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

Percentage Accurate: 41.9% → 90.6%

Time: 12.6s

Alternatives: 8

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% 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: 90.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ (* im 0.5) (sqrt re))
   (sqrt (* 0.5 (- (hypot im re) 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(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 = (im * 0.5) / Math.sqrt(re);
	} 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 = (im * 0.5) / math.sqrt(re)
	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(Float64(im * 0.5) / sqrt(re));
	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 = (im * 0.5) / sqrt(re);
	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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 7.8%

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

   3. Taylor expanded in re around inf 46.3%

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

      1. *-commutative 46.3%

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

      2. sqrt-div 49.7%

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

      3. sqrt-pow1 99.8%

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

      4. metadata-eval 99.8%

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

      5. pow1 99.8%

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

      6. associate-*l/ 99.8%

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

   5. Applied egg-rr 99.8%

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

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

   1. Initial program 52.6%

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

      1. pow1 52.6%

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

   4. Applied egg-rr 92.2%

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

      1. unpow1 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*r* 92.2%

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

      4. metadata-eval 92.2%

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

      5. hypot-undefine 52.6%

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

      6. unpow2 52.6%

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

      7. unpow2 52.6%

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

      8. +-commutative 52.6%

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

      9. unpow2 52.6%

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

      10. unpow2 52.6%

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

      11. hypot-undefine 92.2%

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

   6. Simplified 92.2%

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

Alternative 2: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.033:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{im \cdot \left(0.5 + -0.5 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.033)
   (sqrt (- re))
   (if (<= re 3.2e+46)
     (sqrt (* im (+ 0.5 (* -0.5 (/ re im)))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.033) {
		tmp = sqrt(-re);
	} else if (re <= 3.2e+46) {
		tmp = sqrt((im * (0.5 + (-0.5 * (re / im)))));
	} 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 <= (-0.033d0)) then
        tmp = sqrt(-re)
    else if (re <= 3.2d+46) then
        tmp = sqrt((im * (0.5d0 + ((-0.5d0) * (re / im)))))
    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 <= -0.033) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.2e+46) {
		tmp = Math.sqrt((im * (0.5 + (-0.5 * (re / im)))));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.033:
		tmp = math.sqrt(-re)
	elif re <= 3.2e+46:
		tmp = math.sqrt((im * (0.5 + (-0.5 * (re / im)))))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.033)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.2e+46)
		tmp = sqrt(Float64(im * Float64(0.5 + Float64(-0.5 * Float64(re / im)))));
	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 <= -0.033)
		tmp = sqrt(-re);
	elseif (re <= 3.2e+46)
		tmp = sqrt((im * (0.5 + (-0.5 * (re / im)))));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.033], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.2e+46], N[Sqrt[N[(im * N[(0.5 + N[(-0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.033000000000000002

   1. Initial program 43.4%

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

      1. pow1 43.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. +-commutative 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 77.2%

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

      1. neg-mul-1 77.2%

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

   9. Simplified 77.2%

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

   if -0.033000000000000002 < re < 3.1999999999999998e46

   1. Initial program 59.5%

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

      1. pow1 59.5%

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

   4. Applied egg-rr 86.5%

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

      1. unpow1 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. metadata-eval 86.5%

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

      5. hypot-undefine 59.5%

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

      6. unpow2 59.5%

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

      7. unpow2 59.5%

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

      8. +-commutative 59.5%

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

      9. unpow2 59.5%

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

      10. unpow2 59.5%

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

      11. hypot-undefine 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in im around inf 76.0%

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

   if 3.1999999999999998e46 < re

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf 49.5%

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

      1. *-commutative 49.5%

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

      2. sqrt-div 59.2%

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

      3. sqrt-pow1 73.4%

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

      4. metadata-eval 73.4%

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

      5. pow1 73.4%

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

      6. associate-*l/ 73.4%

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

   5. Applied egg-rr 73.4%

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

Alternative 3: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0105:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0105)
   (sqrt (- re))
   (if (<= re 8.2e+46) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0105) {
		tmp = sqrt(-re);
	} else if (re <= 8.2e+46) {
		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 <= (-0.0105d0)) then
        tmp = sqrt(-re)
    else if (re <= 8.2d+46) 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 <= -0.0105) {
		tmp = Math.sqrt(-re);
	} else if (re <= 8.2e+46) {
		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 <= -0.0105:
		tmp = math.sqrt(-re)
	elif re <= 8.2e+46:
		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 <= -0.0105)
		tmp = sqrt(Float64(-re));
	elseif (re <= 8.2e+46)
		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 <= -0.0105)
		tmp = sqrt(-re);
	elseif (re <= 8.2e+46)
		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, -0.0105], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 8.2e+46], 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 < -0.0105000000000000007

