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

Percentage Accurate: 41.7% → 87.7%

Time: 6.9s

Alternatives: 9

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: 41.7% 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: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.1e-42)
   (* 0.5 (sqrt (* 2.0 (- (hypot im re) re))))
   (* (sqrt (/ 1.0 re)) (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.1e-42) {
		tmp = 0.5 * sqrt((2.0 * (hypot(im, re) - re)));
	} else {
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.1e-42) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(im, re) - re)));
	} else {
		tmp = Math.sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.1e-42:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(im, re) - re)))
	else:
		tmp = math.sqrt((1.0 / re)) * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.1e-42)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(im, re) - re))));
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.1e-42)
		tmp = 0.5 * sqrt((2.0 * (hypot(im, re) - re)));
	else
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.1e-42], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.1e-42

   1. Initial program 48.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 48.1

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

      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 48.1

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

      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 94.9

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

   4. Applied rewrites 94.9%

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

   if 5.1e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\frac{re}{im} \cdot re}{im}, 0.5, 1\right) \cdot im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.9e+59)
   (* (sqrt (fma (- im) (/ im re) (* -4.0 re))) 0.5)
   (if (<= re -9.2e-162)
     (* (sqrt (* (- (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
     (if (<= re 1.5e-42)
       (*
        (sqrt (* (- (* (fma (/ (* (/ re im) re) im) 0.5 1.0) im) re) 2.0))
        0.5)
       (* (sqrt (/ 1.0 re)) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+59) {
		tmp = sqrt(fma(-im, (im / re), (-4.0 * re))) * 0.5;
	} else if (re <= -9.2e-162) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) - re) * 2.0)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt((((fma((((re / im) * re) / im), 0.5, 1.0) * im) - re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.9e+59)
		tmp = Float64(sqrt(fma(Float64(-im), Float64(im / re), Float64(-4.0 * re))) * 0.5);
	elseif (re <= -9.2e-162)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) - re) * 2.0)) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(Float64(Float64(Float64(re / im) * re) / im), 0.5, 1.0) * im) - re) * 2.0)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(0.5 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.9e+59], N[(N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision] + N[(-4.0 * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -9.2e-162], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * im), $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.90000000000000021e59

   1. Initial program 34.0%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot re\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-re\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{{re}^{2}} + 4\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \left(\frac{\color{blue}{im \cdot im}}{{re}^{2}} + 4\right)} \]

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{im}{re}}, \frac{im}{re}, 4\right)} \]

      11. lower-/.f64 83.1

          \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right)} \]

   5. Applied rewrites 83.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 83.2%

      \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(-im, \color{blue}{\frac{im}{re}}, -4 \cdot re\right)} \]

   if -3.90000000000000021e59 < re < -9.1999999999999992e-162

   1. Initial program 80.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 80.0

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

   4. Applied rewrites 80.0%

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

   if -9.1999999999999992e-162 < re < 1.50000000000000014e-42

   1. Initial program 41.9%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\frac{re}{im} \cdot re}{im}, 0.5, 1\right) \cdot im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.9e+59)
   (* (sqrt (fma (- im) (/ im re) (* -4.0 re))) 0.5)
   (if (<= re -9.2e-162)
     (* (sqrt (* (- (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
     (if (<= re 1.5e-42)
       (* (sqrt (fma (- (/ re im) 2.0) re (* 2.0 im))) 0.5)
       (* (sqrt (/ 1.0 re)) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.9e+59) {
		tmp = sqrt(fma(-im, (im / re), (-4.0 * re))) * 0.5;
	} else if (re <= -9.2e-162) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) - re) * 2.0)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt(fma(((re / im) - 2.0), re, (2.0 * im))) * 0.5;
	} else {
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.9e+59)
		tmp = Float64(sqrt(fma(Float64(-im), Float64(im / re), Float64(-4.0 * re))) * 0.5);
	elseif (re <= -9.2e-162)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) - re) * 2.0)) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im) - 2.0), re, Float64(2.0 * im))) * 0.5);
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(0.5 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.9e+59], N[(N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision] + N[(-4.0 * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -9.2e-162], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], 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[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.90000000000000021e59

   1. Initial program 34.0%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot re\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-re\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{{re}^{2}} + 4\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \left(\frac{\color{blue}{im \cdot im}}{{re}^{2}} + 4\right)} \]

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{im}{re}}, \frac{im}{re}, 4\right)} \]

