math.sqrt on complex, real part

Percentage Accurate: 41.6% → 84.7%

Time: 6.8s

Alternatives: 6

Speedup: 1.8×

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: 41.6% 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: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (* (/ im -1.0) (/ im re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * sqrt(((im / -1.0) * (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 (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * Math.sqrt(((im / -1.0) * (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 math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * math.sqrt(((im / -1.0) * (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 (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * 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 (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(im / re), $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 (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 13.5%

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

      1. sqr-neg 13.5%

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

      2. +-commutative 13.5%

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

      3. sqr-neg 13.5%

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

      4. +-commutative 13.5%

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

      5. distribute-rgt-in 13.5%

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

      6. cancel-sign-sub 13.5%

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

      7. distribute-rgt-out-- 13.5%

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

      8. sub-neg 13.5%

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

      9. remove-double-neg 13.5%

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

      10. +-commutative 13.5%

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

      11. hypot-define 13.5%

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

   3. Simplified 13.5%

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

   5. Taylor expanded in re around -inf 54.4%

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

      1. mul-1-neg 54.4%

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

      2. distribute-neg-frac2 54.4%

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

   7. Simplified 54.4%

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

      1. unpow2 54.4%

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

      2. neg-mul-1 54.4%

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

      3. times-frac 64.7%

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

   9. Applied egg-rr 64.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 44.5%

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

      1. sqr-neg 44.5%

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

      2. +-commutative 44.5%

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

      3. sqr-neg 44.5%

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

      4. +-commutative 44.5%

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

      5. distribute-rgt-in 44.4%

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

      6. cancel-sign-sub 44.4%

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

      7. distribute-rgt-out-- 44.5%

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

      8. sub-neg 44.5%

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

      9. remove-double-neg 44.5%

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

      10. +-commutative 44.5%

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

      11. hypot-define 88.1%

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

   3. Simplified 88.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+115)
   (* 0.5 (sqrt (* (/ im -1.0) (/ im re))))
   (if (<= re 1.05e+38)
     (* 0.5 (sqrt (+ (* 2.0 im) (* re (+ 2.0 (/ re im))))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+115) {
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 1.05e+38) {
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (2.0 + (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 <= (-3.6d+115)) then
        tmp = 0.5d0 * sqrt(((im / (-1.0d0)) * (im / re)))
    else if (re <= 1.05d+38) then
        tmp = 0.5d0 * sqrt(((2.0d0 * im) + (re * (2.0d0 + (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 <= -3.6e+115) {
		tmp = 0.5 * Math.sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 1.05e+38) {
		tmp = 0.5 * Math.sqrt(((2.0 * im) + (re * (2.0 + (re / im)))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e+115:
		tmp = 0.5 * math.sqrt(((im / -1.0) * (im / re)))
	elif re <= 1.05e+38:
		tmp = 0.5 * math.sqrt(((2.0 * im) + (re * (2.0 + (re / im)))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+115)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * Float64(im / re))));
	elseif (re <= 1.05e+38)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * im) + Float64(re * Float64(2.0 + Float64(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 <= -3.6e+115)
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	elseif (re <= 1.05e+38)
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (2.0 + (re / im)))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+115], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+38], N[(0.5 * N[Sqrt[N[(N[(2.0 * im), $MachinePrecision] + N[(re * N[(2.0 + N[(re / im), $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 < -3.6000000000000001e115

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. +-commutative 7.9%

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

      5. distribute-rgt-in 7.8%

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

      6. cancel-sign-sub 7.8%

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

      7. distribute-rgt-out-- 7.9%

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

      8. sub-neg 7.9%

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

      9. remove-double-neg 7.9%

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

      10. +-commutative 7.9%

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

      11. hypot-define 32.9%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in re around -inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. distribute-neg-frac2 50.4%

