math.sqrt on complex, real part

Percentage Accurate: 41.1% → 84.4%

Time: 10.9s

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: 41.1% 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: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \left({\left(e^{0.25 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+94)
   (*
    0.5
    (*
     (pow (exp (* 0.25 (+ (log (* 0.5 (pow im 2.0))) (log (/ -1.0 re))))) 2.0)
     (sqrt 2.0)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+94) {
		tmp = 0.5 * (pow(exp((0.25 * (log((0.5 * pow(im, 2.0))) + log((-1.0 / re))))), 2.0) * sqrt(2.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.05e+94) {
		tmp = 0.5 * (Math.pow(Math.exp((0.25 * (Math.log((0.5 * Math.pow(im, 2.0))) + Math.log((-1.0 / re))))), 2.0) * Math.sqrt(2.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.05e+94:
		tmp = 0.5 * (math.pow(math.exp((0.25 * (math.log((0.5 * math.pow(im, 2.0))) + math.log((-1.0 / re))))), 2.0) * math.sqrt(2.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.05e+94)
		tmp = Float64(0.5 * Float64((exp(Float64(0.25 * Float64(log(Float64(0.5 * (im ^ 2.0))) + log(Float64(-1.0 / re))))) ^ 2.0) * sqrt(2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.05e+94)
		tmp = 0.5 * ((exp((0.25 * (log((0.5 * (im ^ 2.0))) + log((-1.0 / re))))) ^ 2.0) * sqrt(2.0));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.05e+94], N[(0.5 * N[(N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.04999999999999995e94

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. +-commutative 7.0%

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

      5. distribute-rgt-in 7.0%

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

      6. cancel-sign-sub 7.0%

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

      7. distribute-rgt-out-- 7.0%

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

      8. sub-neg 7.0%

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

      9. remove-double-neg 7.0%

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

      10. +-commutative 7.0%

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

      11. hypot-define 34.0%

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

   3. Simplified 34.0%

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

      1. sqrt-prod 36.1%

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

      2. hypot-define 7.0%

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

      3. +-commutative 7.0%

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

      4. *-commutative 7.0%

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

      5. +-commutative 7.0%

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

      6. hypot-define 36.1%

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

   6. Applied egg-rr 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. pow2 36.1%

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

      3. pow1/2 36.1%

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

      4. sqrt-pow1 36.1%

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

      5. metadata-eval 36.1%

         \[\leadsto 0.5 \cdot \left({\left({\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sqrt{2}\right) \]

   8. Applied egg-rr 36.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{{\left({\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{0.25}\right)}^{2}} \cdot \sqrt{2}\right) \]

   9. Taylor expanded in re around -inf 60.0%

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

   if -1.04999999999999995e94 < re

   1. Initial program 50.2%

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

      1. sqr-neg 50.2%

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

      2. +-commutative 50.2%

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

      3. sqr-neg 50.2%

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

      4. +-commutative 50.2%

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

      5. distribute-rgt-in 50.2%

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

      6. cancel-sign-sub 50.2%

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

      7. distribute-rgt-out-- 50.2%

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

      8. sub-neg 50.2%

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

      9. remove-double-neg 50.2%

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

      10. +-commutative 50.2%

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

      11. hypot-define 89.2%

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

   3. Simplified 89.2%

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

Alternative 2: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.3 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.3e+98)
   (* 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 (re <= -4.3e+98) {
		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 (re <= -4.3e+98) {
		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 re <= -4.3e+98:
		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 (re <= -4.3e+98)
		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 (re <= -4.3e+98)
		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[re, -4.3e+98], 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 re < -4.3000000000000001e98

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. +-commutative 7.0%

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

      5. distribute-rgt-in 7.0%

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

      6. cancel-sign-sub 7.0%

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

      7. distribute-rgt-out-- 7.0%

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

      8. sub-neg 7.0%

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

      9. remove-double-neg 7.0%

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

      10. +-commutative 7.0%

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

      11. hypot-define 34.7%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in re around -inf 55.0%

