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

Percentage Accurate: 41.5% → 90.3%

Time: 10.4s

Alternatives: 6

Speedup: 2.1×

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: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = (im * 0.5) / math.sqrt(re)
	else:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - 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(Float64(im * 0.5) / sqrt(re));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - 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 = (im * 0.5) / sqrt(re);
	else
		tmp = sqrt((0.5 * (hypot(re, im) - 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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 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 5.0%

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

   3. Taylor expanded in im around 0 92.0%

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

      1. associate-*l* 92.2%

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

      2. *-commutative 92.2%

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

   5. Simplified 92.2%

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

      1. associate-*r* 92.2%

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

      2. sqrt-unprod 93.4%

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

      3. metadata-eval 93.4%

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

      4. metadata-eval 93.4%

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

      5. *-un-lft-identity 93.4%

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

      6. sqrt-div 93.4%

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

      7. metadata-eval 93.4%

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

      8. un-div-inv 93.4%

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

      9. *-commutative 93.4%

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

   7. Applied egg-rr 93.4%

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

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

   1. Initial program 46.7%

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

      1. add-sqr-sqrt 46.4%

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

      2. sqrt-unprod 46.7%

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

      3. *-commutative 46.7%

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

      4. *-commutative 46.7%

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

      5. 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. add-sqr-sqrt 46.7%

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

      7. *-commutative 46.7%

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

      8. hypot-define 90.2%

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

      9. metadata-eval 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. associate-*l* 90.2%

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

      2. metadata-eval 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 90.8%

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

Alternative 2: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-51} \lor \neg \left(re \leq 7.2 \cdot 10^{+81}\right):\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e+71)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.6e-67)
     (sqrt (* 0.5 (- im re)))
     (if (or (<= re 1.4e-51) (not (<= re 7.2e+81)))
       (* im (/ 0.5 (sqrt re)))
       (sqrt (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+71) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.6e-67) {
		tmp = sqrt((0.5 * (im - re)));
	} else if ((re <= 1.4e-51) || !(re <= 7.2e+81)) {
		tmp = im * (0.5 / sqrt(re));
	} else {
		tmp = sqrt((im * 0.5));
	}
	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+71)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.6d-67) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if ((re <= 1.4d-51) .or. (.not. (re <= 7.2d+81))) then
        tmp = im * (0.5d0 / sqrt(re))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+71) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.6e-67) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if ((re <= 1.4e-51) || !(re <= 7.2e+81)) {
		tmp = im * (0.5 / Math.sqrt(re));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e+71:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.6e-67:
		tmp = math.sqrt((0.5 * (im - re)))
	elif (re <= 1.4e-51) or not (re <= 7.2e+81):
		tmp = im * (0.5 / math.sqrt(re))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e+71)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.6e-67)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif ((re <= 1.4e-51) || !(re <= 7.2e+81))
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e+71)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.6e-67)
		tmp = sqrt((0.5 * (im - re)));
	elseif ((re <= 1.4e-51) || ~((re <= 7.2e+81)))
		tmp = im * (0.5 / sqrt(re));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e+71], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.6e-67], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[re, 1.4e-51], N[Not[LessEqual[re, 7.2e+81]], $MachinePrecision]], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.19999999999999978e71

   1. Initial program 37.4%

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

      1. sub-neg 37.4%

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

      2. sqr-neg 37.4%

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

      3. sub-neg 37.4%

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

      4. sqr-neg 37.4%

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

      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. Taylor expanded in re around -inf 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -4.19999999999999978e71 < re < 3.59999999999999999e-67

   1. Initial program 56.8%

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

      1. add-sqr-sqrt 56.4%

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

      2. sqrt-unprod 56.8%

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

      3. *-commutative 56.8%

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

      4. *-commutative 56.8%

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

      5. swap-sqr 56.8%

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

      6. add-sqr-sqrt 56.8%

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

      7. *-commutative 56.8%

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

      8. hypot-define 90.5%

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

      9. metadata-eval 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. associate-*l* 90.5%

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

      2. metadata-eval 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in re around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. unsub-neg 77.0%

