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

Percentage Accurate: 40.5% → 89.8%

Time: 6.6s

Alternatives: 8

Speedup: 1.7×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $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 7.2%

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

      1. 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 7.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 7.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 12.8

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

   4. Applied rewrites 12.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 11.8

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

   7. Applied rewrites 11.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 94.3

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

   10. Applied rewrites 94.3%

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

   12. Applied rewrites 95.0%

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

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

   1. Initial program 43.9%

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

      1. 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 43.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 43.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.6

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

   4. Applied rewrites 90.6%

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

4. Final simplification 91.3%

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

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{{re}^{-1}} \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.9e-58)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 5.3e-51)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* (sqrt (pow re -1.0)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.9e-58) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 5.3e-51) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (sqrt(pow(re, -1.0)) * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.9d-58)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 5.3d-51) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (sqrt((re ** (-1.0d0))) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.9e-58) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 5.3e-51) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (Math.sqrt(Math.pow(re, -1.0)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.9e-58:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 5.3e-51:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (math.sqrt(math.pow(re, -1.0)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.9e-58)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 5.3e-51)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(sqrt((re ^ -1.0)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.9e-58)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 5.3e-51)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (sqrt((re ^ -1.0)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.9e-58], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 5.3e-51], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.8999999999999999e-58

   1. Initial program 45.7%

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

      1. 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 45.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 45.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.8999999999999999e-58 < re < 5.29999999999999974e-51

   1. Initial program 48.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 5.29999999999999974e-51 < re

   1. Initial program 15.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{2} \cdot \frac{1}{re}\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\frac{{im}^{2}}{{re}^{3}} \cdot \frac{-1}{8}} + \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{{re}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right)} \cdot {im}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{{re}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \color{blue}{\frac{im}{{re}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{{re}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{re}\right) \cdot {im}^{2}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \color{blue}{\frac{\frac{1}{2} \cdot 1}{re}}\right) \cdot {im}^{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\color{blue}{\frac{1}{2}}}{re}\right) \cdot {im}^{2}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \color{blue}{\frac{\frac{1}{2}}{re}}\right) \cdot {im}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)} \]

      14. lower-*.f64 43.4

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, -0.125, \frac{0.5}{re}\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)} \]

   5. Applied rewrites 43.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, -0.125, \frac{0.5}{re}\right) \cdot \left(im \cdot im\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{{\left(2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)\right)}^{\frac{1}{2}}} \]

      3. sqr-pow N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{fma}\left(im \cdot \frac{im}{{re}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{re}\right) \cdot \left(im \cdot im\right)\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}} \]

   7. Applied rewrites 46.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(\left(\mathsf{fma}\left(-0.125, \frac{im}{{re}^{3}} \cdot im, \frac{0.5}{re}\right) \cdot im\right) \cdot im\right) \cdot 2\right)}^{0.25}\right)}^{2}} \]

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 73.1

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

   10. Applied rewrites 73.1%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{{re}^{-1}} \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im} - re\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* re re) (* im im))) re)))
   (if (<= t_0 0.0)
     (* (/ im (sqrt re)) 0.5)
     (if (<= t_0 2e+154)
       (* 0.5 (sqrt (* 2.0 t_0)))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im))) - re;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im / sqrt(re)) * 0.5;
	} else if (t_0 <= 2e+154) {
		tmp = 0.5 * sqrt((2.0 * t_0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im))) - re
    if (t_0 <= 0.0d0) then
        tmp = (im / sqrt(re)) * 0.5d0
    else if (t_0 <= 2d+154) then
        tmp = 0.5d0 * sqrt((2.0d0 * t_0))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im))) - re;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im / Math.sqrt(re)) * 0.5;
	} else if (t_0 <= 2e+154) {
		tmp = 0.5 * Math.sqrt((2.0 * t_0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im))) - re
	tmp = 0
	if t_0 <= 0.0:
		tmp = (im / math.sqrt(re)) * 0.5
	elif t_0 <= 2e+154:
		tmp = 0.5 * math.sqrt((2.0 * t_0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	elseif (t_0 <= 2e+154)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * t_0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im))) - re;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (im / sqrt(re)) * 0.5;
	elseif (t_0 <= 2e+154)
		tmp = 0.5 * sqrt((2.0 * t_0));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+154], N[(0.5 * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.2%

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

      1. 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 7.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 7.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 12.8

