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

Percentage Accurate: 42.0% → 88.2%

Time: 8.5s

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: 42.0% 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: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.4e-24)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (sqrt (/ 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.4e-24) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.4e-24) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.4e-24:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.4e-24)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.4e-24)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.4e-24], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.40000000000000025e-24

   1. Initial program 52.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 94.2

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

   4. Applied rewrites 94.2%

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

   if 6.40000000000000025e-24 < re

   1. Initial program 11.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 78.5%

      \[\leadsto \left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e+126)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re -1.8e-39)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 3.4e-55)
       (* 0.5 (sqrt (fma re (+ -2.0 (/ re im)) (* 2.0 im))))
       (* 0.5 (* im (sqrt (/ 1.0 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+126) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= -1.8e-39) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 3.4e-55) {
		tmp = 0.5 * sqrt(fma(re, (-2.0 + (re / im)), (2.0 * im)));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75e+126)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= -1.8e-39)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 3.4e-55)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(-2.0 + Float64(re / im)), Float64(2.0 * im))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75e+126], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.8e-39], 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, 3.4e-55], N[(0.5 * N[Sqrt[N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision] + N[(2.0 * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.7500000000000001e126

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   if -1.7500000000000001e126 < re < -1.8e-39

   1. Initial program 83.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 83.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 83.1

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

   4. Applied rewrites 83.1%

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

   if -1.8e-39 < re < 3.39999999999999973e-55

   1. Initial program 59.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)}} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   if 3.39999999999999973e-55 < re

   1. Initial program 11.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 77.6%

      \[\leadsto \left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.8 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, -2 + \frac{re}{im}, 2 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6e-5)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.4e-55)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.4e-55) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-1.6d-5)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.4d-55) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.4e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6e-5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.4e-55:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6e-5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.4e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6e-5)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.4e-55)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6e-5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e-55], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.59999999999999993e-5

   1. Initial program 39.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.59999999999999993e-5 < re < 3.39999999999999973e-55

   1. Initial program 60.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 3.39999999999999973e-55 < re

   1. Initial program 11.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 77.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.6e-5)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.4e-55)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.4e-55) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.6d-5)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.4d-55) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.6e-5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.4e-55) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.6e-5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.4e-55:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.6e-5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.4e-55)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.6e-5)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.4e-55)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.6e-5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.4e-55], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.59999999999999993e-5

   1. Initial program 39.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.59999999999999993e-5 < re < 3.39999999999999973e-55

   1. Initial program 60.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 3.39999999999999973e-55 < re

   1. Initial program 11.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 77.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.32e-5) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.32e-5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * 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 <= (-1.32d-5)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.32e-5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.32e-5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.32e-5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.32e-5)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.32e-5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.32000000000000007e-5

   1. Initial program 39.9%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.32000000000000007e-5 < re

   1. Initial program 44.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 54.4

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

5. Applied rewrites 54.4%

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

6. Final simplification 54.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))))