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

   9. Simplified 77.0%

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

   if 3.59999999999999999e-67 < re < 1.4e-51 or 7.20000000000000011e81 < re

   1. Initial program 3.5%

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

   3. Taylor expanded in im around 0 91.2%

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

      1. associate-*l* 91.3%

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

      2. *-commutative 91.3%

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

   5. Simplified 91.3%

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

      1. associate-*r* 91.4%

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

      2. sqrt-unprod 92.4%

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

      3. metadata-eval 92.4%

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

      4. metadata-eval 92.4%

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

      5. *-un-lft-identity 92.4%

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

      6. sqrt-div 92.3%

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

      7. metadata-eval 92.3%

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

      8. un-div-inv 92.5%

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

      9. *-commutative 92.5%

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

   7. Applied egg-rr 92.5%

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

      1. associate-/l* 92.3%

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

      2. *-commutative 92.3%

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

   9. Applied egg-rr 92.3%

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

   if 1.4e-51 < re < 7.20000000000000011e81

   1. Initial program 38.4%

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

      1. add-sqr-sqrt 38.1%

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

      2. sqrt-unprod 38.4%

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

      3. *-commutative 38.4%

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

      4. *-commutative 38.4%

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

      5. swap-sqr 38.4%

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

      6. add-sqr-sqrt 38.4%

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

      7. *-commutative 38.4%

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

      8. hypot-define 71.2%

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

      9. metadata-eval 71.2%

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

   4. Applied egg-rr 71.2%

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

      1. associate-*l* 71.2%

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

      2. metadata-eval 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in re around 0 69.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.6 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-51} \lor \neg \left(re \leq 7.2 \cdot 10^{+81}\right):\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-51}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e+71)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.7e-64)
     (sqrt (* 0.5 (- im re)))
     (if (<= re 1.4e-51)
       (* im (/ 0.5 (sqrt re)))
       (if (<= re 5.4e+81) (sqrt (* im 0.5)) (/ (* im 0.5) (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+71) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.7e-64) {
		tmp = sqrt((0.5 * (im - re)));
	} else if (re <= 1.4e-51) {
		tmp = im * (0.5 / sqrt(re));
	} else if (re <= 5.4e+81) {
		tmp = sqrt((im * 0.5));
	} 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 <= (-4.2d+71)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.7d-64) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if (re <= 1.4d-51) then
        tmp = im * (0.5d0 / sqrt(re))
    else if (re <= 5.4d+81) then
        tmp = sqrt((im * 0.5d0))
    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 <= -4.2e+71) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.7e-64) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if (re <= 1.4e-51) {
		tmp = im * (0.5 / Math.sqrt(re));
	} else if (re <= 5.4e+81) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e+71:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.7e-64:
		tmp = math.sqrt((0.5 * (im - re)))
	elif re <= 1.4e-51:
		tmp = im * (0.5 / math.sqrt(re))
	elif re <= 5.4e+81:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e+71)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.7e-64)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif (re <= 1.4e-51)
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	elseif (re <= 5.4e+81)
		tmp = sqrt(Float64(im * 0.5));
	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 <= -4.2e+71)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.7e-64)
		tmp = sqrt((0.5 * (im - re)));
	elseif (re <= 1.4e-51)
		tmp = im * (0.5 / sqrt(re));
	elseif (re <= 5.4e+81)
		tmp = sqrt((im * 0.5));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e+71], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e-64], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 1.4e-51], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.4e+81], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -4.19999999999999978e71

   1. Initial program 37.4%

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

      1. sub-neg 37.4%

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

      2. sqr-neg 37.4%

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

      3. sub-neg 37.4%

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

      4. sqr-neg 37.4%

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

      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. Taylor expanded in re around -inf 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -4.19999999999999978e71 < re < 2.69999999999999986e-64

   1. Initial program 56.8%

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

      1. add-sqr-sqrt 56.4%

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

      2. sqrt-unprod 56.8%

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

      3. *-commutative 56.8%

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

      4. *-commutative 56.8%

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

      5. swap-sqr 56.8%

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

      6. add-sqr-sqrt 56.8%

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

      7. *-commutative 56.8%

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

      8. hypot-define 90.5%

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

      9. metadata-eval 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. associate-*l* 90.5%

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

      2. metadata-eval 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in re around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. unsub-neg 77.0%

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

   9. Simplified 77.0%

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

   if 2.69999999999999986e-64 < re < 1.4e-51

   1. Initial program 5.5%

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

   3. Taylor expanded in im around 0 97.6%

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

      1. associate-*l* 98.2%

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

      2. *-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. associate-*r* 98.2%

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

      2. sqrt-unprod 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. sqrt-div 100.0%

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

      7. metadata-eval 100.0%

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

      8. un-div-inv 99.7%

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

      9. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.4e-51 < re < 5.3999999999999999e81