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

   7. Simplified 50.4%

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

      1. unpow2 50.4%

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

      2. neg-mul-1 50.4%

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

      3. times-frac 63.4%

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

   9. Applied egg-rr 63.4%

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

   if -3.6000000000000001e115 < re < 1.05e38

   1. Initial program 47.6%

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

      1. sqr-neg 47.6%

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

      2. +-commutative 47.6%

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

      3. sqr-neg 47.6%

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

      4. +-commutative 47.6%

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

      5. distribute-rgt-in 47.6%

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

      6. cancel-sign-sub 47.6%

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

      7. distribute-rgt-out-- 47.6%

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

      8. sub-neg 47.6%

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

      9. remove-double-neg 47.6%

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

      10. +-commutative 47.6%

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

      11. hypot-define 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in re around 0 34.4%

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

   if 1.05e38 < re

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. +-commutative 39.9%

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

      3. sqr-neg 39.9%

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

      4. +-commutative 39.9%

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

      5. distribute-rgt-in 39.9%

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

      6. cancel-sign-sub 39.9%

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

      7. distribute-rgt-out-- 39.9%

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

      8. sub-neg 39.9%

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

      9. remove-double-neg 39.9%

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

      10. +-commutative 39.9%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. rem-square-sqrt 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5e+115)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 1.6e+38)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e+115) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 1.6e+38) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-3.5d+115)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 1.6d+38) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -3.5e+115) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 1.6e+38) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.5e+115:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 1.6e+38:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5e+115)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 1.6e+38)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 <= -3.5e+115)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 1.6e+38)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5e+115], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.6e+38], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $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 < -3.50000000000000005e115

   1. Initial program 7.9%

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

   3. Taylor expanded in re around -inf 23.0%

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

      1. mul-1-neg 23.0%

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

   5. Simplified 23.0%

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

   if -3.50000000000000005e115 < re < 1.59999999999999993e38

   1. Initial program 47.6%

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

      1. sqr-neg 47.6%

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

      2. +-commutative 47.6%

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

      3. sqr-neg 47.6%

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

      4. +-commutative 47.6%

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

      5. distribute-rgt-in 47.6%

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

      6. cancel-sign-sub 47.6%

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

      7. distribute-rgt-out-- 47.6%

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

      8. sub-neg 47.6%

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

      9. remove-double-neg 47.6%

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

      10. +-commutative 47.6%

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

      11. hypot-define 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in re around 0 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. *-commutative 34.9%

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

   7. Simplified 34.9%

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

   if 1.59999999999999993e38 < re

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. +-commutative 39.9%

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

      3. sqr-neg 39.9%

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

      4. +-commutative 39.9%

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

      5. distribute-rgt-in 39.9%

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

      6. cancel-sign-sub 39.9%

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

      7. distribute-rgt-out-- 39.9%

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

      8. sub-neg 39.9%

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

      9. remove-double-neg 39.9%

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

      10. +-commutative 39.9%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. rem-square-sqrt 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.6 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.6e+114)
   (* 0.5 (sqrt (* (/ im -1.0) (/ im re))))
   (if (<= re 2e+39)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.6e+114) {
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 2e+39) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-6.6d+114)) then
        tmp = 0.5d0 * sqrt(((im / (-1.0d0)) * (im / re)))
    else if (re <= 2d+39) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -6.6e+114) {
		tmp = 0.5 * Math.sqrt(((im / -1.0) * (im / re)));
	} else if (re <= 2e+39) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.6e+114:
		tmp = 0.5 * math.sqrt(((im / -1.0) * (im / re)))
	elif re <= 2e+39:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.6e+114)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / -1.0) * Float64(im / re))));
	elseif (re <= 2e+39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 <= -6.6e+114)
		tmp = 0.5 * sqrt(((im / -1.0) * (im / re)));
	elseif (re <= 2e+39)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.6e+114], N[(0.5 * N[Sqrt[N[(N[(im / -1.0), $MachinePrecision] * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+39], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $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 < -6.6000000000000001e114

