math.sqrt on complex, real part

Percentage Accurate: 41.5% → 86.9%

Time: 8.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.5% 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: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{hypot}\left(im, \frac{\sqrt{{im}^{4} \cdot -0.25}}{re}\right) \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (*
    0.5
    (* (hypot im (/ (sqrt (* (pow im 4.0) -0.25)) re)) (sqrt (/ -1.0 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 * (hypot(im, (sqrt((pow(im, 4.0) * -0.25)) / re)) * sqrt((-1.0 / 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.hypot(im, (Math.sqrt((Math.pow(im, 4.0) * -0.25)) / re)) * Math.sqrt((-1.0 / 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.hypot(im, (math.sqrt((math.pow(im, 4.0) * -0.25)) / re)) * math.sqrt((-1.0 / 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 * Float64(hypot(im, Float64(sqrt(Float64((im ^ 4.0) * -0.25)) / re)) * sqrt(Float64(-1.0 / 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 * (hypot(im, (sqrt(((im ^ 4.0) * -0.25)) / re)) * sqrt((-1.0 / 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[(N[Sqrt[im ^ 2 + N[(N[Sqrt[N[(N[Power[im, 4.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / re), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(-1.0 / 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 13.2%

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

      1. sqr-neg 13.2%

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

      2. +-commutative 13.2%

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

      3. sqr-neg 13.2%

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

      4. +-commutative 13.2%

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

      5. distribute-rgt-in 13.2%

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

      6. cancel-sign-sub 13.2%

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

      7. distribute-rgt-out-- 13.2%

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

      8. sub-neg 13.2%

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

      9. remove-double-neg 13.2%

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

      10. +-commutative 13.2%

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

      11. hypot-define 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in re around -inf 59.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 59.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\sqrt{-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}} \cdot \sqrt{-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}}}} \]

      2. sqrt-unprod 31.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\sqrt{\left(-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right) \cdot \left(-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right)}}} \]

      3. mul-1-neg 31.6%

         \[\leadsto 0.5 \cdot \sqrt{\sqrt{\color{blue}{\left(-\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right)} \cdot \left(-1 \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right)}} \]

      4. mul-1-neg 31.6%

         \[\leadsto 0.5 \cdot \sqrt{\sqrt{\left(-\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right) \cdot \color{blue}{\left(-\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}\right)}}} \]

      5. sqr-neg 31.6%

         \[\leadsto 0.5 \cdot \sqrt{\sqrt{\color{blue}{\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re} \cdot \frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}}}} \]

      6. sqrt-unprod 11.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\sqrt{\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}} \cdot \sqrt{\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}}}} \]

      7. add-sqr-sqrt 11.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}}{re}}} \]

      8. div-inv 11.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}\right) \cdot \frac{1}{re}}} \]

      9. sqrt-prod 0.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{-0.25 \cdot \frac{{im}^{4}}{{re}^{2}} + {im}^{2}} \cdot \sqrt{\frac{1}{re}}\right)} \]

   7. Applied egg-rr 72.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{hypot}\left(im, \frac{\sqrt{{im}^{4} \cdot -0.25}}{re}\right) \cdot \sqrt{\frac{-1}{re}}\right)} \]

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

   1. Initial program 47.0%

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

      1. sqr-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. sqr-neg 47.0%

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

      4. +-commutative 47.0%

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

      5. distribute-rgt-in 47.0%

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

      6. cancel-sign-sub 47.0%

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

      7. distribute-rgt-out-- 47.0%

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

      8. sub-neg 47.0%

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

      9. remove-double-neg 47.0%

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

      10. +-commutative 47.0%

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

      11. hypot-define 88.8%

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

   3. Simplified 88.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{hypot}\left(im, \frac{\sqrt{{im}^{4} \cdot -0.25}}{re}\right) \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (* re (* (pow (* im (/ (sqrt 2.0) re)) 2.0) -0.5))))
   (* 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((re * (pow((im * (sqrt(2.0) / re)), 2.0) * -0.5)));
	} 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((re * (Math.pow((im * (Math.sqrt(2.0) / re)), 2.0) * -0.5)));
	} 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((re * (math.pow((im * (math.sqrt(2.0) / re)), 2.0) * -0.5)))
	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(re * Float64((Float64(im * Float64(sqrt(2.0) / re)) ^ 2.0) * -0.5))));
	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((re * (((im * (sqrt(2.0) / re)) ^ 2.0) * -0.5)));
	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[(re * N[(N[Power[N[(im * N[(N[Sqrt[2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $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 13.2%

