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

Percentage Accurate: 42.0% → 89.9%

Time: 9.7s

Alternatives: 9

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: 42.0% 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: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (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((2.0 * (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((2.0 * (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((2.0 * (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 (sqrt(Float64(2.0 * 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((2.0 * (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[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $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 (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 8.7%

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

   3. Taylor expanded in re around inf 98.4%

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

      1. associate-*l* 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 98.7%

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

   5. Simplified 98.7%

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

      1. pow1 98.7%

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

      2. add-log-exp 11.0%

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

      3. associate-*l* 11.0%

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

      4. exp-prod 11.0%

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

      5. sqrt-unprod 11.0%

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

      6. metadata-eval 11.0%

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

      7. metadata-eval 11.0%

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

      8. pow1 11.0%

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

      9. add-log-exp 99.4%

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

      10. inv-pow 99.4%

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

      11. sqrt-pow1 99.4%

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

      12. metadata-eval 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. unpow1 99.4%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

   9. Simplified 99.4%

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

       1. *-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. rem-cube-cbrt 97.6%

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

       4. rem-cube-cbrt 99.4%

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

       5. add-sqr-sqrt 99.4%

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

       6. sqrt-unprod 60.0%

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

       7. swap-sqr 49.4%

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

       8. unpow2 49.4%

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

       9. pow-prod-up 49.4%

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

       10. metadata-eval 49.4%

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

       11. inv-pow 49.4%

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

       12. div-inv 49.4%

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

       13. sqrt-div 53.6%

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

       14. sqrt-pow1 99.6%

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

       15. metadata-eval 99.6%

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

       16. pow1 99.6%

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

       17. associate-*r/ 99.6%

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

   11. Applied egg-rr 99.6%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 42.5%

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

      1. pow1 42.5%

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

   4. Applied egg-rr 91.9%

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

      1. unpow1 91.9%

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

      2. *-commutative 91.9%

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

      3. associate-*r* 91.9%

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

      4. metadata-eval 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e+104)
   (sqrt (- re))
   (if (<= re 4e-44) (sqrt (* 0.5 (- im re))) (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+104) {
		tmp = sqrt(-re);
	} else if (re <= 4e-44) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) * pow(re, -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 <= (-3.8d+104)) then
        tmp = sqrt(-re)
    else if (re <= 4d-44) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+104) {
		tmp = Math.sqrt(-re);
	} else if (re <= 4e-44) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8e+104:
		tmp = math.sqrt(-re)
	elif re <= 4e-44:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8e+104)
		tmp = sqrt(Float64(-re));
	elseif (re <= 4e-44)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.8e+104)
		tmp = sqrt(-re);
	elseif (re <= 4e-44)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8e+104], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 4e-44], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.79999999999999969e104

   1. Initial program 26.2%

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

      1. pow1 26.2%

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

   4. Applied egg-rr 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   if -3.79999999999999969e104 < re < 3.99999999999999981e-44

   1. Initial program 55.6%

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

      1. pow1 55.6%

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

   4. Applied egg-rr 94.0%

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

      1. unpow1 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*r* 94.0%

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

      4. metadata-eval 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in re around 0 77.4%

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

      1. neg-mul-1 77.4%

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

      2. sub-neg 77.4%

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

   9. Simplified 77.4%

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

   if 3.99999999999999981e-44 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 73.4%

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

      1. associate-*l* 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.5%

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

   5. Simplified 73.5%

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

      1. pow1 73.5%

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

      2. add-log-exp 19.8%

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

      3. associate-*l* 19.8%

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

      4. exp-prod 19.8%

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

      5. sqrt-unprod 19.8%

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

      6. metadata-eval 19.8%

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

      7. metadata-eval 19.8%

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

      8. pow1 19.8%

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

      9. add-log-exp 74.0%

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

      10. inv-pow 74.0%

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

      11. sqrt-pow1 74.0%

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

      12. metadata-eval 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. unpow1 74.0%

