math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.6% → 87.6%

Time: 9.7s

Alternatives: 8

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.5 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.5e+91)
   (sqrt (* 0.5 (- (hypot im re) re)))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+91) {
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.5e+91) {
		tmp = Math.sqrt((0.5 * (Math.hypot(im, re) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.5e+91:
		tmp = math.sqrt((0.5 * (math.hypot(im, re) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.5e+91)
		tmp = sqrt(Float64(0.5 * Float64(hypot(im, re) - re)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.5e+91)
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.5e+91], N[Sqrt[N[(0.5 * N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.50000000000000003e91

   1. Initial program 53.7%

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

      1. pow1 53.7%

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

   4. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 93.6%

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

      2. *-commutative 93.6%

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

      3. associate-*r* 93.6%

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

      4. metadata-eval 93.6%

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

      5. hypot-undefine 53.7%

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

      6. unpow2 53.7%

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

      7. unpow2 53.7%

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

      8. +-commutative 53.7%

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

      9. unpow2 53.7%

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

      10. unpow2 53.7%

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

      11. hypot-undefine 93.6%

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

   6. Simplified 93.6%

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

   if 1.50000000000000003e91 < re

   1. Initial program 6.2%

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

      1. pow1 6.2%

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

   4. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 33.0%

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

      2. *-commutative 33.0%

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

      3. associate-*r* 33.0%

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

      4. metadata-eval 33.0%

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

      5. hypot-undefine 6.2%

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

      6. unpow2 6.2%

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

      7. unpow2 6.2%

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

      8. +-commutative 6.2%

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

      9. unpow2 6.2%

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

      10. unpow2 6.2%

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

      11. hypot-undefine 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in im around 0 84.3%

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

      1. associate-*l* 84.2%

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

      2. unpow2 84.2%

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

      3. rem-square-sqrt 85.2%

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

   9. Simplified 85.2%

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

       1. sqrt-div 85.1%

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

       2. metadata-eval 85.1%

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

       3. div-inv 85.1%

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

       4. clear-num 85.1%

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

       5. un-div-inv 85.2%

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

       6. div-inv 85.2%

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

       7. metadata-eval 85.2%

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

   11. Applied egg-rr 85.2%

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

       1. *-rgt-identity 85.2%

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

       2. times-frac 85.2%

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

       3. metadata-eval 85.2%

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

       4. *-commutative 85.2%

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

       5. associate-*r/ 85.2%

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

   13. Simplified 85.2%

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

Alternative 2: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e-23)
   (sqrt (- re))
   (if (<= re 7.4e+90) (sqrt (* 0.5 (- im re))) (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-23) {
		tmp = sqrt(-re);
	} else if (re <= 7.4e+90) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (0.5 * im) / 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 <= (-1.5d-23)) then
        tmp = sqrt(-re)
    else if (re <= 7.4d+90) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-23) {
		tmp = Math.sqrt(-re);
	} else if (re <= 7.4e+90) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.5e-23:
		tmp = math.sqrt(-re)
	elif re <= 7.4e+90:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.5e-23)
		tmp = sqrt(Float64(-re));
	elseif (re <= 7.4e+90)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e-23)
		tmp = sqrt(-re);
	elseif (re <= 7.4e+90)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.5e-23], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 7.4e+90], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.50000000000000001e-23

   1. Initial program 49.9%

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

      1. pow1 49.9%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 49.9%

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

      6. unpow2 49.9%

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

      7. unpow2 49.9%

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

      8. +-commutative 49.9%

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

      9. unpow2 49.9%

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

      10. unpow2 49.9%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 73.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 73.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 73.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -1.50000000000000001e-23 < re < 7.4e90

   1. Initial program 55.5%

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

      1. pow1 55.5%

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

   4. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*r* 90.7%

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

      4. metadata-eval 90.7%

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

      5. hypot-undefine 55.5%

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

      6. unpow2 55.5%

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

      7. unpow2 55.5%

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

      8. +-commutative 55.5%

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

      9. unpow2 55.5%

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

      10. unpow2 55.5%

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

      11. hypot-undefine 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in re around 0 79.1%