   1. Initial program 43.4%

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

      1. pow1 43.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. +-commutative 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 77.2%

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

      1. neg-mul-1 77.2%

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

   9. Simplified 77.2%

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

   if -0.0105000000000000007 < re < 8.19999999999999999e46

   1. Initial program 59.5%

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

      1. pow1 59.5%

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

   4. Applied egg-rr 86.5%

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

      1. unpow1 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. metadata-eval 86.5%

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

      5. hypot-undefine 59.5%

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

      6. unpow2 59.5%

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

      7. unpow2 59.5%

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

      8. +-commutative 59.5%

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

      9. unpow2 59.5%

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

      10. unpow2 59.5%

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

      11. hypot-undefine 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in re around 0 76.0%

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

      1. neg-mul-1 76.0%

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

      2. unsub-neg 76.0%

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

   9. Simplified 76.0%

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

   if 8.19999999999999999e46 < re

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf 49.5%

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

      1. *-commutative 49.5%

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

      2. sqrt-div 59.2%

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

      3. sqrt-pow1 73.4%

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

      4. metadata-eval 73.4%

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

      5. pow1 73.4%

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

      6. associate-*l/ 73.4%

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

   5. Applied egg-rr 73.4%

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

Alternative 4: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00029:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00029)
   (sqrt (- re))
   (if (<= re 2.9e+46) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00029) {
		tmp = sqrt(-re);
	} else if (re <= 2.9e+46) {
		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 <= (-0.00029d0)) then
        tmp = sqrt(-re)
    else if (re <= 2.9d+46) 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 <= -0.00029) {
		tmp = Math.sqrt(-re);
	} else if (re <= 2.9e+46) {
		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 <= -0.00029:
		tmp = math.sqrt(-re)
	elif re <= 2.9e+46:
		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 <= -0.00029)
		tmp = sqrt(Float64(-re));
	elseif (re <= 2.9e+46)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.00029)
		tmp = sqrt(-re);
	elseif (re <= 2.9e+46)
		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, -0.00029], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 2.9e+46], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9e-4

   1. Initial program 43.4%

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

      1. pow1 43.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. +-commutative 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 77.2%

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

      1. neg-mul-1 77.2%

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

   9. Simplified 77.2%

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

   if -2.9e-4 < re < 2.9000000000000002e46

   1. Initial program 59.5%

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

      1. pow1 59.5%

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

   4. Applied egg-rr 86.5%

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

      1. unpow1 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. metadata-eval 86.5%

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

      5. hypot-undefine 59.5%

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

      6. unpow2 59.5%

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

      7. unpow2 59.5%

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

      8. +-commutative 59.5%

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

      9. unpow2 59.5%

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

      10. unpow2 59.5%

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

      11. hypot-undefine 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in re around 0 76.0%

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

      1. neg-mul-1 76.0%

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

      2. unsub-neg 76.0%

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

   9. Simplified 76.0%

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

   if 2.9000000000000002e46 < re

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf 49.5%

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

      1. sqrt-div 59.2%

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

      2. sqrt-pow1 73.4%

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

      3. metadata-eval 73.4%

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

      4. pow1 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.3%

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

   7. Simplified 73.3%

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

Alternative 5: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.033:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.033)
   (sqrt (- re))
   (if (<= re 2.7e+46) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.033) {
		tmp = sqrt(-re);
	} else if (re <= 2.7e+46) {
		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 <= (-0.033d0)) then
        tmp = sqrt(-re)
    else if (re <= 2.7d+46) 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 <= -0.033) {
		tmp = Math.sqrt(-re);
	} else if (re <= 2.7e+46) {
		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 <= -0.033:
		tmp = math.sqrt(-re)
	elif re <= 2.7e+46:
		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 <= -0.033)
		tmp = sqrt(Float64(-re));
	elseif (re <= 2.7e+46)
		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 <= -0.033)
		tmp = sqrt(-re);
	elseif (re <= 2.7e+46)
		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, -0.033], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 2.7e+46], 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 < -0.033000000000000002

   1. Initial program 43.4%

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

      1. pow1 43.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. +-commutative 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 77.2%