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

      1. mul-1-neg 55.0%

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

      2. distribute-neg-frac2 55.0%

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

   7. Simplified 55.0%

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

   if -4.3000000000000001e98 < re

   1. Initial program 49.9%

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

      1. sqr-neg 49.9%

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

      2. +-commutative 49.9%

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

      3. sqr-neg 49.9%

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

      4. +-commutative 49.9%

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

      5. distribute-rgt-in 49.9%

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

      6. cancel-sign-sub 49.9%

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

      7. distribute-rgt-out-- 49.9%

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

      8. sub-neg 49.9%

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

      9. remove-double-neg 49.9%

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

      10. +-commutative 49.9%

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

      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. Add Preprocessing

Alternative 3: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im + re \cdot \left(1 + 0.5 \cdot \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e+93)
   (* 0.5 (sqrt (/ (pow im 2.0) (- re))))
   (if (<= re 5.4e+68)
     (sqrt (* 0.5 (+ im (* re (+ 1.0 (* 0.5 (/ re im)))))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e+93) {
		tmp = 0.5 * sqrt((pow(im, 2.0) / -re));
	} else if (re <= 5.4e+68) {
		tmp = sqrt((0.5 * (im + (re * (1.0 + (0.5 * (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 <= (-3.3d+93)) then
        tmp = 0.5d0 * sqrt(((im ** 2.0d0) / -re))
    else if (re <= 5.4d+68) then
        tmp = sqrt((0.5d0 * (im + (re * (1.0d0 + (0.5d0 * (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 <= -3.3e+93) {
		tmp = 0.5 * Math.sqrt((Math.pow(im, 2.0) / -re));
	} else if (re <= 5.4e+68) {
		tmp = Math.sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e+93:
		tmp = 0.5 * math.sqrt((math.pow(im, 2.0) / -re))
	elif re <= 5.4e+68:
		tmp = math.sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e+93)
		tmp = Float64(0.5 * sqrt(Float64((im ^ 2.0) / Float64(-re))));
	elseif (re <= 5.4e+68)
		tmp = sqrt(Float64(0.5 * Float64(im + Float64(re * Float64(1.0 + Float64(0.5 * 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 <= -3.3e+93)
		tmp = 0.5 * sqrt(((im ^ 2.0) / -re));
	elseif (re <= 5.4e+68)
		tmp = sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e+93], N[(0.5 * N[Sqrt[N[(N[Power[im, 2.0], $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.4e+68], N[Sqrt[N[(0.5 * N[(im + N[(re * N[(1.0 + N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.30000000000000009e93

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. +-commutative 7.0%

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

      5. distribute-rgt-in 7.0%

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

      6. cancel-sign-sub 7.0%

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

      7. distribute-rgt-out-- 7.0%

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

      8. sub-neg 7.0%

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

      9. remove-double-neg 7.0%

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

      10. +-commutative 7.0%

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

      11. hypot-define 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in re around -inf 53.9%

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

      1. mul-1-neg 53.9%

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

      2. distribute-neg-frac2 53.9%

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

   7. Simplified 53.9%

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

   if -3.30000000000000009e93 < re < 5.39999999999999982e68

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. +-commutative 53.1%

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

      3. sqr-neg 53.1%

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

      4. +-commutative 53.1%

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

      5. distribute-rgt-in 53.1%

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

      6. cancel-sign-sub 53.1%

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

      7. distribute-rgt-out-- 53.1%

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

      8. sub-neg 53.1%

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

      9. remove-double-neg 53.1%

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

      10. +-commutative 53.1%

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

      11. hypot-define 85.5%

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

   3. Simplified 85.5%

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

      1. sqrt-prod 84.9%

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

      2. hypot-define 52.8%

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

      3. +-commutative 52.8%

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

      4. *-commutative 52.8%

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

      5. +-commutative 52.8%

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

      6. hypot-define 84.9%

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

   6. Applied egg-rr 84.9%

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

   7. Taylor expanded in re around 0 41.2%

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

      1. add-sqr-sqrt 41.1%

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

      2. sqrt-unprod 41.2%

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

      3. swap-sqr 41.2%

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

      4. metadata-eval 41.2%

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

      5. sqrt-unprod 41.4%

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

      6. sqrt-unprod 41.5%

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

      7. add-sqr-sqrt 41.5%

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

      8. *-commutative 41.5%

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

      9. +-commutative 41.5%

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

      10. fma-define 41.5%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(re, 1 + 0.5 \cdot \frac{re}{im}, im\right)}\right)} \]