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

      1. sqr-neg 13.2%

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

      2. +-commutative 13.2%

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

      3. sqr-neg 13.2%

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

      4. +-commutative 13.2%

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

      5. distribute-rgt-in 13.2%

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

      6. cancel-sign-sub 13.2%

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

      7. distribute-rgt-out-- 13.2%

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

      8. sub-neg 13.2%

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

      9. remove-double-neg 13.2%

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

      10. +-commutative 13.2%

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

      11. hypot-define 13.2%

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

   3. Simplified 13.2%

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

      1. hypot-define 13.2%

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

      2. +-commutative 13.2%

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

      3. distribute-rgt-in 13.2%

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

      4. add-sqr-sqrt 10.2%

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

      5. *-commutative 10.2%

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

      6. fma-define 2.7%

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

      7. *-commutative 2.7%

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

      8. hypot-define 2.7%

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

      9. *-commutative 2.7%

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

      10. hypot-define 2.7%

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

      11. *-commutative 2.7%

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

   6. Applied egg-rr 2.7%

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

   7. Taylor expanded in re around -inf 5.5%

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

      1. mul-1-neg 5.5%

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

      2. *-commutative 5.5%

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

      3. distribute-rgt-neg-in 5.5%

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

      4. associate--l+ 5.5%

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

      5. fma-define 5.5%

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

      6. associate-/l* 5.5%

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

      7. unpow2 5.5%

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

      8. rem-square-sqrt 5.5%

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

      9. unpow2 5.5%

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

      10. rem-square-sqrt 59.8%

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

      11. metadata-eval 59.8%

          \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(0.5, {im}^{2} \cdot \frac{2}{{re}^{2}}, \color{blue}{0}\right) \cdot \left(-re\right)} \]

   9. Simplified 59.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {im}^{2} \cdot \frac{2}{{re}^{2}}, 0\right) \cdot \left(-re\right)}} \]
   10. Step-by-step derivation

       1. distribute-rgt-neg-out 59.8%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\mathsf{fma}\left(0.5, {im}^{2} \cdot \frac{2}{{re}^{2}}, 0\right) \cdot re}} \]

       2. neg-sub0 59.8%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \mathsf{fma}\left(0.5, {im}^{2} \cdot \frac{2}{{re}^{2}}, 0\right) \cdot re}} \]

       3. *-commutative 59.8%

          \[\leadsto 0.5 \cdot \sqrt{0 - \color{blue}{re \cdot \mathsf{fma}\left(0.5, {im}^{2} \cdot \frac{2}{{re}^{2}}, 0\right)}} \]

       4. fma-undefine 59.8%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot \frac{2}{{re}^{2}}\right) + 0\right)}} \]

       5. +-rgt-identity 59.8%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot \frac{2}{{re}^{2}}\right)\right)}} \]

       6. add-sqr-sqrt 59.4%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot \color{blue}{\left(\sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}} \cdot \sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}}\right)}\right)} \]

       7. pow2 59.4%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot \color{blue}{{\left(\sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}}\right)}^{2}}\right)} \]

       8. sqrt-prod 59.4%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\color{blue}{\left(\sqrt{{im}^{2}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}}^{2}\right)} \]

       9. sqrt-pow1 72.4%

          \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(\color{blue}{{im}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2}\right)} \]

       10. metadata-eval 72.4%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left({im}^{\color{blue}{1}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2}\right)} \]

       11. pow1 72.4%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(\color{blue}{im} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2}\right)} \]

       12. sqrt-div 72.5%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(im \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{{re}^{2}}}}\right)}^{2}\right)} \]

       13. sqrt-pow1 72.5%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(im \cdot \frac{\sqrt{2}}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}}\right)}^{2}\right)} \]