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. +-commutative 7.9%

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

      5. distribute-rgt-in 7.8%

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

      6. cancel-sign-sub 7.8%

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

      7. distribute-rgt-out-- 7.9%

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

      8. sub-neg 7.9%

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

      9. remove-double-neg 7.9%

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

      10. +-commutative 7.9%

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

      11. hypot-define 32.9%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in re around -inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. distribute-neg-frac2 50.4%

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

   7. Simplified 50.4%

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

      1. unpow2 50.4%

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

      2. neg-mul-1 50.4%

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

      3. times-frac 63.4%

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

   9. Applied egg-rr 63.4%

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

   if -6.6000000000000001e114 < re < 1.99999999999999988e39

   1. Initial program 47.6%

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

      1. sqr-neg 47.6%

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

      2. +-commutative 47.6%

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

      3. sqr-neg 47.6%

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

      4. +-commutative 47.6%

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

      5. distribute-rgt-in 47.6%

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

      6. cancel-sign-sub 47.6%

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

      7. distribute-rgt-out-- 47.6%

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

      8. sub-neg 47.6%

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

      9. remove-double-neg 47.6%

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

      10. +-commutative 47.6%

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

      11. hypot-define 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in re around 0 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. *-commutative 34.9%

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

   7. Simplified 34.9%

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

   if 1.99999999999999988e39 < re

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. +-commutative 39.9%

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

      3. sqr-neg 39.9%

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

      4. +-commutative 39.9%

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

      5. distribute-rgt-in 39.9%

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

      6. cancel-sign-sub 39.9%

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

      7. distribute-rgt-out-- 39.9%

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

      8. sub-neg 39.9%

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

      9. remove-double-neg 39.9%

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

      10. +-commutative 39.9%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. rem-square-sqrt 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.6 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{-1} \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.9e+36) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.9e+36) {
		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 <= 1.9d+36) 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 <= 1.9e+36) {
		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 <= 1.9e+36:
		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 <= 1.9e+36)
		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 <= 1.9e+36)
		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, 1.9e+36], 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 < 1.90000000000000012e36

   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. sqr-neg 40.3%

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

      2. +-commutative 40.3%

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

      3. sqr-neg 40.3%

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

      4. +-commutative 40.3%

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

      5. distribute-rgt-in 40.3%

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

      6. cancel-sign-sub 40.3%

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

      7. distribute-rgt-out-- 40.3%

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

      8. sub-neg 40.3%

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

      9. remove-double-neg 40.3%

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

      10. +-commutative 40.3%

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

      11. hypot-define 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in re around 0 29.6%

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

      1. *-commutative 29.6%

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

   7. Simplified 29.6%

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

   if 1.90000000000000012e36 < re

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. +-commutative 39.9%

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

      3. sqr-neg 39.9%

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

      4. +-commutative 39.9%

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

      5. distribute-rgt-in 39.9%

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

      6. cancel-sign-sub 39.9%

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

      7. distribute-rgt-out-- 39.9%

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

      8. sub-neg 39.9%

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

      9. remove-double-neg 39.9%

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

      10. +-commutative 39.9%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. rem-square-sqrt 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 25.4% 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 40.2%

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

   1. sqr-neg 40.2%

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

   2. +-commutative 40.2%

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

   3. sqr-neg 40.2%

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

   4. +-commutative 40.2%

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

   5. distribute-rgt-in 40.2%

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

   6. cancel-sign-sub 40.2%

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

   7. distribute-rgt-out-- 40.2%

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

   8. sub-neg 40.2%

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

   9. remove-double-neg 40.2%

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

   10. +-commutative 40.2%

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

   11. hypot-define 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in re around 0 25.9%

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

   1. *-commutative 25.9%

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

7. Simplified 25.9%

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

8. Final simplification 25.9%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
9. Add Preprocessing

Developer target: 48.6% 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 2024071 
(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)))))