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

Percentage Accurate: 42.2% → 90.3%

Time: 9.8s

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	else:
		tmp = (0.5 * math.sqrt((math.hypot(re, im) - re))) * math.sqrt(2.0)
	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((re ^ -0.5) * Float64(im * 0.5));
	else
		tmp = Float64(Float64(0.5 * sqrt(Float64(hypot(re, im) - re))) * sqrt(2.0));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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 7.5%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 15.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 15.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 93.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 93.3%

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

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

   1. Initial program 45.0%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}} \cdot \frac{1}{2}\right), \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \left({\color{blue}{2}}^{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right)\right), \left({\color{blue}{2}}^{\frac{1}{2}}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

      12. accelerator-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \left(\sqrt{2}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 92.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   4. Applied egg-rr 92.1%

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

4. Final simplification 92.2%

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

Alternative 2: 90.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	else:
		tmp = 0.5 * math.sqrt(((math.hypot(re, im) - re) * 2.0))
	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((re ^ -0.5) * Float64(im * 0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(hypot(re, im) - re) * 2.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $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 7.5%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 15.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 15.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 93.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 93.3%

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

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

   1. Initial program 45.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), 2\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), 2\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 91.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), 2\right)\right)\right) \]

   4. Applied egg-rr 91.9%

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

4. Final simplification 92.1%

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

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.5 \cdot \sqrt{re \cdot -2}\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.08e+90)
   (* (sqrt 2.0) (* 0.5 (sqrt (* re -2.0))))
   (if (<= re 1.05e-22)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.08e+90) {
		tmp = sqrt(2.0) * (0.5 * sqrt((re * -2.0)));
	} else if (re <= 1.05e-22) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = pow(re, -0.5) * (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 <= (-1.08d+90)) then
        tmp = sqrt(2.0d0) * (0.5d0 * sqrt((re * (-2.0d0))))
    else if (re <= 1.05d-22) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.08e+90) {
		tmp = Math.sqrt(2.0) * (0.5 * Math.sqrt((re * -2.0)));
	} else if (re <= 1.05e-22) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.08e+90:
		tmp = math.sqrt(2.0) * (0.5 * math.sqrt((re * -2.0)))
	elif re <= 1.05e-22:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.08e+90)
		tmp = Float64(sqrt(2.0) * Float64(0.5 * sqrt(Float64(re * -2.0))));
	elseif (re <= 1.05e-22)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.08e+90)
		tmp = sqrt(2.0) * (0.5 * sqrt((re * -2.0)));
	elseif (re <= 1.05e-22)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.08e+90], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 * N[Sqrt[N[(re * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e-22], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.08e90

   1. Initial program 22.6%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}} \cdot \frac{1}{2}\right), \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \left({\color{blue}{2}}^{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right)\right), \left({\color{blue}{2}}^{\frac{1}{2}}\right)\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

      12. accelerator-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \left({2}^{\frac{1}{2}}\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \left(\sqrt{2}\right)\right) \]

      14. sqrt-lowering-sqrt.f64 99.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot re\right)}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

      2. *-lowering-*.f64 85.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -2\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   7. Simplified 85.8%

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

   if -1.08e90 < re < 1.05000000000000004e-22

   1. Initial program 56.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 82.7%

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

   if 1.05000000000000004e-22 < re

   1. Initial program 11.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 75.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.5 \cdot \sqrt{re \cdot -2}\right)\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.15 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + \frac{re \cdot -2}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e+25)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.15e-23)
     (* 0.5 (sqrt (* im (+ 2.0 (/ (* re -2.0) im)))))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e+25) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.15e-23) {
		tmp = 0.5 * sqrt((im * (2.0 + ((re * -2.0) / im))));
	} else {
		tmp = pow(re, -0.5) * (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 <= (-1d+25)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.15d-23) then
        tmp = 0.5d0 * sqrt((im * (2.0d0 + ((re * (-2.0d0)) / im))))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e+25) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.15e-23) {
		tmp = 0.5 * Math.sqrt((im * (2.0 + ((re * -2.0) / im))));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e+25:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.15e-23:
		tmp = 0.5 * math.sqrt((im * (2.0 + ((re * -2.0) / im))))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e+25)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.15e-23)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(2.0 + Float64(Float64(re * -2.0) / im)))));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e+25)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.15e-23)
		tmp = 0.5 * sqrt((im * (2.0 + ((re * -2.0) / im))));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e+25], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.15e-23], N[(0.5 * N[Sqrt[N[(im * N[(2.0 + N[(N[(re * -2.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.00000000000000009e25