       14. metadata-eval 72.5%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(im \cdot \frac{\sqrt{2}}{{re}^{\color{blue}{1}}}\right)}^{2}\right)} \]

       15. pow1 72.5%

           \[\leadsto 0.5 \cdot \sqrt{0 - re \cdot \left(0.5 \cdot {\left(im \cdot \frac{\sqrt{2}}{\color{blue}{re}}\right)}^{2}\right)} \]

   11. Applied egg-rr 72.5%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - re \cdot \left(0.5 \cdot {\left(im \cdot \frac{\sqrt{2}}{re}\right)}^{2}\right)}} \]
   12. Step-by-step derivation

       1. neg-sub0 72.5%

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

       2. distribute-rgt-neg-in 72.5%

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

       3. *-commutative 72.5%

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

       4. distribute-rgt-neg-in 72.5%

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

       5. metadata-eval 72.5%

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

   13. Simplified 72.5%

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

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

   1. Initial program 47.0%

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

      1. sqr-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. sqr-neg 47.0%

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

      4. +-commutative 47.0%

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

      5. distribute-rgt-in 47.0%

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

      6. cancel-sign-sub 47.0%

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

      7. distribute-rgt-out-- 47.0%

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

      8. sub-neg 47.0%

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

      9. remove-double-neg 47.0%

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

      10. +-commutative 47.0%

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

      11. hypot-define 88.8%

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

   3. Simplified 88.8%

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

4. Final simplification 86.7%

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

Alternative 3: 84.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (/ (pow im 2.0) (- 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((pow(im, 2.0) / -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((Math.pow(im, 2.0) / -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((math.pow(im, 2.0) / -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((im ^ 2.0) / Float64(-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 ^ 2.0) / -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[Power[im, 2.0], $MachinePrecision] / (-re)), $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 13.2%

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

      1. sqr-neg 13.2%

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

      2. +-commutative 13.2%

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

      3. sqr-neg 13.2%

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

      4. +-commutative 13.2%

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

      5. distribute-rgt-in 13.2%

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

      6. cancel-sign-sub 13.2%

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

      7. distribute-rgt-out-- 13.2%

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

      8. sub-neg 13.2%

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

      9. remove-double-neg 13.2%

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

      10. +-commutative 13.2%

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

      11. hypot-define 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in re around -inf 62.7%

