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

Percentage Accurate: 41.9% → 90.2%

Time: 30.5s

Alternatives: 7

Speedup: 1.7×

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: 90.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* im (/ 0.5 (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = im * (0.5 / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = im * (0.5 / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = im * (0.5 / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = im * (0.5 / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $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 7.0%

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

   3. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 93.1

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

   5. Simplified 93.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sqrt-lowering-sqrt.f64 94.1

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

   7. Applied egg-rr 94.1%

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

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

   1. Initial program 50.0%

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

      1. accelerator-lowering-hypot.f64 89.2

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

   4. Applied egg-rr 89.2%

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

4. Final simplification 89.8%

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

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -6.6 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+95)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -6.6e-116)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
     (if (<= re 6.8e+40) (* 0.5 (sqrt (* im 2.0))) (/ (* im 0.5) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+95) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -6.6e-116) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else if (re <= 6.8e+40) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+95)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -6.6e-116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	elseif (re <= 6.8e+40)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.8e+95], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.6e-116], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+40], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.7999999999999998e95

   1. Initial program 23.4%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 90.7

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

   5. Simplified 90.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 90.7

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

   7. Applied egg-rr 90.7%

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

   if -2.7999999999999998e95 < re < -6.60000000000000002e-116

   1. Initial program 80.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} - re\right)} \cdot \frac{1}{2} \]

      9. *-lowering-*.f64 80.8

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

   4. Applied egg-rr 80.8%

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

   if -6.60000000000000002e-116 < re < 6.79999999999999977e40

   1. Initial program 57.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{im}} \]
   4. Step-by-step derivation

   5. Simplified 80.2%

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

   if 6.79999999999999977e40 < re

   1. Initial program 4.7%

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

   3. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 80.9

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

   5. Simplified 80.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. un-div-inv N/A

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

      10. associate-*r/ N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 81.7

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

   7. Applied egg-rr 81.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -6.6 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+70)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.9e+49)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+70) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.9e+49) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-2.7d+70)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.9d+49) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -2.7e+70) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.9e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+70:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.9e+49:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+70)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.9e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -2.7e+70)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.9e+49)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+70], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.9e+49], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $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 < -2.7e70

   1. Initial program 27.0%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.2

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

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 91.2

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

   7. Applied egg-rr 91.2%

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

   if -2.7e70 < re < 2.9e49

   1. Initial program 63.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.2

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

   5. Simplified 75.2%

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

   if 2.9e49 < re

   1. Initial program 4.7%

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

   3. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 80.9

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

   5. Simplified 80.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. un-div-inv N/A

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

      10. associate-*r/ N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 81.7

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

   7. Applied egg-rr 81.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+70)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6e+58)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+70) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6e+58) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-3.2d+70)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6d+58) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -3.2e+70) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6e+58) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+70:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6e+58:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+70)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6e+58)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -3.2e+70)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6e+58)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+70], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+58], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.2000000000000002e70

   1. Initial program 27.0%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.2

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

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 91.2

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

   7. Applied egg-rr 91.2%

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

   if -3.2000000000000002e70 < re < 6.0000000000000005e58

   1. Initial program 63.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.2

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

   5. Simplified 75.2%

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

   if 6.0000000000000005e58 < re

   1. Initial program 4.7%

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

   3. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 80.9

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

   5. Simplified 80.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sqrt-lowering-sqrt.f64 81.6

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+62) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+62) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.8d+62)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+62) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.8e+62:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+62)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.8e+62)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.8e+62], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.80000000000000014e62

   1. Initial program 30.3%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 89.6

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

   5. Simplified 89.6%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 89.6

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

   7. Applied egg-rr 89.6%

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

   if -2.80000000000000014e62 < re

   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 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{im}} \]
   4. Step-by-step derivation

   5. Simplified 61.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.6 \cdot 10^{-269}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.6e-269) 0.0 (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.6e-269) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.6d-269) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.6e-269) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.6e-269:
		tmp = 0.0
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.6e-269)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.6e-269)
		tmp = 0.0;
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.6e-269], 0.0, N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.5999999999999999e-269

   1. Initial program 33.8%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 38.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 38.3

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

   7. Applied egg-rr 38.3%

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

   if 6.5999999999999999e-269 < im

   1. Initial program 45.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{im}} \]
   4. Step-by-step derivation

   5. Simplified 56.8%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.6 \cdot 10^{-269}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.0% accurate, 47.0× 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 44.6%

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

3. Taylor expanded in re around inf

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

5. Simplified 6.5%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

   6. metadata-eval 6.5

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

7. Applied egg-rr 6.5%

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

Reproduce

herbie shell --seed 2024198 
(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)))))