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

      1. neg-mul-1 79.1%

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

      2. unsub-neg 79.1%

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

   9. Simplified 79.1%

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

   if 7.4e90 < re

   1. Initial program 6.2%

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

      1. pow1 6.2%

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

   4. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 33.0%

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

      2. *-commutative 33.0%

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

      3. associate-*r* 33.0%

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

      4. metadata-eval 33.0%

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

      5. hypot-undefine 6.2%

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

      6. unpow2 6.2%

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

      7. unpow2 6.2%

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

      8. +-commutative 6.2%

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

      9. unpow2 6.2%

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

      10. unpow2 6.2%

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

      11. hypot-undefine 33.0%

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

   6. Simplified 33.0%

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

   7. Taylor expanded in im around 0 84.3%

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

      1. associate-*l* 84.2%

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

      2. unpow2 84.2%

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

      3. rem-square-sqrt 85.2%

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

   9. Simplified 85.2%

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

       1. sqrt-div 85.1%

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

       2. metadata-eval 85.1%

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

       3. div-inv 85.1%

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

       4. clear-num 85.1%

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

       5. un-div-inv 85.2%

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

       6. div-inv 85.2%

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

       7. metadata-eval 85.2%

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

   11. Applied egg-rr 85.2%

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

       1. *-rgt-identity 85.2%

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

       2. times-frac 85.2%

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

       3. metadata-eval 85.2%

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

       4. *-commutative 85.2%

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

       5. associate-*r/ 85.2%

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

   13. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot im}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.2e-24)
   (sqrt (- re))
   (if (<= re 4.2e+90) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.2e-24) {
		tmp = sqrt(-re);
	} else if (re <= 4.2e+90) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / 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 <= (-5.2d-24)) then
        tmp = sqrt(-re)
    else if (re <= 4.2d+90) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.2e-24) {
		tmp = Math.sqrt(-re);
	} else if (re <= 4.2e+90) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.2e-24:
		tmp = math.sqrt(-re)
	elif re <= 4.2e+90:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.2e-24)
		tmp = sqrt(Float64(-re));
	elseif (re <= 4.2e+90)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.2e-24)
		tmp = sqrt(-re);
	elseif (re <= 4.2e+90)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.2e-24], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 4.2e+90], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.2e-24

   1. Initial program 49.9%

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

      1. pow1 49.9%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 49.9%

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

      6. unpow2 49.9%

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

      7. unpow2 49.9%

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

      8. +-commutative 49.9%

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

      9. unpow2 49.9%

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

      10. unpow2 49.9%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 73.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 73.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 73.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -5.2e-24 < re < 4.19999999999999961e90

   1. Initial program 55.5%

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

      1. pow1 55.5%

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

   4. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*r* 90.7%

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

      4. metadata-eval 90.7%

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

      5. hypot-undefine 55.5%

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

      6. unpow2 55.5%

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

      7. unpow2 55.5%

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

      8. +-commutative 55.5%

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

      9. unpow2 55.5%

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

      10. unpow2 55.5%

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

      11. hypot-undefine 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in re around 0 79.1%

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

      1. neg-mul-1 79.1%

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

      2. unsub-neg 79.1%

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

   9. Simplified 79.1%

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

   if 4.19999999999999961e90 < re

   1. Initial program 6.2%

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

   3. Taylor expanded in re around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l* 84.3%

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

      3. *-commutative 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l* 84.5%

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

   5. Simplified 84.5%

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

      1. add-cube-cbrt 83.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)} \cdot \sqrt[3]{\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right) \cdot \sqrt[3]{\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}} \]

      2. pow3 83.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3}} \]

      3. associate-*r* 83.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot im\right)} \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3} \]

      4. sqrt-unprod 83.6%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{\sqrt{2 \cdot 0.5}} \cdot im\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3} \]

      5. metadata-eval 83.6%

         \[\leadsto {\left(\sqrt[3]{\left(\sqrt{\color{blue}{1}} \cdot im\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3} \]

      6. metadata-eval 83.6%

         \[\leadsto {\left(\sqrt[3]{\left(\color{blue}{1} \cdot im\right) \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3} \]

