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

Percentage Accurate: 41.5% → 89.8%

Time: 8.0s

Alternatives: 6

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: 41.5% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot {\left({2}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* (* (sqrt (/ 1.0 re)) (* im (sqrt 0.5))) (pow (pow 2.0 0.25) 2.0)))
   (* 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 * ((sqrt((1.0 / re)) * (im * sqrt(0.5))) * pow(pow(2.0, 0.25), 2.0));
	} 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 * ((Math.sqrt((1.0 / re)) * (im * Math.sqrt(0.5))) * Math.pow(Math.pow(2.0, 0.25), 2.0));
	} 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 * ((math.sqrt((1.0 / re)) * (im * math.sqrt(0.5))) * math.pow(math.pow(2.0, 0.25), 2.0))
	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(Float64(sqrt(Float64(1.0 / re)) * Float64(im * sqrt(0.5))) * ((2.0 ^ 0.25) ^ 2.0)));
	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 * ((sqrt((1.0 / re)) * (im * sqrt(0.5))) * ((2.0 ^ 0.25) ^ 2.0));
	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[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $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 9.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 9.8

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

   4. Applied rewrites 9.8%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-approx N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 7.2

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

   6. Applied rewrites 7.2%

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

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 92.2

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

   9. Applied rewrites 92.2%

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

       1. pow1/2 N/A

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

       2. sqr-pow N/A

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

       3. pow2 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. metadata-eval 92.9

          \[\leadsto \left(\left(\sqrt{\frac{1}{re}} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot {\left({2}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot 0.5 \]

   11. Applied rewrites 92.9%

       \[\leadsto \left(\left(\sqrt{\frac{1}{re}} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}\right) \cdot 0.5 \]

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

   1. Initial program 46.2%

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

      1. lower-hypot.f64 88.0

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

   4. Applied rewrites 88.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{\frac{1}{re}} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot {\left({2}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% 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 9.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 9.8

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

   4. Applied rewrites 9.8%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-approx N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 7.2

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

   6. Applied rewrites 7.2%

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

   7. Taylor expanded in re around 0

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

      1. lower-sqrt.f64 13.6

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

   9. Applied rewrites 13.6%

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

       1. lift-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-prod N/A

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

       5. lift-*.f64 N/A

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

       6. lift-approx 92.1

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

       7. lift-approx N/A

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

       8. lift-approx 92.9

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

   11. Applied rewrites 92.9%

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

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

   1. Initial program 46.2%

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

      1. lower-hypot.f64 88.0

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

   4. Applied rewrites 88.0%

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

4. Final simplification 88.8%

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

Alternative 3: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -5 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 655000000000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e+108)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -5e-158)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 655000000000.0)
       (* 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 <= -5e+108) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -5e-158) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 655000000000.0) {
		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 <= -5e+108)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -5e-158)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 655000000000.0)
		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, -5e+108], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -5e-158], 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, 655000000000.0], 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 < -4.99999999999999991e108

   1. Initial program 20.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   if -4.99999999999999991e108 < re < -4.99999999999999972e-158

   1. Initial program 76.4%

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

      1. lift-*.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

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

      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-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

   4. Applied rewrites 76.4%

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

   if -4.99999999999999972e-158 < re < 6.55e11

   1. Initial program 52.4%

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

   3. 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 80.9

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

   5. Applied rewrites 80.9%

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

   if 6.55e11 < re

   1. Initial program 11.2%

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

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

      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-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 11.2

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

   4. Applied rewrites 11.2%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-approx N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 25.5

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

   6. Applied rewrites 25.5%

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

   7. Taylor expanded in re around 0

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

      1. lower-sqrt.f64 29.6

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

   9. Applied rewrites 29.6%

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

       1. lift-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-prod N/A

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

       5. lift-*.f64 N/A

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

       6. lift-approx 75.5

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

       7. lift-approx N/A

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

       8. lift-approx 75.9

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

   11. Applied rewrites 75.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -5 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 655000000000:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, im \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 655000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-44)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 655000000000.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-44) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 655000000000.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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 <= (-2d-44)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 655000000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    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 <= -2e-44) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 655000000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-44:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 655000000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-44)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 655000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -2e-44)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 655000000000.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-44], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 655000000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $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 < -1.99999999999999991e-44

   1. Initial program 43.7%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.99999999999999991e-44 < re < 6.55e11

   1. Initial program 55.7%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   if 6.55e11 < re

   1. Initial program 11.2%

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

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

      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-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 11.2

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

   4. Applied rewrites 11.2%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-approx N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 25.5

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

   6. Applied rewrites 25.5%

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

   7. Taylor expanded in re around 0

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

      1. lower-sqrt.f64 29.6

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

   9. Applied rewrites 29.6%

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

       1. lift-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-prod N/A

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

       5. lift-*.f64 N/A

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

       6. lift-approx 75.5

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

       7. lift-approx N/A

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

       8. lift-approx 75.9

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

   11. Applied rewrites 75.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 655000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-44) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-44) {
		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 <= (-2d-44)) 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 <= -2e-44) {
		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 <= -2e-44:
		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 <= -2e-44)
		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 <= -2e-44)
		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, -2e-44], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.99999999999999991e-44

   1. Initial program 43.7%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.99999999999999991e-44 < re

   1. Initial program 39.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

Alternative 6: 51.9% 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 40.3%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 50.8

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

5. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2024212 
(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)))))