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

Percentage Accurate: 41.1% → 90.2%

Time: 8.0s

Alternatives: 7

Speedup: 2.2×

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: 41.1% 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

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (/ im (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 = 0.5 * (im / 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 = 0.5 * (im / 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 = 0.5 * (im / 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(0.5 * Float64(im / 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 = 0.5 * (im / 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[(0.5 * N[(im / 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 12.0%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 90.0

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

   5. Applied rewrites 90.0%

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

   7. Applied rewrites 90.9%

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

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

   1. Initial program 44.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 87.8

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

   4. Applied rewrites 87.8%

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

4. Final simplification 88.4%

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

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -4.8 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 150:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e+134)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -4.8e-99)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 150.0)
       (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* im 2.0))))
       (* 0.5 (/ im (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+134) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -4.8e-99) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 150.0) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (im * 2.0)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75e+134)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -4.8e-99)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 150.0)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(im * 2.0))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75e+134], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.8e-99], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 150.0], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(im * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.75000000000000001e134

   1. Initial program 10.3%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -1.75000000000000001e134 < re < -4.8000000000000001e-99

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 78.1

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

   4. Applied rewrites 78.1%

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

   if -4.8000000000000001e-99 < re < 150

   1. Initial program 46.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if 150 < re

   1. Initial program 13.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -4.8 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 150:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 150:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-37)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 150.0)
     (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* im 2.0))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-37) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 150.0) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (im * 2.0)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-37)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 150.0)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(im * 2.0))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-37], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 150.0], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(im * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.39999999999999991e-37

   1. Initial program 44.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.39999999999999991e-37 < re < 150

   1. Initial program 50.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if 150 < re

   1. Initial program 13.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 150:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-37)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6.8e+41) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-37) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6.8e+41) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / 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 <= (-2.4d-37)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6.8d+41) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-37) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6.8e+41) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e-37:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6.8e+41:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e-37)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6.8e+41)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-37)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6.8e+41)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e-37], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+41], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.39999999999999991e-37

   1. Initial program 44.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.39999999999999991e-37 < re < 6.79999999999999996e41

   1. Initial program 48.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 6.79999999999999996e41 < re

   1. Initial program 13.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-37)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6.8e+41) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-37) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6.8e+41) {
		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 <= (-2.4d-37)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6.8d+41) 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 <= -2.4e-37) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6.8e+41) {
		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 <= -2.4e-37:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6.8e+41:
		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 <= -2.4e-37)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6.8e+41)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e-37)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6.8e+41)
		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, -2.4e-37], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.8e+41], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.39999999999999991e-37

   1. Initial program 44.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.39999999999999991e-37 < re < 6.79999999999999996e41

   1. Initial program 48.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if 6.79999999999999996e41 < re

   1. Initial program 13.7%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 84.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e-37) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e-37) {
		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 <= (-2.4d-37)) 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 <= -2.4e-37) {
		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 <= -2.4e-37:
		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 <= -2.4e-37)
		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 <= -2.4e-37)
		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, -2.4e-37], 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 < -2.39999999999999991e-37

   1. Initial program 44.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.39999999999999991e-37 < re

   1. Initial program 36.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

Alternative 7: 52.7% accurate, 2.2× 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 38.3%

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

3. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 49.8

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

5. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2024232 
(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)))))