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

Percentage Accurate: 41.4% → 90.4%

Time: 10.7s

Alternatives: 7

Speedup: 1.9×

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: 41.4% 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: 90.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 8.4%

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

      1. sub-neg 8.4%

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

      2. sqr-neg 8.4%

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

      3. sub-neg 8.4%

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

      4. sqr-neg 8.4%

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

      5. hypot-define 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in re around inf 38.9%

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

      1. div-inv 39.0%

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

      2. sqrt-prod 45.5%

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

      3. sqrt-pow1 97.3%

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

      4. metadata-eval 97.3%

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

      5. pow1 97.3%

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

      6. *-commutative 97.3%

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

      7. inv-pow 97.3%

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

      8. sqrt-pow1 97.4%

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sqr-neg 43.3%

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

      3. sub-neg 43.3%

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

      4. sqr-neg 43.3%

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

      5. hypot-define 88.6%

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

   3. Simplified 88.6%

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

      1. *-commutative 88.6%

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

      2. hypot-define 43.3%

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

      3. *-commutative 43.3%

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

      4. add-sqr-sqrt 42.9%

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

      5. sqrt-unprod 43.3%

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

      6. *-commutative 43.3%

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

      7. *-commutative 43.3%

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

      8. swap-sqr 43.3%

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

   6. Applied egg-rr 88.6%

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

      1. associate-*l* 88.6%

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

      2. hypot-undefine 43.3%

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

      3. unpow2 43.3%

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

      4. unpow2 43.3%

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

      5. +-commutative 43.3%

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

      6. unpow2 43.3%

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

      7. unpow2 43.3%

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

      8. hypot-undefine 88.6%

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

      9. metadata-eval 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 89.9%

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

Alternative 2: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6e+87)
   (sqrt (- re))
   (if (<= re 1.1e-49) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6e+87) {
		tmp = sqrt(-re);
	} else if (re <= 1.1e-49) {
		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 <= (-6d+87)) then
        tmp = sqrt(-re)
    else if (re <= 1.1d-49) 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 <= -6e+87) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.1e-49) {
		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 <= -6e+87:
		tmp = math.sqrt(-re)
	elif re <= 1.1e-49:
		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 <= -6e+87)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.1e-49)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6e+87)
		tmp = sqrt(-re);
	elseif (re <= 1.1e-49)
		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, -6e+87], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.1e-49], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.9999999999999998e87

   1. Initial program 26.2%

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

      1. sub-neg 26.2%

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

      2. sqr-neg 26.2%

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

      3. sub-neg 26.2%

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

      4. sqr-neg 26.2%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 26.2%

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

      3. *-commutative 26.2%

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

      4. add-sqr-sqrt 26.1%

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

      5. sqrt-unprod 26.2%

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

      6. *-commutative 26.2%

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

      7. *-commutative 26.2%

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

      8. swap-sqr 26.2%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. hypot-undefine 26.2%

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

      3. unpow2 26.2%

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

      4. unpow2 26.2%

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

      5. +-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. unpow2 26.2%

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

      8. hypot-undefine 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 86.8%

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

       1. neg-mul-1 86.8%

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

   11. Simplified 86.8%

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

   if -5.9999999999999998e87 < re < 1.09999999999999995e-49

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. sqr-neg 55.9%

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

      3. sub-neg 55.9%

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

      4. sqr-neg 55.9%

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

      5. hypot-define 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. hypot-define 55.9%

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

      3. *-commutative 55.9%

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

      4. add-sqr-sqrt 55.5%

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

      5. sqrt-unprod 55.9%

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

      6. *-commutative 55.9%

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

      7. *-commutative 55.9%

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

      8. swap-sqr 55.9%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*l* 90.7%

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

      2. hypot-undefine 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. +-commutative 55.9%

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

      6. unpow2 55.9%

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

      7. unpow2 55.9%

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

      8. hypot-undefine 90.7%

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

      9. metadata-eval 90.7%

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

   8. Simplified 90.7%

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

   9. Taylor expanded in re around 0 75.7%

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

       1. neg-mul-1 75.7%

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

       2. unsub-neg 75.7%

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

   11. Simplified 75.7%

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

   if 1.09999999999999995e-49 < re

   1. Initial program 14.6%

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

      1. sub-neg 14.6%

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

      2. sqr-neg 14.6%

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

      3. sub-neg 14.6%

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

      4. sqr-neg 14.6%

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

      5. hypot-define 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in re around inf 45.5%