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

      1. mul-1-neg 62.7%

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

      2. distribute-neg-frac2 62.7%

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

   7. Simplified 62.7%

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

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

   1. Initial program 47.0%

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

      1. sqr-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. sqr-neg 47.0%

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

      4. +-commutative 47.0%

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

      5. distribute-rgt-in 47.0%

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

      6. cancel-sign-sub 47.0%

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

      7. distribute-rgt-out-- 47.0%

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

      8. sub-neg 47.0%

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

      9. remove-double-neg 47.0%

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

      10. +-commutative 47.0%

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

      11. hypot-define 88.8%

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

   3. Simplified 88.8%

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

4. Final simplification 85.4%

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

Alternative 4: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{if}\;re \leq -2.7 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq -520000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (/ (pow im 2.0) (- re))))))
   (if (<= re -2.7e+56)
     t_0
     (if (<= re -520000.0)
       (* 0.5 (sqrt (+ (* 2.0 im) (* re (+ 2.0 (/ re im))))))
       (if (<= re -9e-43)
         t_0
         (if (<= re 5.3e+57)
           (* 0.5 (sqrt (* 2.0 (+ re im))))
           (* 0.5 (* 2.0 (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((pow(im, 2.0) / -re));
	double tmp;
	if (re <= -2.7e+56) {
		tmp = t_0;
	} else if (re <= -520000.0) {
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (2.0 + (re / im)))));
	} else if (re <= -9e-43) {
		tmp = t_0;
	} else if (re <= 5.3e+57) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt(((im ** 2.0d0) / -re))
    if (re <= (-2.7d+56)) then
        tmp = t_0
    else if (re <= (-520000.0d0)) then
        tmp = 0.5d0 * sqrt(((2.0d0 * im) + (re * (2.0d0 + (re / im)))))
    else if (re <= (-9d-43)) then
        tmp = t_0
    else if (re <= 5.3d+57) 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 t_0 = 0.5 * Math.sqrt((Math.pow(im, 2.0) / -re));
	double tmp;
	if (re <= -2.7e+56) {
		tmp = t_0;
	} else if (re <= -520000.0) {
		tmp = 0.5 * Math.sqrt(((2.0 * im) + (re * (2.0 + (re / im)))));
	} else if (re <= -9e-43) {
		tmp = t_0;
	} else if (re <= 5.3e+57) {
		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):
	t_0 = 0.5 * math.sqrt((math.pow(im, 2.0) / -re))
	tmp = 0
	if re <= -2.7e+56:
		tmp = t_0
	elif re <= -520000.0:
		tmp = 0.5 * math.sqrt(((2.0 * im) + (re * (2.0 + (re / im)))))
	elif re <= -9e-43:
		tmp = t_0
	elif re <= 5.3e+57:
		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)
	t_0 = Float64(0.5 * sqrt(Float64((im ^ 2.0) / Float64(-re))))
	tmp = 0.0
	if (re <= -2.7e+56)
		tmp = t_0;
	elseif (re <= -520000.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * im) + Float64(re * Float64(2.0 + Float64(re / im))))));
	elseif (re <= -9e-43)
		tmp = t_0;
	elseif (re <= 5.3e+57)
		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)
	t_0 = 0.5 * sqrt(((im ^ 2.0) / -re));
	tmp = 0.0;
	if (re <= -2.7e+56)
		tmp = t_0;
	elseif (re <= -520000.0)
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (2.0 + (re / im)))));
	elseif (re <= -9e-43)
		tmp = t_0;
	elseif (re <= 5.3e+57)
		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_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(N[Power[im, 2.0], $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.7e+56], t$95$0, If[LessEqual[re, -520000.0], 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], If[LessEqual[re, -9e-43], t$95$0, If[LessEqual[re, 5.3e+57], 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 4 regimes

2. if re < -2.7000000000000001e56 or -5.2e5 < re < -9.0000000000000005e-43

   1. Initial program 11.6%

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

      1. sqr-neg 11.6%

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

      2. +-commutative 11.6%

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

      3. sqr-neg 11.6%

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

      4. +-commutative 11.6%

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

      5. distribute-rgt-in 11.6%

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

      6. cancel-sign-sub 11.6%

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

      7. distribute-rgt-out-- 11.6%

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

      8. sub-neg 11.6%

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

      9. remove-double-neg 11.6%

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

      10. +-commutative 11.6%

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

      11. hypot-define 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in re around -inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. distribute-neg-frac2 65.6%

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

   7. Simplified 65.6%

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

   if -2.7000000000000001e56 < re < -5.2e5

   1. Initial program 29.9%

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

      1. sqr-neg 29.9%

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

      2. +-commutative 29.9%

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

      3. sqr-neg 29.9%

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

      4. +-commutative 29.9%

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

      5. distribute-rgt-in 29.9%

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

      6. cancel-sign-sub 29.9%

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

      7. distribute-rgt-out-- 29.9%

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

      8. sub-neg 29.9%

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

      9. remove-double-neg 29.9%

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

      10. +-commutative 29.9%

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

      11. hypot-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in re around 0 33.7%

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

   if -9.0000000000000005e-43 < re < 5.29999999999999986e57

   1. Initial program 61.2%

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

      1. sqr-neg 61.2%

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

      2. +-commutative 61.2%

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

      3. sqr-neg 61.2%

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

      4. +-commutative 61.2%

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

      5. distribute-rgt-in 61.2%

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

      6. cancel-sign-sub 61.2%

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

      7. distribute-rgt-out-- 61.2%

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

      8. sub-neg 61.2%

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

      9. remove-double-neg 61.2%

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

      10. +-commutative 61.2%

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

      11. hypot-define 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in re around 0 37.0%