      11. +-commutative 41.5%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{0.5 \cdot \frac{re}{im} + 1}, im\right)\right)} \]

      12. fma-define 41.5%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, \frac{re}{im}, 1\right)}, im\right)\right)} \]

   9. Applied egg-rr 41.5%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{\color{blue}{0.5} \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)} \]

   11. Simplified 41.5%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

   12. Taylor expanded in re around 0 41.5%

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

   if 5.39999999999999982e68 < re

   1. Initial program 41.5%

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

      1. sqr-neg 41.5%

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

      2. +-commutative 41.5%

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

      3. sqr-neg 41.5%

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

      4. +-commutative 41.5%

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

      5. distribute-rgt-in 41.5%

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

      6. cancel-sign-sub 41.5%

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

      7. distribute-rgt-out-- 41.5%

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

      8. sub-neg 41.5%

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

      9. remove-double-neg 41.5%

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

      10. +-commutative 41.5%

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

      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 84.3%

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

      1. *-commutative 84.3%

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

      2. unpow2 84.3%

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

      3. rem-square-sqrt 85.8%

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

   7. Simplified 85.8%

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

Alternative 4: 40.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.7 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im + re \cdot \left(1 + 0.5 \cdot \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.7e+69)
   (sqrt (* 0.5 (+ im (* re (+ 1.0 (* 0.5 (/ re im)))))))
   (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.7e+69) {
		tmp = sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.7d+69) then
        tmp = sqrt((0.5d0 * (im + (re * (1.0d0 + (0.5d0 * (re / im)))))))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.7e+69) {
		tmp = Math.sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.7e+69:
		tmp = math.sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.7e+69)
		tmp = sqrt(Float64(0.5 * Float64(im + Float64(re * Float64(1.0 + Float64(0.5 * 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 <= 1.7e+69)
		tmp = sqrt((0.5 * (im + (re * (1.0 + (0.5 * (re / im)))))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.7e+69], N[Sqrt[N[(0.5 * N[(im + N[(re * N[(1.0 + N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.69999999999999993e69

   1. Initial program 43.1%

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

      1. sqr-neg 43.1%

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

      2. +-commutative 43.1%

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

      3. sqr-neg 43.1%

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

      4. +-commutative 43.1%

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

      5. distribute-rgt-in 43.1%

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

      6. cancel-sign-sub 43.1%

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

      7. distribute-rgt-out-- 43.1%

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

      8. sub-neg 43.1%

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

      9. remove-double-neg 43.1%

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

      10. +-commutative 43.1%

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

      11. hypot-define 74.3%

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

   3. Simplified 74.3%

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

      1. sqrt-prod 74.3%

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

      2. hypot-define 42.8%

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

      3. +-commutative 42.8%

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

      4. *-commutative 42.8%

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

      5. +-commutative 42.8%

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

      6. hypot-define 74.3%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in re around 0 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 34.2%

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

      3. swap-sqr 33.8%

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

      4. metadata-eval 33.8%

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

      5. sqrt-unprod 33.9%

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

      6. sqrt-unprod 34.0%

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

      7. add-sqr-sqrt 34.0%

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

      8. *-commutative 34.0%

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

      9. +-commutative 34.0%

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

      10. fma-define 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(re, 1 + 0.5 \cdot \frac{re}{im}, im\right)}\right)} \]

      11. +-commutative 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{0.5 \cdot \frac{re}{im} + 1}, im\right)\right)} \]

      12. fma-define 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, \frac{re}{im}, 1\right)}, im\right)\right)} \]

   9. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 34.5%

          \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

       2. metadata-eval 34.5%

          \[\leadsto \sqrt{\color{blue}{0.5} \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)} \]