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

      1. *-commutative 45.5%

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

      2. sqrt-div 58.1%

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

      3. sqrt-pow1 76.9%

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

      4. metadata-eval 76.9%

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

      5. pow1 76.9%

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

      6. associate-*l/ 76.9%

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

   7. Applied egg-rr 76.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e+86)
   (sqrt (- re))
   (if (<= re 8.2e-50) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+86) {
		tmp = sqrt(-re);
	} else if (re <= 8.2e-50) {
		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 <= (-4.2d+86)) then
        tmp = sqrt(-re)
    else if (re <= 8.2d-50) 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 <= -4.2e+86) {
		tmp = Math.sqrt(-re);
	} else if (re <= 8.2e-50) {
		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 <= -4.2e+86:
		tmp = math.sqrt(-re)
	elif re <= 8.2e-50:
		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 <= -4.2e+86)
		tmp = sqrt(Float64(-re));
	elseif (re <= 8.2e-50)
		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 <= -4.2e+86)
		tmp = sqrt(-re);
	elseif (re <= 8.2e-50)
		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, -4.2e+86], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 8.2e-50], 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 < -4.1999999999999998e86

   1. Initial program 26.2%

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

      1. sub-neg 26.2%

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

      2. sqr-neg 26.2%

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

      3. sub-neg 26.2%

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

      4. sqr-neg 26.2%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 26.2%

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

      3. *-commutative 26.2%

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

      4. add-sqr-sqrt 26.1%

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

      5. sqrt-unprod 26.2%

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

      6. *-commutative 26.2%

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

      7. *-commutative 26.2%

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

      8. swap-sqr 26.2%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. hypot-undefine 26.2%

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

      3. unpow2 26.2%

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

      4. unpow2 26.2%

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

      5. +-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. unpow2 26.2%

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

      8. hypot-undefine 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 86.8%

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

       1. neg-mul-1 86.8%

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

   11. Simplified 86.8%

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

   if -4.1999999999999998e86 < re < 8.19999999999999971e-50

   1. Initial program 55.9%

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

      1. sub-neg 55.9%

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

      2. sqr-neg 55.9%

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

      3. sub-neg 55.9%

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

      4. sqr-neg 55.9%

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

      5. hypot-define 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. hypot-define 55.9%

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

      3. *-commutative 55.9%

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

      4. add-sqr-sqrt 55.5%

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

      5. sqrt-unprod 55.9%

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

      6. *-commutative 55.9%

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

      7. *-commutative 55.9%

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

      8. swap-sqr 55.9%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*l* 90.7%

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

      2. hypot-undefine 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

      5. +-commutative 55.9%

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

      6. unpow2 55.9%

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

      7. unpow2 55.9%

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

      8. hypot-undefine 90.7%

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

      9. metadata-eval 90.7%

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

   8. Simplified 90.7%

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

   9. Taylor expanded in re around 0 75.7%

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

       1. neg-mul-1 75.7%

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

       2. unsub-neg 75.7%

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

   11. Simplified 75.7%

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

   if 8.19999999999999971e-50 < re

   1. Initial program 14.6%

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

      1. sub-neg 14.6%

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

      2. sqr-neg 14.6%

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

      3. sub-neg 14.6%

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

      4. sqr-neg 14.6%

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

      5. hypot-define 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in re around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-*l* 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*r* 75.9%

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

      5. associate-*l* 75.9%

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

      6. *-commutative 75.9%

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

   7. Simplified 75.9%

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

      1. add-sqr-sqrt 75.8%

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

      2. sqrt-unprod 75.9%

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

      3. sqrt-unprod 76.4%

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

      4. metadata-eval 76.4%

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

      5. metadata-eval 76.4%

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

      6. metadata-eval 76.4%

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

      7. sqrt-unprod 76.9%

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

      8. metadata-eval 76.9%

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

      9. metadata-eval 76.9%

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

      10. metadata-eval 76.9%

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

      11. swap-sqr 76.9%

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

      12. add-sqr-sqrt 76.9%

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

      13. metadata-eval 76.9%

          \[\leadsto im \cdot \sqrt{\frac{1}{re} \cdot \color{blue}{0.25}} \]