   1. Initial program 38.4%

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

      1. add-sqr-sqrt 38.1%

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

      2. sqrt-unprod 38.4%

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

      3. *-commutative 38.4%

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

      4. *-commutative 38.4%

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

      5. swap-sqr 38.4%

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

      6. add-sqr-sqrt 38.4%

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

      7. *-commutative 38.4%

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

      8. hypot-define 71.2%

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

      9. metadata-eval 71.2%

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

   4. Applied egg-rr 71.2%

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

      1. associate-*l* 71.2%

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

      2. metadata-eval 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in re around 0 69.7%

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

   if 5.3999999999999999e81 < re

   1. Initial program 3.2%

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

   3. Taylor expanded in im around 0 90.4%

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

      1. associate-*l* 90.4%

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

      2. *-commutative 90.4%

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

   5. Simplified 90.4%

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

      1. associate-*r* 90.4%

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

      2. sqrt-unprod 91.4%

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

      3. metadata-eval 91.4%

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

      4. metadata-eval 91.4%

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

      5. *-un-lft-identity 91.4%

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

      6. sqrt-div 91.3%

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

      7. metadata-eval 91.3%

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

      8. un-div-inv 91.5%

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

      9. *-commutative 91.5%

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

   7. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\frac{im \cdot 0.5}{\sqrt{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-51}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 5.4 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e-293) (sqrt (* 0.5 (- im re))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e-293) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = sqrt((im * 0.5));
	}
	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-293) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e-293) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e-293:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e-293)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e-293)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e-293], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.0000000000000003e-293

   1. Initial program 51.0%

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

      1. add-sqr-sqrt 50.7%

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

      2. sqrt-unprod 51.0%

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

      3. *-commutative 51.0%

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

      4. *-commutative 51.0%

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

      5. swap-sqr 51.0%

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

      6. add-sqr-sqrt 51.0%

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

      7. *-commutative 51.0%

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

      8. hypot-define 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right) \cdot 0.25}} \]
   5. 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. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 57.7%

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

      1. mul-1-neg 57.7%

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

      2. unsub-neg 57.7%

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

   9. Simplified 57.7%

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

   if 5.0000000000000003e-293 < re

   1. Initial program 29.6%

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

      1. add-sqr-sqrt 29.4%

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

      2. sqrt-unprod 29.6%

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

      3. *-commutative 29.6%

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

      4. *-commutative 29.6%

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

      5. swap-sqr 29.6%

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

      6. add-sqr-sqrt 29.6%

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

      7. *-commutative 29.6%

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

      8. hypot-define 57.1%

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

      9. metadata-eval 57.1%

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

   4. Applied egg-rr 57.1%

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

      1. associate-*l* 57.1%

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

      2. metadata-eval 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in re around 0 50.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e+71) (* 0.5 (sqrt (* re -4.0))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+71) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.75d+71)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+71) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.75e+71:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75e+71)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.75e+71)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75e+71], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.75e71

   1. Initial program 37.4%

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

      1. sub-neg 37.4%

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

      2. sqr-neg 37.4%

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

      3. sub-neg 37.4%

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

      4. sqr-neg 37.4%

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

      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. Taylor expanded in re around -inf 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   6. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 89.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.75e71 < re

   1. Initial program 40.3%

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

      1. add-sqr-sqrt 40.0%

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

      2. sqrt-unprod 40.3%

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

      3. *-commutative 40.3%

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

      4. *-commutative 40.3%

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

      5. swap-sqr 40.3%

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

      6. add-sqr-sqrt 40.3%

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

      7. *-commutative 40.3%

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

      8. hypot-define 71.6%

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

      9. metadata-eval 71.6%

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

   4. Applied egg-rr 71.6%

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

      1. associate-*l* 71.6%

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

      2. metadata-eval 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in re around 0 59.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

   1. add-sqr-sqrt 39.4%

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

   2. sqrt-unprod 39.7%

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

   3. *-commutative 39.7%

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

   4. *-commutative 39.7%

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

   5. swap-sqr 39.7%

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

   6. add-sqr-sqrt 39.7%

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

   7. *-commutative 39.7%

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

   8. hypot-define 77.4%

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

   9. metadata-eval 77.4%

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

4. Applied egg-rr 77.4%

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

   1. associate-*l* 77.4%

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

   2. metadata-eval 77.4%

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

6. Simplified 77.4%

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

7. Taylor expanded in re around 0 50.4%

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

8. Final simplification 50.4%

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

Reproduce

herbie shell --seed 2024043 
(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)))))