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

      1. distribute-lft-out 37.0%

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

      2. *-commutative 37.0%

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

   7. Simplified 37.0%

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

   if 5.29999999999999986e57 < re

   1. Initial program 30.4%

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

      1. sqr-neg 30.4%

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

      2. +-commutative 30.4%

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

      3. sqr-neg 30.4%

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

      4. +-commutative 30.4%

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

      5. distribute-rgt-in 30.4%

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

      6. cancel-sign-sub 30.4%

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

      7. distribute-rgt-out-- 30.4%

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

      8. sub-neg 30.4%

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

      9. remove-double-neg 30.4%

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

      10. +-commutative 30.4%

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

      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 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. rem-square-sqrt 83.3%

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

   7. Simplified 83.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot \sqrt{re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{elif}\;re \leq -520000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}\\ \mathbf{elif}\;re \leq -9 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{+216}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 0}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.8e+216)
   (* 0.5 (sqrt (* re 0.0)))
   (if (<= re 2.3e+49) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.8e+216) {
		tmp = 0.5 * sqrt((re * 0.0));
	} else if (re <= 2.3e+49) {
		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 <= (-8.8d+216)) then
        tmp = 0.5d0 * sqrt((re * 0.0d0))
    else if (re <= 2.3d+49) 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 <= -8.8e+216) {
		tmp = 0.5 * Math.sqrt((re * 0.0));
	} else if (re <= 2.3e+49) {
		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 <= -8.8e+216:
		tmp = 0.5 * math.sqrt((re * 0.0))
	elif re <= 2.3e+49:
		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 <= -8.8e+216)
		tmp = Float64(0.5 * sqrt(Float64(re * 0.0)));
	elseif (re <= 2.3e+49)
		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 <= -8.8e+216)
		tmp = 0.5 * sqrt((re * 0.0));
	elseif (re <= 2.3e+49)
		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, -8.8e+216], N[(0.5 * N[Sqrt[N[(re * 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e+49], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.8e216

   1. Initial program 2.4%

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

      1. sqr-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. sqr-neg 2.4%

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

      4. +-commutative 2.4%

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

      5. distribute-rgt-in 2.4%

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

      6. cancel-sign-sub 2.4%

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

      7. distribute-rgt-out-- 2.4%

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

      8. sub-neg 2.4%

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

      9. remove-double-neg 2.4%

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

      10. +-commutative 2.4%

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

      11. hypot-define 35.4%

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

   3. Simplified 35.4%

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

      1. hypot-define 2.4%

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

      2. +-commutative 2.4%

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

      3. distribute-rgt-in 2.4%

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

      4. add-sqr-sqrt 2.4%

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

      5. *-commutative 2.4%

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

      6. fma-define 2.4%

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

      7. *-commutative 2.4%

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

      8. hypot-define 2.4%

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

      9. *-commutative 2.4%

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

      10. hypot-define 7.3%

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

      11. *-commutative 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in re around -inf 4.1%

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

      1. mul-1-neg 4.1%

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

      2. distribute-rgt-neg-in 4.1%

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

      3. unpow2 4.1%

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

      4. rem-square-sqrt 29.9%

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

      5. metadata-eval 29.9%

         \[\leadsto 0.5 \cdot \sqrt{re \cdot \left(-\color{blue}{0}\right)} \]

      6. metadata-eval 29.9%

         \[\leadsto 0.5 \cdot \sqrt{re \cdot \color{blue}{0}} \]

   9. Simplified 29.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot 0}} \]

   if -8.8e216 < re < 2.30000000000000002e49

   1. Initial program 50.3%

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

      1. sqr-neg 50.3%

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

      2. +-commutative 50.3%

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

      3. sqr-neg 50.3%

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

      4. +-commutative 50.3%

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

      5. distribute-rgt-in 50.3%

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

      6. cancel-sign-sub 50.3%

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

      7. distribute-rgt-out-- 50.3%

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

      8. sub-neg 50.3%

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

      9. remove-double-neg 50.3%

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

      10. +-commutative 50.3%

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

      11. hypot-define 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in re around 0 30.8%