      11. lower-/.f64 83.1

          \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right)} \]

   5. Applied rewrites 83.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 83.2%

      \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(-im, \color{blue}{\frac{im}{re}}, -4 \cdot re\right)} \]

   if -3.90000000000000021e59 < re < -9.1999999999999992e-162

   1. Initial program 80.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 80.0

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

   4. Applied rewrites 80.0%

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

   if -9.1999999999999992e-162 < re < 1.50000000000000014e-42

   1. Initial program 41.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, 2 \cdot im\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+41)
   (* (sqrt (fma (- im) (/ im re) (* -4.0 re))) 0.5)
   (if (<= re 1.5e-42)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (* (sqrt (/ 1.0 re)) (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = sqrt(fma(-im, (im / re), (-4.0 * re))) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e+41)
		tmp = Float64(sqrt(fma(Float64(-im), Float64(im / re), Float64(-4.0 * re))) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(0.5 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e+41], N[(N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision] + N[(-4.0 * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9999999999999998e41

   1. Initial program 37.1%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot re\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(re\right)\right) \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)}} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-re\right)} \cdot \left(4 + \frac{{im}^{2}}{{re}^{2}}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\left(\frac{{im}^{2}}{{re}^{2}} + 4\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \left(\frac{\color{blue}{im \cdot im}}{{re}^{2}} + 4\right)} \]

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{im}{re}}, \frac{im}{re}, 4\right)} \]

      11. lower-/.f64 83.9

          \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \color{blue}{\frac{im}{re}}, 4\right)} \]

   5. Applied rewrites 83.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-re\right) \cdot \mathsf{fma}\left(\frac{im}{re}, \frac{im}{re}, 4\right)}} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(-im, \color{blue}{\frac{im}{re}}, -4 \cdot re\right)} \]

   if -2.9999999999999998e41 < re < 1.50000000000000014e-42

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-im, \frac{im}{re}, -4 \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+41)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 1.5e-42)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (* (sqrt (/ 1.0 re)) (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt((1.0 / re)) * (0.5 * 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 <= (-3d+41)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 1.5d-42) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt((1.0d0 / re)) * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt((1.0 / re)) * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e+41:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 1.5e-42:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt((1.0 / re)) * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e+41)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e+41)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 1.5e-42)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = sqrt((1.0 / re)) * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e+41], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9999999999999998e41

   1. Initial program 37.1%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.9999999999999998e41 < re < 1.50000000000000014e-42

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+41)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 1.5e-42)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} 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 <= (-3d+41)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 1.5d-42) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    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 <= -3e+41) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e+41:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 1.5e-42:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e+41)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e+41)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 1.5e-42)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e+41], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9999999999999998e41

   1. Initial program 37.1%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.9999999999999998e41 < re < 1.50000000000000014e-42

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{re}} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+41)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 1.5e-42)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (* (/ 0.5 (sqrt re)) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 / sqrt(re)) * 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 <= (-3d+41)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 1.5d-42) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = (0.5d0 / sqrt(re)) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3e+41) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 1.5e-42) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (0.5 / Math.sqrt(re)) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e+41:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 1.5e-42:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = (0.5 / math.sqrt(re)) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e+41)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 1.5e-42)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(0.5 / sqrt(re)) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e+41)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 1.5e-42)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = (0.5 / sqrt(re)) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e+41], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.5e-42], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.9999999999999998e41

   1. Initial program 37.1%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.9999999999999998e41 < re < 1.50000000000000014e-42

   1. Initial program 53.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   if 1.50000000000000014e-42 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 8.1

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

      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 8.1

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

      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 35.5

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 83.8

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 84.5%

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

   11. Applied rewrites 84.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{re}} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e+41) (* (sqrt (* -4.0 re)) 0.5) (* (sqrt (* 2.0 im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+41) {
		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 <= (-2.5d+41)) 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 <= -2.5e+41) {
		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 <= -2.5e+41:
		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 <= -2.5e+41)
		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 <= -2.5e+41)
		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, -2.5e+41], 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 < -2.50000000000000011e41

   1. Initial program 37.1%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -2.50000000000000011e41 < re

   1. Initial program 39.7%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot im} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.3% 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 39.0%

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

3. Taylor expanded in re around -inf

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

   1. lower-*.f64 30.5

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

5. Applied rewrites 30.5%

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

6. Final simplification 30.5%

   \[\leadsto \sqrt{-4 \cdot re} \cdot 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(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)))))