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 78.5%

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

   if -1.00000000000000009e25 < re < 3.15000000000000012e-23

   1. Initial program 55.8%

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

   3. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(im \cdot \left(2 + -2 \cdot \frac{re}{im}\right)\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \left(2 + -2 \cdot \frac{re}{im}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{re}{im}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(\frac{-2 \cdot re}{im}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot re}{im}\right)\right)\right)\right)\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(2 \cdot re\right)}{im}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot re\right)\right), im\right)\right)\right)\right)\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot re\right), im\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot re\right), im\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(re \cdot -2\right), im\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 85.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, -2\right), im\right)\right)\right)\right)\right) \]

   5. Simplified 85.2%

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

   if 3.15000000000000012e-23 < re

   1. Initial program 11.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 75.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.15 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + \frac{re \cdot -2}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+90)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.5e-23)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+90) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.5e-23) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = pow(re, -0.5) * (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 <= (-2.45d+90)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9.5d-23) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+90) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9.5e-23) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.45e+90:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9.5e-23:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+90)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.5e-23)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.45e+90)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9.5e-23)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+90], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-23], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4500000000000001e90

   1. Initial program 22.6%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 84.5%

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

   if -2.4500000000000001e90 < re < 9.50000000000000058e-23

   1. Initial program 56.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 82.7%

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

   if 9.50000000000000058e-23 < re

   1. Initial program 11.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 75.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e+25)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.9e-23)
     (* 0.5 (sqrt (* im 2.0)))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+25) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.9e-23) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = pow(re, -0.5) * (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 <= (-2.5d+25)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.9d-23) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+25) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.9e-23) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e+25:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.9e-23:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e+25)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.9e-23)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e+25)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.9e-23)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e+25], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.9e-23], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.50000000000000012e25

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 78.5%

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

   if -2.50000000000000012e25 < re < 4.8999999999999998e-23

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 84.4%

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

   if 4.8999999999999998e-23 < re

   1. Initial program 11.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}}\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{re}\right)}^{\frac{1}{2}}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({re}^{-1}\right)}^{\frac{1}{2}}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({re}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \left(-1 \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot im\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(re, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{im}\right)\right) \]

   9. Applied egg-rr 75.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+24)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 9.2e-23) (* 0.5 (sqrt (* im 2.0))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+24) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 9.2e-23) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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+24)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 9.2d-23) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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+24) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 9.2e-23) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.8e+24:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 9.2e-23:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+24)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 9.2e-23)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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+24)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 9.2e-23)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.8e+24], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.2e-23], 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 3 regimes

2. if re < -2.8000000000000002e24

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 78.5%

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

   if -2.8000000000000002e24 < re < 9.2000000000000004e-23

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 84.4%

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

   if 9.2000000000000004e-23 < re

   1. Initial program 11.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(e^{\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right) \cdot \frac{1}{2}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\left(\frac{1}{2} \cdot \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \log \left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right)\right)\right) \]

      9. accelerator-lowering-hypot.f64 32.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{log.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in re around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 75.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right)\right) \]

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot im\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \left(\sqrt{\color{blue}{re}}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 75.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, im\right), \mathsf{sqrt.f64}\left(re\right)\right) \]

   9. Applied egg-rr 75.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+23) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+23) {
		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 <= (-1.15d+23)) 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 <= -1.15e+23) {
		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 <= -1.15e+23:
		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 <= -1.15e+23)
		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 <= -1.15e+23)
		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, -1.15e+23], 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 < -1.15e23

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 78.5%

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

   if -1.15e23 < re

   1. Initial program 42.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 66.9%

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

Alternative 9: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{im \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.7%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

   2. *-lowering-*.f64 57.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

5. Simplified 57.1%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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)))))