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

      2. associate-*r* 74.1%

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

      3. *-commutative 74.1%

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

   9. Simplified 74.1%

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

Alternative 3: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+103)
   (sqrt (- re))
   (if (<= re 1.25e-43) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+103) {
		tmp = sqrt(-re);
	} else if (re <= 1.25e-43) {
		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 <= (-2.8d+103)) then
        tmp = sqrt(-re)
    else if (re <= 1.25d-43) 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 <= -2.8e+103) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.25e-43) {
		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 <= -2.8e+103:
		tmp = math.sqrt(-re)
	elif re <= 1.25e-43:
		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 <= -2.8e+103)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.25e-43)
		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 <= -2.8e+103)
		tmp = sqrt(-re);
	elseif (re <= 1.25e-43)
		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, -2.8e+103], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.25e-43], 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 < -2.80000000000000008e103

   1. Initial program 26.2%

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

      1. pow1 26.2%

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

   4. Applied egg-rr 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   if -2.80000000000000008e103 < re < 1.25000000000000005e-43

   1. Initial program 55.6%

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

      1. pow1 55.6%

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

   4. Applied egg-rr 94.0%

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

      1. unpow1 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*r* 94.0%

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

      4. metadata-eval 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in re around 0 77.4%

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

      1. neg-mul-1 77.4%

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

      2. sub-neg 77.4%

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

   9. Simplified 77.4%

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

   if 1.25000000000000005e-43 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 73.4%

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

      1. associate-*l* 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.5%

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

   5. Simplified 73.5%

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

      1. pow1 73.5%

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

      2. add-log-exp 19.8%

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

      3. associate-*l* 19.8%

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

      4. exp-prod 19.8%

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

      5. sqrt-unprod 19.8%

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

      6. metadata-eval 19.8%

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

      7. metadata-eval 19.8%

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

      8. pow1 19.8%

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

      9. add-log-exp 74.0%

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

      10. inv-pow 74.0%

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

      11. sqrt-pow1 74.0%

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

      12. metadata-eval 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. unpow1 74.0%

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

      2. associate-*r* 74.1%

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

      3. *-commutative 74.1%

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

   9. Simplified 74.1%

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

       1. *-commutative 74.1%

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

       2. associate-*r* 74.0%

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

       3. rem-cube-cbrt 73.1%

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

       4. rem-cube-cbrt 74.0%

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

       5. add-sqr-sqrt 73.9%

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

       6. sqrt-unprod 47.5%

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

       7. swap-sqr 39.7%

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

       8. unpow2 39.7%

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

       9. pow-prod-up 39.7%

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

       10. metadata-eval 39.7%

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

       11. inv-pow 39.7%

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

       12. div-inv 39.8%

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

       13. sqrt-div 48.5%

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

       14. sqrt-pow1 74.0%

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

       15. metadata-eval 74.0%

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

       16. pow1 74.0%

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

       17. associate-*r/ 74.0%

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

   11. Applied egg-rr 74.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+103)
   (sqrt (- re))
   (if (<= re 5e-46) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+103) {
		tmp = sqrt(-re);
	} else if (re <= 5e-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 <= (-2.8d+103)) then
        tmp = sqrt(-re)
    else if (re <= 5d-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 <= -2.8e+103) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5e-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 <= -2.8e+103:
		tmp = math.sqrt(-re)
	elif re <= 5e-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 <= -2.8e+103)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5e-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 <= -2.8e+103)
		tmp = sqrt(-re);
	elseif (re <= 5e-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, -2.8e+103], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5e-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 < -2.80000000000000008e103

   1. Initial program 26.2%

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

      1. pow1 26.2%

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

   4. Applied egg-rr 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   if -2.80000000000000008e103 < re < 4.99999999999999992e-46

   1. Initial program 55.6%

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

      1. pow1 55.6%

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

   4. Applied egg-rr 94.0%

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

      1. unpow1 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*r* 94.0%

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

      4. metadata-eval 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in re around 0 77.4%

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

      1. neg-mul-1 77.4%

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

      2. sub-neg 77.4%

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

   9. Simplified 77.4%

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

   if 4.99999999999999992e-46 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 73.4%

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

      1. associate-*l* 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 73.5%

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

   5. Simplified 73.5%

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

      1. pow1 73.5%

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

      2. add-log-exp 19.8%

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

      3. associate-*l* 19.8%

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

      4. exp-prod 19.8%

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

      5. sqrt-unprod 19.8%

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

      6. metadata-eval 19.8%

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

      7. metadata-eval 19.8%

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

      8. pow1 19.8%

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

      9. add-log-exp 74.0%

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

      10. inv-pow 74.0%

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

      11. sqrt-pow1 74.0%

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

      12. metadata-eval 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. unpow1 74.0%