   11. Simplified 34.5%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

   12. Taylor expanded in re around 0 34.5%

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

   if 1.69999999999999993e69 < re

   1. Initial program 41.5%

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

      1. sqr-neg 41.5%

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

      2. +-commutative 41.5%

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

      3. sqr-neg 41.5%

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

      4. +-commutative 41.5%

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

      5. distribute-rgt-in 41.5%

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

      6. cancel-sign-sub 41.5%

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

      7. distribute-rgt-out-- 41.5%

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

      8. sub-neg 41.5%

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

      9. remove-double-neg 41.5%

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

      10. +-commutative 41.5%

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

      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 84.3%

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

      1. *-commutative 84.3%

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

      2. unpow2 84.3%

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

      3. rem-square-sqrt 85.8%

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

   7. Simplified 85.8%

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

Alternative 5: 41.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.65e+84) (sqrt (* 0.5 (+ re im))) (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+84) {
		tmp = sqrt((0.5 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.65d+84) then
        tmp = sqrt((0.5d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.65e+84) {
		tmp = Math.sqrt((0.5 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.65e+84:
		tmp = math.sqrt((0.5 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.65e+84)
		tmp = sqrt(Float64(0.5 * 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 <= 1.65e+84)
		tmp = sqrt((0.5 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.65e+84], N[Sqrt[N[(0.5 * N[(re + 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 < 1.65000000000000008e84

   1. Initial program 44.2%

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

      1. sqr-neg 44.2%

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

      2. +-commutative 44.2%

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

      3. sqr-neg 44.2%

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

      4. +-commutative 44.2%

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

      5. distribute-rgt-in 44.2%

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

      6. cancel-sign-sub 44.2%

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

      7. distribute-rgt-out-- 44.2%

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

      8. sub-neg 44.2%

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

      9. remove-double-neg 44.2%

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

      10. +-commutative 44.2%

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

      11. hypot-define 75.0%

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

   3. Simplified 75.0%

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

      1. sqrt-prod 75.0%

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

      2. hypot-define 44.0%

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

      3. +-commutative 44.0%

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

      4. *-commutative 44.0%

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

      5. +-commutative 44.0%

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

      6. hypot-define 75.0%

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

   6. Applied egg-rr 75.0%

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

   7. Taylor expanded in re around 0 34.2%

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

      1. add-sqr-sqrt 34.1%

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

      2. sqrt-unprod 34.2%

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

      3. swap-sqr 33.8%

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

      4. metadata-eval 33.8%

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

      5. sqrt-unprod 33.9%

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

      6. sqrt-unprod 34.0%

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

      7. add-sqr-sqrt 34.0%

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

      8. *-commutative 34.0%

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

      9. +-commutative 34.0%

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

      10. fma-define 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(re, 1 + 0.5 \cdot \frac{re}{im}, im\right)}\right)} \]

      11. +-commutative 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{0.5 \cdot \frac{re}{im} + 1}, im\right)\right)} \]

      12. fma-define 34.0%

          \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, \frac{re}{im}, 1\right)}, im\right)\right)} \]

   9. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 34.5%

          \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

       2. metadata-eval 34.5%

          \[\leadsto \sqrt{\color{blue}{0.5} \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)} \]

   11. Simplified 34.5%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

   12. Taylor expanded in re around 0 34.9%

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

   if 1.65000000000000008e84 < re

   1. Initial program 36.2%

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

      1. sqr-neg 36.2%

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

      2. +-commutative 36.2%

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

      3. sqr-neg 36.2%

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

      4. +-commutative 36.2%

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

      5. distribute-rgt-in 36.2%

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

      6. cancel-sign-sub 36.2%

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

      7. distribute-rgt-out-- 36.2%

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

      8. sub-neg 36.2%

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

      9. remove-double-neg 36.2%

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

      10. +-commutative 36.2%

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

      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 88.2%

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

      1. *-commutative 88.2%

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

      2. unpow2 88.2%

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

      3. rem-square-sqrt 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.65 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