      7. *-un-lft-identity 83.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{im} \cdot \left(\sqrt{\frac{1}{re}} \cdot 0.5\right)}\right)}^{3} \]

      8. *-commutative 83.6%

         \[\leadsto {\left(\sqrt[3]{im \cdot \color{blue}{\left(0.5 \cdot \sqrt{\frac{1}{re}}\right)}}\right)}^{3} \]

      9. sqrt-div 83.5%

         \[\leadsto {\left(\sqrt[3]{im \cdot \left(0.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}\right)}\right)}^{3} \]

      10. metadata-eval 83.5%

          \[\leadsto {\left(\sqrt[3]{im \cdot \left(0.5 \cdot \frac{\color{blue}{1}}{\sqrt{re}}\right)}\right)}^{3} \]

      11. un-div-inv 83.5%

          \[\leadsto {\left(\sqrt[3]{im \cdot \color{blue}{\frac{0.5}{\sqrt{re}}}}\right)}^{3} \]

   7. Applied egg-rr 83.5%

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

      1. rem-cube-cbrt 85.1%

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

      2. *-commutative 85.1%

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

      3. add-sqr-sqrt 84.9%

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

      4. sqrt-unprod 85.1%

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

      5. frac-times 85.0%

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

      6. metadata-eval 85.0%

         \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot im \]

      7. add-sqr-sqrt 85.2%

         \[\leadsto \sqrt{\frac{0.25}{\color{blue}{re}}} \cdot im \]

   9. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}} \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.2 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.2e-23)
   (sqrt (- re))
   (if (<= re 4.2e+90) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-23) {
		tmp = sqrt(-re);
	} else if (re <= 4.2e+90) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / 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 <= (-1.2d-23)) then
        tmp = sqrt(-re)
    else if (re <= 4.2d+90) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.2e-23) {
		tmp = Math.sqrt(-re);
	} else if (re <= 4.2e+90) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.2e-23:
		tmp = math.sqrt(-re)
	elif re <= 4.2e+90:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.2e-23)
		tmp = sqrt(Float64(-re));
	elseif (re <= 4.2e+90)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.2e-23)
		tmp = sqrt(-re);
	elseif (re <= 4.2e+90)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.2e-23], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 4.2e+90], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.19999999999999998e-23

   1. Initial program 49.9%

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

      1. pow1 49.9%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 49.9%

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

      6. unpow2 49.9%

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

      7. unpow2 49.9%

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

      8. +-commutative 49.9%

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

      9. unpow2 49.9%

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

      10. unpow2 49.9%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 73.1%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 73.1%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 73.1%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -1.19999999999999998e-23 < re < 4.19999999999999961e90

   1. Initial program 55.5%

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

      1. pow1 55.5%

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

   4. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*r* 90.7%

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

      4. metadata-eval 90.7%

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

      5. hypot-undefine 55.5%

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

      6. unpow2 55.5%

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

      7. unpow2 55.5%

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

      8. +-commutative 55.5%

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

      9. unpow2 55.5%

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

      10. unpow2 55.5%

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

      11. hypot-undefine 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in re around 0 79.1%

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

      1. neg-mul-1 79.1%

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

      2. unsub-neg 79.1%

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

   9. Simplified 79.1%

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

   if 4.19999999999999961e90 < re

   1. Initial program 6.2%

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

   3. Taylor expanded in re around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l* 84.3%

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

      3. *-commutative 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*l* 84.5%

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

   5. Simplified 84.5%

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

      1. pow1 84.5%

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

      2. associate-*r* 84.3%

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

      3. sqrt-unprod 85.2%

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

      4. metadata-eval 85.2%

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

      5. metadata-eval 85.2%

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

      6. *-un-lft-identity 85.2%

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

      7. *-commutative 85.2%

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

      8. sqrt-div 85.1%

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

      9. metadata-eval 85.1%

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

      10. un-div-inv 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. unpow1 85.1%