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

      1. *-commutative 30.8%

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

   7. Simplified 30.8%

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

   if 2.30000000000000002e49 < re

   1. Initial program 31.1%

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

      1. sqr-neg 31.1%

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

      2. +-commutative 31.1%

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

      3. sqr-neg 31.1%

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

      4. +-commutative 31.1%

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

      5. distribute-rgt-in 31.1%

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

      6. cancel-sign-sub 31.1%

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

      7. distribute-rgt-out-- 31.1%

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

      8. sub-neg 31.1%

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

      9. remove-double-neg 31.1%

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

      10. +-commutative 31.1%

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

      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 80.7%

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

      1. *-commutative 80.7%

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

      2. unpow2 80.7%

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

      3. rem-square-sqrt 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{+216}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 0}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.5e-211) (* 0.5 (sqrt (* re 0.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-211) {
		tmp = 0.5 * sqrt((re * 0.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-211) then
        tmp = 0.5d0 * sqrt((re * 0.0d0))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.5e-211) {
		tmp = 0.5 * Math.sqrt((re * 0.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.5e-211:
		tmp = 0.5 * math.sqrt((re * 0.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.5e-211)
		tmp = Float64(0.5 * sqrt(Float64(re * 0.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.5e-211)
		tmp = 0.5 * sqrt((re * 0.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.5e-211], N[(0.5 * N[Sqrt[N[(re * 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.5000000000000001e-211

   1. Initial program 41.7%

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

      1. sqr-neg 41.7%

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

      2. +-commutative 41.7%

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

      3. sqr-neg 41.7%

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

      4. +-commutative 41.7%

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

      5. distribute-rgt-in 41.7%

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

      6. cancel-sign-sub 41.7%

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

      7. distribute-rgt-out-- 41.7%

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

      8. sub-neg 41.7%

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

      9. remove-double-neg 41.7%

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

      10. +-commutative 41.7%

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

      11. hypot-define 77.7%

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

   3. Simplified 77.7%

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

      1. hypot-define 41.7%

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

      2. +-commutative 41.7%

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

      3. distribute-rgt-in 41.7%

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

      4. add-sqr-sqrt 41.0%

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

      5. *-commutative 41.0%

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

      6. fma-define 39.3%

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

      7. *-commutative 39.3%

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

      8. hypot-define 39.4%

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

      9. *-commutative 39.4%

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

      10. hypot-define 71.9%

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

      11. *-commutative 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in re around -inf 3.0%

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

      1. mul-1-neg 3.0%

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

      2. distribute-rgt-neg-in 3.0%

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

      3. unpow2 3.0%

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

      4. rem-square-sqrt 8.5%

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

      5. metadata-eval 8.5%

         \[\leadsto 0.5 \cdot \sqrt{re \cdot \left(-\color{blue}{0}\right)} \]

      6. metadata-eval 8.5%

         \[\leadsto 0.5 \cdot \sqrt{re \cdot \color{blue}{0}} \]

   9. Simplified 8.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot 0}} \]

   if 2.5000000000000001e-211 < im

   1. Initial program 43.8%

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

      1. sqr-neg 43.8%

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

      2. +-commutative 43.8%

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

      3. sqr-neg 43.8%

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

      4. +-commutative 43.8%

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

      5. distribute-rgt-in 43.8%

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

      6. cancel-sign-sub 43.8%

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

      7. distribute-rgt-out-- 43.8%

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

      8. sub-neg 43.8%

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

      9. remove-double-neg 43.8%

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

      10. +-commutative 43.8%

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

      11. hypot-define 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in re around 0 54.8%

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

      1. *-commutative 54.8%

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

   7. Simplified 54.8%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.1% 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 42.6%

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

   1. sqr-neg 42.6%

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

   2. +-commutative 42.6%

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

   3. sqr-neg 42.6%

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

   4. +-commutative 42.6%

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

   5. distribute-rgt-in 42.6%

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

   6. cancel-sign-sub 42.6%

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

   7. distribute-rgt-out-- 42.6%

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

   8. sub-neg 42.6%

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

   9. remove-double-neg 42.6%

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

   10. +-commutative 42.6%

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

   11. hypot-define 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in re around 0 24.3%

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

   1. *-commutative 24.3%

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

7. Simplified 24.3%

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

8. Final simplification 24.3%

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

Developer target: 48.1% 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 2024087 
(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)))))