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

   4. Applied rewrites 12.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 11.8

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

   7. Applied rewrites 11.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 94.3

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

   10. Applied rewrites 94.3%

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

   12. Applied rewrites 95.0%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 2.00000000000000007e154

   1. Initial program 93.0%

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

   if 2.00000000000000007e154 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 3.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im} - re\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* re re) (* im im))) re)))
   (if (<= t_0 0.0)
     (* (/ im (sqrt re)) 0.5)
     (if (<= t_0 2e+154)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
       (* 0.5 (sqrt (* 2.0 (- im re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im))) - re;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im / sqrt(re)) * 0.5;
	} else if (t_0 <= 2e+154) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	elseif (t_0 <= 2e+154)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+154], 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], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.2%

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

      1. 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 7.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 7.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 12.8

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

   4. Applied rewrites 12.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 11.8

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

   7. Applied rewrites 11.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 94.3

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

   10. Applied rewrites 94.3%

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

   12. Applied rewrites 95.0%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 2.00000000000000007e154

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      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. lower-fma.f64 93.0

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

   4. Applied rewrites 93.0%

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

   if 2.00000000000000007e154 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 3.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \leq 2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.9e-58)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 5.3e-51)
     (* (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))) 0.5)
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.9e-58) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 5.3e-51) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (2.0 * im))) * 0.5;
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.9e-58)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 5.3e-51)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))) * 0.5);
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.9e-58], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 5.3e-51], N[(N[Sqrt[N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.8999999999999999e-58

   1. Initial program 45.7%

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

      1. 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 45.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 45.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.8999999999999999e-58 < re < 5.29999999999999974e-51

   1. Initial program 48.8%

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

      1. 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 48.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 48.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.1

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

   7. Applied rewrites 80.1%

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

   if 5.29999999999999974e-51 < re

   1. Initial program 15.8%

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

      1. 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 15.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 40.4

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

   4. Applied rewrites 40.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 72.7

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

   10. Applied rewrites 72.7%

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

   12. Applied rewrites 73.2%

       \[\leadsto \frac{1 \cdot \left(1 \cdot im\right)}{\color{blue}{\sqrt{re}}} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.9e-58)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 5.3e-51)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.9e-58) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 5.3e-51) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.9d-58)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 5.3d-51) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im / sqrt(re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.9e-58) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 5.3e-51) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.9e-58:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 5.3e-51:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.9e-58)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 5.3e-51)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.9e-58)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 5.3e-51)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.9e-58], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 5.3e-51], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.8999999999999999e-58

   1. Initial program 45.7%

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

      1. 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 45.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 45.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -2.8999999999999999e-58 < re < 5.29999999999999974e-51

   1. Initial program 48.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 5.29999999999999974e-51 < re

   1. Initial program 15.8%

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

      1. 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 15.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 40.4

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

   4. Applied rewrites 40.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 30.8

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

   7. Applied rewrites 30.8%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 72.7

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

   10. Applied rewrites 72.7%

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

   12. Applied rewrites 73.2%

       \[\leadsto \frac{1 \cdot \left(1 \cdot im\right)}{\color{blue}{\sqrt{re}}} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e-58) (* (sqrt (* -4.0 re)) 0.5) (* (sqrt (* 2.0 im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e-58) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt((2.0 * 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 <= (-1.85d-58)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt((2.0d0 * im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.85e-58) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt((2.0 * im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.85e-58:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt((2.0 * im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e-58)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(2.0 * im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.85e-58)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt((2.0 * im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e-58], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.8500000000000001e-58

   1. Initial program 45.7%

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

      1. 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 45.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 45.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

   if -1.8500000000000001e-58 < re

   1. Initial program 34.5%

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

      1. 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 34.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 34.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

Alternative 8: 26.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{-4 \cdot re} \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sqrt (* -4.0 re)) 0.5))

C

double code(double re, double im) {
	return sqrt((-4.0 * re)) * 0.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(((-4.0d0) * re)) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return Math.sqrt((-4.0 * re)) * 0.5;
}

Python

def code(re, im):
	return math.sqrt((-4.0 * re)) * 0.5

Julia

function code(re, im)
	return Float64(sqrt(Float64(-4.0 * re)) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((-4.0 * re)) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 38.1%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 38.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 38.1

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-hypot.f64 78.4

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in re around -inf

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

   1. lower-*.f64 30.0

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

7. Applied rewrites 30.0%

   \[\leadsto \sqrt{\color{blue}{-4 \cdot re}} \cdot 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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)))))