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

   9. Simplified 85.1%

      \[\leadsto \color{blue}{im \cdot \frac{0.5}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.5e-152) (sqrt (- re)) (sqrt (* 0.5 (- im re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.5e-152) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((0.5 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.5d-152) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((0.5d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.5e-152) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.5e-152:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.5e-152)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.5e-152)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.5e-152], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.5000000000000004e-152

   1. Initial program 45.3%

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

      1. pow1 45.3%

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

   4. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-*r* 79.9%

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

      4. metadata-eval 79.9%

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

      5. hypot-undefine 45.3%

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

      6. unpow2 45.3%

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

      7. unpow2 45.3%

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

      8. +-commutative 45.3%

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

      9. unpow2 45.3%

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

      10. unpow2 45.3%

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

      11. hypot-undefine 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in re around -inf 57.3%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 57.3%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 57.3%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if 4.5000000000000004e-152 < im

   1. Initial program 44.7%

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

      1. pow1 44.7%

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

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*r* 83.1%

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

      4. metadata-eval 83.1%

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

      5. hypot-undefine 44.7%

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

      6. unpow2 44.7%

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

      7. unpow2 44.7%

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

      8. +-commutative 44.7%

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

      9. unpow2 44.7%

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

      10. unpow2 44.7%

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

      11. hypot-undefine 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in re around 0 70.8%

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

      1. neg-mul-1 70.8%

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

      2. unsub-neg 70.8%

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

   9. Simplified 70.8%

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

Alternative 6: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.6 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.6e-151) (sqrt (- re)) (sqrt (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.6e-151) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((0.5 * 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 <= 8.6d-151) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.6e-151) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.6e-151:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.6e-151)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.6e-151)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.6e-151], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.60000000000000035e-151

   1. Initial program 45.3%

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

      1. pow1 45.3%

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

   4. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-*r* 79.9%

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

      4. metadata-eval 79.9%

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

      5. hypot-undefine 45.3%

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

      6. unpow2 45.3%

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

      7. unpow2 45.3%

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

      8. +-commutative 45.3%

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

      9. unpow2 45.3%

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

      10. unpow2 45.3%

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

      11. hypot-undefine 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in re around -inf 57.3%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 57.3%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 57.3%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if 8.60000000000000035e-151 < im

   1. Initial program 44.7%

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

      1. pow1 44.7%

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

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*r* 83.1%

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

      4. metadata-eval 83.1%

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

      5. hypot-undefine 44.7%

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

      6. unpow2 44.7%

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

      7. unpow2 44.7%

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

      8. +-commutative 44.7%

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

      9. unpow2 44.7%

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

      10. unpow2 44.7%

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

      11. hypot-undefine 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in im around inf 69.7%

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

      1. *-commutative 69.7%

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

   9. Simplified 69.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.6 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d-310)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 59.4%

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

      1. pow1 59.4%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 59.4%

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

      6. unpow2 59.4%

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

      7. unpow2 59.4%

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

      8. +-commutative 59.4%

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

      9. unpow2 59.4%

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

      10. unpow2 59.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 50.2%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 50.2%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 50.2%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   if -4.999999999999985e-310 < re

   1. Initial program 28.1%

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

      1. pow1 28.1%

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

   4. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*r* 61.8%

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

      4. metadata-eval 61.8%

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

      5. hypot-undefine 28.1%

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

      6. unpow2 28.1%

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

      7. unpow2 28.1%

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

      8. +-commutative 28.1%

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

      9. unpow2 28.1%

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

      10. unpow2 28.1%

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

      11. hypot-undefine 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in re around -inf 0.0%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 0.0%

         \[\leadsto \sqrt{\color{blue}{-re}} \]

   9. Simplified 0.0%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

   10. Taylor expanded in re around 0 0.0%

       \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   11. Step-by-step derivation

       1. mul-1-neg 0.0%

          \[\leadsto \sqrt{\color{blue}{-re}} \]

       2. rem-square-sqrt 0.0%

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

       3. distribute-rgt-neg-out 0.0%

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

       4. neg-mul-1 0.0%

          \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left(-1 \cdot \sqrt{re}\right)}} \]

