

      1. sqr-neg 47.8%

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

      2. +-commutative 47.8%

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

      3. sqr-neg 47.8%

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

      4. +-commutative 47.8%

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

      5. distribute-rgt-in 47.8%

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

      6. cancel-sign-sub 47.8%

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

      7. distribute-rgt-out-- 47.8%

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

      8. sub-neg 47.8%

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

      9. remove-double-neg 47.8%

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

      10. +-commutative 47.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. rem-square-sqrt 83.7%

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

      4. associate-*r* 83.7%

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

      5. metadata-eval 83.7%

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

   7. Simplified 83.7%

      \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9.8 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.4%

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

   1. sqr-neg 46.4%

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

   2. +-commutative 46.4%

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

   3. sqr-neg 46.4%

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

   4. +-commutative 46.4%

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

   5. distribute-rgt-in 46.4%

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

   6. cancel-sign-sub 46.4%

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

   7. distribute-rgt-out-- 46.4%

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

   8. sub-neg 46.4%

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

   9. remove-double-neg 46.4%

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

   10. +-commutative 46.4%

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

3. Simplified 82.0%

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

   1. *-commutative 82.0%

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

   2. hypot-define 46.4%

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

   3. +-commutative 46.4%

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

   4. *-commutative 46.4%

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

   5. add-sqr-sqrt 46.1%

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

   6. sqrt-unprod 46.4%

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

   7. *-commutative 46.4%

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

   8. *-commutative 46.4%

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

   9. swap-sqr 46.4%

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

6. Applied egg-rr 82.0%

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

   1. *-commutative 82.0%

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

   2. associate-*r* 82.0%

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

   3. metadata-eval 82.0%

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

8. Simplified 82.0%

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

9. Taylor expanded in re around 0 24.1%

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

10. Final simplification 24.1%

    \[\leadsto \sqrt{0.5 \cdot im} \]
11. Add Preprocessing

Developer target: 49.2% 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 2024095 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (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)))))