   9. Applied egg-rr 76.9%

      \[\leadsto im \cdot \color{blue}{\sqrt{\frac{1}{re} \cdot 0.25}} \]
   10. Step-by-step derivation

       1. associate-*l/ 76.9%

          \[\leadsto im \cdot \sqrt{\color{blue}{\frac{1 \cdot 0.25}{re}}} \]

       2. metadata-eval 76.9%

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

   11. Simplified 76.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6e+86)
   (sqrt (- re))
   (if (<= re 7e+237) (sqrt (* 0.5 (- im re))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6e+86) {
		tmp = sqrt(-re);
	} else if (re <= 7e+237) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.6d+86)) then
        tmp = sqrt(-re)
    else if (re <= 7d+237) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6e+86) {
		tmp = Math.sqrt(-re);
	} else if (re <= 7e+237) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6e+86:
		tmp = math.sqrt(-re)
	elif re <= 7e+237:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6e+86)
		tmp = sqrt(Float64(-re));
	elseif (re <= 7e+237)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6e+86)
		tmp = sqrt(-re);
	elseif (re <= 7e+237)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6e+86], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 7e+237], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -1.6e86

   1. Initial program 26.2%

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

      1. sub-neg 26.2%

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

      2. sqr-neg 26.2%

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

      3. sub-neg 26.2%

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

      4. sqr-neg 26.2%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 26.2%

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

      3. *-commutative 26.2%

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

      4. add-sqr-sqrt 26.1%

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

      5. sqrt-unprod 26.2%

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

      6. *-commutative 26.2%

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

      7. *-commutative 26.2%

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

      8. swap-sqr 26.2%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. hypot-undefine 26.2%

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

      3. unpow2 26.2%

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

      4. unpow2 26.2%

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

      5. +-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. unpow2 26.2%

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

      8. hypot-undefine 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 86.8%

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

       1. neg-mul-1 86.8%

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

   11. Simplified 86.8%

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

   if -1.6e86 < re < 6.99999999999999976e237

   1. Initial program 45.3%

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

      1. sub-neg 45.3%

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

      2. sqr-neg 45.3%

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

      3. sub-neg 45.3%

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

      4. sqr-neg 45.3%

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

      5. hypot-define 75.9%

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

   3. Simplified 75.9%

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

      1. *-commutative 75.9%

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

      2. hypot-define 45.3%

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

      3. *-commutative 45.3%

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

      4. add-sqr-sqrt 45.0%

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

      5. sqrt-unprod 45.3%

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

      6. *-commutative 45.3%

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

      7. *-commutative 45.3%

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

      8. swap-sqr 45.3%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*l* 75.9%

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

      2. hypot-undefine 45.3%

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

      3. unpow2 45.3%

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

      4. unpow2 45.3%

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

      5. +-commutative 45.3%

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

      6. unpow2 45.3%

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

      7. unpow2 45.3%

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

      8. hypot-undefine 75.9%

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

      9. metadata-eval 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in re around 0 63.0%

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

       1. neg-mul-1 63.0%

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

       2. unsub-neg 63.0%

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

   11. Simplified 63.0%

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

   if 6.99999999999999976e237 < re

   1. Initial program 2.6%

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

   3. Taylor expanded in re around inf 31.3%

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

   4. Taylor expanded in re around 0 31.3%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]
   5. Step-by-step derivation

      1. metadata-eval 31.3%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+237}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+60)
   (sqrt (- re))
   (if (<= re 6.8e+238) (sqrt (* im 0.5)) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+60) {
		tmp = sqrt(-re);
	} else if (re <= 6.8e+238) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.2d+60)) then
        tmp = sqrt(-re)
    else if (re <= 6.8d+238) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+60) {
		tmp = Math.sqrt(-re);
	} else if (re <= 6.8e+238) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+60:
		tmp = math.sqrt(-re)
	elif re <= 6.8e+238:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+60)
		tmp = sqrt(Float64(-re));
	elseif (re <= 6.8e+238)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e+60)
		tmp = sqrt(-re);
	elseif (re <= 6.8e+238)
		tmp = sqrt((im * 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+60], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 6.8e+238], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -3.19999999999999991e60