       5. rem-square-sqrt 0.0%

          \[\leadsto \sqrt{\sqrt{re} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{re}\right)} \]

       6. unpow1/2 0.0%

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

       7. metadata-eval 0.0%

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

       8. pow-sqr 0.0%

          \[\leadsto \sqrt{\sqrt{re} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)}\right)} \]

       9. unswap-sqr 0.0%

          \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left(\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right)}} \]

       10. fabs-sqr 0.0%

           \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left|\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right|}} \]

       11. unswap-sqr 0.0%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)}\right|} \]

       12. rem-square-sqrt 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{-1} \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)\right|} \]

       13. pow-sqr 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot \color{blue}{{re}^{\left(2 \cdot 0.25\right)}}\right|} \]

       14. metadata-eval 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot {re}^{\color{blue}{0.5}}\right|} \]

       15. unpow1/2 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot \color{blue}{\sqrt{re}}\right|} \]

       16. neg-mul-1 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{-\sqrt{re}}\right|} \]

       17. fabs-neg 5.9%

           \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left|\sqrt{re}\right|}} \]

       18. rem-sqrt-square 5.9%

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

       19. rem-square-sqrt 5.9%

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

       20. rem-square-sqrt 5.9%

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

   12. Simplified 5.9%

       \[\leadsto \sqrt{\color{blue}{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 2.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 44.8%

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

   1. pow1 44.8%

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

4. Applied egg-rr 82.3%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
5. Step-by-step derivation

   1. unpow1 82.3%

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

   2. *-commutative 82.3%

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

   3. associate-*r* 82.3%

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

   4. metadata-eval 82.3%

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

   5. hypot-undefine 44.8%

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

   6. unpow2 44.8%

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

   7. unpow2 44.8%

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

   8. +-commutative 44.8%

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

   9. unpow2 44.8%

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

   10. unpow2 44.8%

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

   11. hypot-undefine 82.3%

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

6. Simplified 82.3%

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

7. Taylor expanded in re around -inf 26.9%

   \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
8. Step-by-step derivation

   1. neg-mul-1 26.9%

      \[\leadsto \sqrt{\color{blue}{-re}} \]

9. Simplified 26.9%

   \[\leadsto \sqrt{\color{blue}{-re}} \]

10. Taylor expanded in re around 0 26.9%

    \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
11. Step-by-step derivation

    1. mul-1-neg 26.9%

       \[\leadsto \sqrt{\color{blue}{-re}} \]

    2. rem-square-sqrt 0.0%

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

    3. distribute-rgt-neg-out 0.0%

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

    4. neg-mul-1 0.0%

       \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left(-1 \cdot \sqrt{re}\right)}} \]

    5. rem-square-sqrt 0.0%

       \[\leadsto \sqrt{\sqrt{re} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{re}\right)} \]

    6. unpow1/2 0.0%

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

    7. metadata-eval 0.0%

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

    8. pow-sqr 0.0%

       \[\leadsto \sqrt{\sqrt{re} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)}\right)} \]

    9. unswap-sqr 0.0%

       \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left(\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right)}} \]

    10. fabs-sqr 0.0%

        \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left|\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right|}} \]

    11. unswap-sqr 0.0%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)}\right|} \]

    12. rem-square-sqrt 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{-1} \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)\right|} \]

    13. pow-sqr 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot \color{blue}{{re}^{\left(2 \cdot 0.25\right)}}\right|} \]

    14. metadata-eval 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot {re}^{\color{blue}{0.5}}\right|} \]

    15. unpow1/2 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|-1 \cdot \color{blue}{\sqrt{re}}\right|} \]

    16. neg-mul-1 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \left|\color{blue}{-\sqrt{re}}\right|} \]

    17. fabs-neg 2.7%

        \[\leadsto \sqrt{\sqrt{re} \cdot \color{blue}{\left|\sqrt{re}\right|}} \]

    18. rem-sqrt-square 2.7%

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

    19. rem-square-sqrt 2.7%

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

    20. rem-square-sqrt 2.7%

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

12. Simplified 2.7%

    \[\leadsto \sqrt{\color{blue}{re}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))