   1. sqr-neg 42.7%

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

   2. +-commutative 42.7%

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

   3. sqr-neg 42.7%

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

   4. +-commutative 42.7%

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

   5. distribute-rgt-in 42.7%

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

   6. cancel-sign-sub 42.7%

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

   7. distribute-rgt-out-- 42.7%

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

   8. sub-neg 42.7%

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

   9. remove-double-neg 42.7%

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

   10. +-commutative 42.7%

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

   11. hypot-define 79.7%

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

3. Simplified 79.7%

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

   1. sqrt-prod 79.6%

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

   2. hypot-define 42.5%

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

   3. +-commutative 42.5%

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

   4. *-commutative 42.5%

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

   5. +-commutative 42.5%

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

   6. hypot-define 79.6%

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

6. Applied egg-rr 79.6%

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

7. Taylor expanded in re around 0 29.1%

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

   1. add-sqr-sqrt 29.0%

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

   2. sqrt-unprod 29.1%

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

   3. swap-sqr 28.7%

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

   4. metadata-eval 28.7%

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

   5. sqrt-unprod 28.9%

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

   6. sqrt-unprod 29.0%

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

   7. add-sqr-sqrt 29.0%

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

   8. *-commutative 29.0%

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

   9. +-commutative 29.0%

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

   10. fma-define 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(re, 1 + 0.5 \cdot \frac{re}{im}, im\right)}\right)} \]

   11. +-commutative 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{0.5 \cdot \frac{re}{im} + 1}, im\right)\right)} \]

   12. fma-define 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, \frac{re}{im}, 1\right)}, im\right)\right)} \]

9. Applied egg-rr 29.0%

   \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 29.3%

       \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

    2. metadata-eval 29.3%

       \[\leadsto \sqrt{\color{blue}{0.5} \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)} \]

11. Simplified 29.3%

    \[\leadsto \color{blue}{\sqrt{0.5 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

12. Taylor expanded in re around 0 32.6%

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

13. Final simplification 32.6%

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

Alternative 7: 25.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

   1. sqr-neg 42.7%

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

   2. +-commutative 42.7%

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

   3. sqr-neg 42.7%

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

   4. +-commutative 42.7%

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

   5. distribute-rgt-in 42.7%

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

   6. cancel-sign-sub 42.7%

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

   7. distribute-rgt-out-- 42.7%

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

   8. sub-neg 42.7%

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

   9. remove-double-neg 42.7%

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

   10. +-commutative 42.7%

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

   11. hypot-define 79.7%

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

3. Simplified 79.7%

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

   1. sqrt-prod 79.6%

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

   2. hypot-define 42.5%

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

   3. +-commutative 42.5%

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

   4. *-commutative 42.5%

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

   5. +-commutative 42.5%

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

   6. hypot-define 79.6%

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

6. Applied egg-rr 79.6%

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

7. Taylor expanded in re around 0 29.1%

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

   1. add-sqr-sqrt 29.0%

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

   2. sqrt-unprod 29.1%

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

   3. swap-sqr 28.7%

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

   4. metadata-eval 28.7%

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

   5. sqrt-unprod 28.9%

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

   6. sqrt-unprod 29.0%

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

   7. add-sqr-sqrt 29.0%

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

   8. *-commutative 29.0%

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

   9. +-commutative 29.0%

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

   10. fma-define 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(re, 1 + 0.5 \cdot \frac{re}{im}, im\right)}\right)} \]

   11. +-commutative 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{0.5 \cdot \frac{re}{im} + 1}, im\right)\right)} \]

   12. fma-define 29.0%

       \[\leadsto \sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(0.5, \frac{re}{im}, 1\right)}, im\right)\right)} \]

9. Applied egg-rr 29.0%

   \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(2 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 29.3%

       \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

    2. metadata-eval 29.3%

       \[\leadsto \sqrt{\color{blue}{0.5} \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)} \]

11. Simplified 29.3%

    \[\leadsto \color{blue}{\sqrt{0.5 \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(0.5, \frac{re}{im}, 1\right), im\right)}} \]

12. Taylor expanded in re around 0 29.2%

    \[\leadsto \sqrt{0.5 \cdot \color{blue}{im}} \]
13. Add Preprocessing

Developer target: 47.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024118 
(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)))))