   1. Initial program 30.0%

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

      1. sub-neg 30.0%

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

      2. sqr-neg 30.0%

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

      3. sub-neg 30.0%

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

      4. sqr-neg 30.0%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 30.0%

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

      3. *-commutative 30.0%

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

      4. add-sqr-sqrt 29.8%

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

      5. sqrt-unprod 30.0%

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

      6. *-commutative 30.0%

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

      7. *-commutative 30.0%

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

      8. swap-sqr 30.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. hypot-undefine 30.0%

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

      3. unpow2 30.0%

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

      4. unpow2 30.0%

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

      5. +-commutative 30.0%

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

      6. unpow2 30.0%

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

      7. unpow2 30.0%

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

      8. hypot-undefine 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 84.3%

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

       1. neg-mul-1 84.3%

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

   11. Simplified 84.3%

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

   if -3.19999999999999991e60 < re < 6.7999999999999995e238

   1. Initial program 44.6%

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

      1. sub-neg 44.6%

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

      2. sqr-neg 44.6%

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

      3. sub-neg 44.6%

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

      4. sqr-neg 44.6%

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

      5. hypot-define 75.3%

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

   3. Simplified 75.3%

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

      1. *-commutative 75.3%

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

      2. hypot-define 44.6%

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

      3. *-commutative 44.6%

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

      4. add-sqr-sqrt 44.3%

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

      5. sqrt-unprod 44.6%

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

      6. *-commutative 44.6%

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

      7. *-commutative 44.6%

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

      8. swap-sqr 44.6%

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

   6. Applied egg-rr 75.3%

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

      1. associate-*l* 75.3%

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

      2. hypot-undefine 44.6%

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

      3. unpow2 44.6%

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

      4. unpow2 44.6%

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

      5. +-commutative 44.6%

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

      6. unpow2 44.6%

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

      7. unpow2 44.6%

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

      8. hypot-undefine 75.3%

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

      9. metadata-eval 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in im around inf 62.9%

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

   if 6.7999999999999995e238 < re

   1. Initial program 2.6%

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

   3. Taylor expanded in re around inf 31.3%

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

   4. Taylor expanded in re around 0 31.3%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]
   5. Step-by-step derivation

      1. metadata-eval 31.3%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 31.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -1e-310) (sqrt (- re)) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-310) {
		tmp = sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1d-310)) then
        tmp = sqrt(-re)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e-310)
		tmp = sqrt(-re);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-310], N[Sqrt[(-re)], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < -9.999999999999969e-311

   1. Initial program 46.7%

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

      1. sub-neg 46.7%

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

      2. sqr-neg 46.7%

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

      3. sub-neg 46.7%

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

      4. sqr-neg 46.7%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 46.7%

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

      3. *-commutative 46.7%

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

      4. add-sqr-sqrt 46.3%

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

      5. sqrt-unprod 46.7%

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

      6. *-commutative 46.7%

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

      7. *-commutative 46.7%

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

      8. swap-sqr 46.7%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. hypot-undefine 46.7%

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

      3. unpow2 46.7%

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

      4. unpow2 46.7%

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

      5. +-commutative 46.7%

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

      6. unpow2 46.7%

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

      7. unpow2 46.7%

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

      8. hypot-undefine 100.0%

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

      9. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 53.1%

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

       1. neg-mul-1 53.1%

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

   11. Simplified 53.1%

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

   if -9.999999999999969e-311 < re

   1. Initial program 30.0%

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

   3. Taylor expanded in re around inf 10.1%

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

   4. Taylor expanded in re around 0 10.1%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]
   5. Step-by-step derivation

      1. metadata-eval 10.1%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 5.9% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in re around inf 6.5%

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

4. Taylor expanded in re around 0 6.5%

   \[\leadsto 0.5 \cdot \color{blue}{0} \]
5. Step-by-step derivation

   1. metadata-eval 6.5%

      \[\leadsto \color{blue}{0} \]

6. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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)))))