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

      2. associate-*r* 74.1%

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

      3. *-commutative 74.1%

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

   9. Simplified 74.1%

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

       1. *-commutative 74.1%

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

       2. associate-*r* 74.0%

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

       3. rem-cube-cbrt 73.1%

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

       4. rem-cube-cbrt 74.0%

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

       5. add-sqr-sqrt 73.9%

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

       6. sqrt-unprod 47.5%

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

       7. swap-sqr 39.7%

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

       8. unpow2 39.7%

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

       9. pow-prod-up 39.7%

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

       10. metadata-eval 39.7%

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

       11. inv-pow 39.7%

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

       12. div-inv 39.8%

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

       13. sqrt-div 48.5%

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

       14. sqrt-pow1 74.0%

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

       15. metadata-eval 74.0%

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

       16. pow1 74.0%

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

       17. associate-*r/ 74.0%

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

   11. Applied egg-rr 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-lft-identity 74.0%

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

       3. associate-*l/ 73.9%

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

       4. associate-/r/ 73.1%

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

       5. associate-*r/ 73.1%

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

       6. metadata-eval 73.1%

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

   13. Simplified 73.1%

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

       1. associate-/r/ 73.9%

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

   15. Applied egg-rr 73.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e+104)
   (sqrt (- re))
   (if (<= re 3.5e+155) (sqrt (* 0.5 (- im re))) (sqrt (* 0.5 (- re re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e+104) {
		tmp = sqrt(-re);
	} else if (re <= 3.5e+155) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = sqrt((0.5 * (re - 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 <= (-4d+104)) then
        tmp = sqrt(-re)
    else if (re <= 3.5d+155) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = sqrt((0.5d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4e+104) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.5e+155) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = Math.sqrt((0.5 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e+104:
		tmp = math.sqrt(-re)
	elif re <= 3.5e+155:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = math.sqrt((0.5 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e+104)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.5e+155)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4e+104)
		tmp = sqrt(-re);
	elseif (re <= 3.5e+155)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = sqrt((0.5 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e+104], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.5e+155], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4e104

   1. Initial program 26.2%

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

      1. pow1 26.2%

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

   4. Applied egg-rr 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   if -4e104 < re < 3.49999999999999985e155

   1. Initial program 47.3%

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

      1. pow1 47.3%

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

   4. Applied egg-rr 83.9%

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

      1. unpow1 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*r* 83.9%

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

      4. metadata-eval 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in re around 0 69.6%

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

      1. neg-mul-1 69.6%

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

      2. sub-neg 69.6%

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

   9. Simplified 69.6%

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

   if 3.49999999999999985e155 < re

   1. Initial program 2.4%

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

      1. pow1 2.4%

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

   4. Applied egg-rr 43.7%

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

      1. unpow1 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*r* 43.7%

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

      4. metadata-eval 43.7%

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

   6. Simplified 43.7%

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

   7. Taylor expanded in re around inf 36.2%

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

Alternative 6: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.5e+105) (sqrt (- re)) (sqrt (* 0.5 (- im re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.5e+105) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((0.5 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5.5d+105)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((0.5d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.5e+105) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.5e+105:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.5e+105)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.5e+105)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.5e+105], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.49999999999999979e105

   1. Initial program 26.2%

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

      1. pow1 26.2%

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

   4. Applied egg-rr 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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   if -5.49999999999999979e105 < re

   1. Initial program 41.7%

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

      1. pow1 41.7%

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

   4. Applied egg-rr 78.8%

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

      1. unpow1 78.8%

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

      2. *-commutative 78.8%

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

      3. associate-*r* 78.8%

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

      4. metadata-eval 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in re around 0 61.8%

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

      1. neg-mul-1 61.8%

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

      2. sub-neg 61.8%

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

   9. Simplified 61.8%

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

Alternative 7: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e+63) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.36e+63) {
		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 <= (-1.36d+63)) 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 <= -1.36e+63) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.36e+63:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.36e+63)
		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 <= -1.36e+63)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.36e+63], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.36000000000000006e63