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

      1. neg-mul-1 77.2%

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

   9. Simplified 77.2%

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

   if -0.033000000000000002 < re < 2.7000000000000002e46

   1. Initial program 59.5%

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

      1. pow1 59.5%

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

   4. Applied egg-rr 86.5%

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

      1. unpow1 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. metadata-eval 86.5%

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

      5. hypot-undefine 59.5%

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

      6. unpow2 59.5%

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

      7. unpow2 59.5%

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

      8. +-commutative 59.5%

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

      9. unpow2 59.5%

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

      10. unpow2 59.5%

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

      11. hypot-undefine 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in re around 0 76.0%

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

      1. neg-mul-1 76.0%

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

      2. unsub-neg 76.0%

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

   9. Simplified 76.0%

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

   if 2.7000000000000002e46 < re

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf 49.5%

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

      1. sqrt-div 59.2%

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

      2. sqrt-pow1 73.4%

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

      3. metadata-eval 73.4%

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

      4. pow1 73.4%

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

      5. associate-*r/ 73.4%

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

   5. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-/l* 73.3%

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

   7. Simplified 73.3%

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

      1. add-sqr-sqrt 73.1%

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

      2. sqrt-unprod 73.3%

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

      3. frac-times 73.3%

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

      4. metadata-eval 73.3%

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

      5. add-sqr-sqrt 73.2%

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

   9. Applied egg-rr 73.2%

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

Alternative 6: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0054:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0054) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0054) {
		tmp = 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 <= (-0.0054d0)) then
        tmp = 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 <= -0.0054) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0054:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0054)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0054)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0054], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0054000000000000003

   1. Initial program 43.4%

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

      1. pow1 43.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. +-commutative 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 77.2%

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

      1. neg-mul-1 77.2%

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

   9. Simplified 77.2%

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

   if -0.0054000000000000003 < re

   1. Initial program 49.0%

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

      1. pow1 49.0%

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

   4. Applied egg-rr 76.7%

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

      1. unpow1 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-*r* 76.7%

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

      4. metadata-eval 76.7%

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

      5. hypot-undefine 49.0%

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

      6. unpow2 49.0%

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

      7. unpow2 49.0%

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

      8. +-commutative 49.0%

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

      9. unpow2 49.0%

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

      10. unpow2 49.0%

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

      11. hypot-undefine 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in im around inf 66.3%

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

      1. *-commutative 66.3%

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

   9. Simplified 66.3%

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

Alternative 7: 29.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2d-310)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 54.8%

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

      1. pow1 54.8%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 54.8%

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

      6. unpow2 54.8%

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

      7. unpow2 54.8%

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

      8. +-commutative 54.8%

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

      9. unpow2 54.8%

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

      10. unpow2 54.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 50.1%

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

      1. neg-mul-1 50.1%

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

   9. Simplified 50.1%

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

   if -1.999999999999994e-310 < re

   1. Initial program 38.7%

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

      1. pow1 38.7%

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

   4. Applied egg-rr 61.6%

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

      1. unpow1 61.6%

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

      2. *-commutative 61.6%

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

      3. associate-*r* 61.6%

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

      4. metadata-eval 61.6%

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

      5. hypot-undefine 38.7%

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

      6. unpow2 38.7%

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

      7. unpow2 38.7%

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

      8. +-commutative 38.7%

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

      9. unpow2 38.7%

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

      10. unpow2 38.7%

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

      11. hypot-undefine 61.6%

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

   6. Simplified 61.6%

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

   7. Taylor expanded in re around -inf 0.0%

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

      1. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

       1. neg-sub0 0.0%

          \[\leadsto \sqrt{\color{blue}{0 - re}} \]

       2. sub-neg 0.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 4.5%

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

       5. sqr-neg 4.5%

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

       6. sqrt-unprod 6.0%

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

       7. add-sqr-sqrt 6.0%

          \[\leadsto \sqrt{0 + \color{blue}{re}} \]

   11. Applied egg-rr 6.0%

       \[\leadsto \sqrt{\color{blue}{0 + re}} \]
   12. Step-by-step derivation

       1. +-lft-identity 6.0%

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

   13. Simplified 6.0%

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

Alternative 8: 6.0% 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 47.6%

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

3. Taylor expanded in re around inf 4.0%

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

4. Taylor expanded in re around 0 4.0%

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

Reproduce

herbie shell --seed 2024153 
(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)))))