   1. Initial program 29.9%

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

      1. pow1 29.9%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 78.6%

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

      1. neg-mul-1 78.6%

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

   9. Simplified 78.6%

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

   if -1.36000000000000006e63 < re

   1. Initial program 41.3%

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

      1. pow1 41.3%

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

   4. Applied egg-rr 78.0%

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

      1. unpow1 78.0%

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

      2. *-commutative 78.0%

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

      3. associate-*r* 78.0%

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

      4. metadata-eval 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in re around 0 61.6%

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

      1. *-commutative 61.6%

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

   9. Simplified 61.6%

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

Alternative 8: 29.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 52.4%

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

      1. pow1 52.4%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 54.4%

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

      1. neg-mul-1 54.4%

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

   9. Simplified 54.4%

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

   if -4.999999999999985e-310 < re

   1. Initial program 25.2%

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

      1. pow1 25.2%

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

   4. Applied egg-rr 65.6%

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

      1. unpow1 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*r* 65.6%

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

      4. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

   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. expm1-log1p-u 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-re}\right)\right)} \]

       2. expm1-undefine 0.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{-re}\right)} - 1} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{-re} \cdot \sqrt{-re}}}\right)} - 1 \]

       4. sqrt-unprod 3.8%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}}\right)} - 1 \]

       5. sqr-neg 3.8%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\sqrt{\color{blue}{re \cdot re}}}\right)} - 1 \]

       6. sqrt-unprod 4.1%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1 \]

       7. add-sqr-sqrt 4.1%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{re}}\right)} - 1 \]

   11. Applied egg-rr 4.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{re}\right)} - 1} \]
   12. Step-by-step derivation

       1. sub-neg 4.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{re}\right)} + \left(-1\right)} \]

       2. metadata-eval 4.1%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{re}\right)} + \color{blue}{-1} \]

       3. +-commutative 4.1%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\sqrt{re}\right)}} \]

       4. log1p-undefine 4.1%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \sqrt{re}\right)}} \]

       5. rem-exp-log 4.1%

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

       6. associate-+r+ 5.6%

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

       7. metadata-eval 5.6%

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

       8. metadata-eval 5.6%

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

       9. remove-double-neg 5.6%

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

       10. distribute-neg-in 5.6%

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

       11. +-lft-identity 5.6%

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

       12. remove-double-neg 5.6%

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

   13. Simplified 5.6%

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

Alternative 9: 2.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. pow1 38.8%

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

4. Applied egg-rr 82.8%

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

   1. unpow1 82.8%

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

   2. *-commutative 82.8%

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

   3. associate-*r* 82.8%

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

   4. metadata-eval 82.8%

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

6. Simplified 82.8%

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

7. Taylor expanded in re around -inf 27.2%

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

   1. neg-mul-1 27.2%

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

9. Simplified 27.2%

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

    1. expm1-log1p-u 25.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-re}\right)\right)} \]

    2. expm1-undefine 18.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{-re}\right)} - 1} \]

    3. add-sqr-sqrt 18.6%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{-re} \cdot \sqrt{-re}}}\right)} - 1 \]

    4. sqrt-unprod 10.9%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}}\right)} - 1 \]

    5. sqr-neg 10.9%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\sqrt{\color{blue}{re \cdot re}}}\right)} - 1 \]

    6. sqrt-unprod 2.1%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\sqrt{re} \cdot \sqrt{re}}}\right)} - 1 \]

    7. add-sqr-sqrt 2.1%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{re}}\right)} - 1 \]

11. Applied egg-rr 2.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{re}\right)} - 1} \]
12. Step-by-step derivation

    1. sub-neg 2.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{re}\right)} + \left(-1\right)} \]

    2. metadata-eval 2.1%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{re}\right)} + \color{blue}{-1} \]

    3. +-commutative 2.1%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\sqrt{re}\right)}} \]

    4. log1p-undefine 2.1%

       \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \sqrt{re}\right)}} \]

    5. rem-exp-log 2.1%

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

    6. associate-+r+ 2.8%

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

    7. metadata-eval 2.8%

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

    8. metadata-eval 2.8%

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

    9. remove-double-neg 2.8%

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

    10. distribute-neg-in 2.8%

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

    11. +-lft-identity 2.8%

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

    12. remove-double-neg 2.8%

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

13. Simplified 2.8%

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

Reproduce

herbie shell --seed 2024145 
(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)))))