math.sqrt on complex, real part

Percentage Accurate: 41.7% → 83.1%

Time: 8.8s

Alternatives: 9

Speedup: 1.7×

Specification

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: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\sqrt{im\_m \cdot im\_m + re \cdot re} + re\right) \cdot 2} \leq 0:\\ \;\;\;\;e^{\left(\log im\_m \cdot 2 - \log \left(-re\right)\right) \cdot 0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(\mathsf{hypot}\left(im\_m, re\right) + re\right) \cdot 2\right) \cdot 0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* (+ (sqrt (+ (* im_m im_m) (* re re))) re) 2.0)) 0.0)
   (* (exp (* (- (* (log im_m) 2.0) (log (- re))) 0.5)) 0.5)
   (* (exp (* (log (* (+ (hypot im_m re) re) 2.0)) 0.5)) 0.5)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(N[(N[Sqrt[N[(N[(im$95$m * im$95$m), $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Exp[N[(N[(N[(N[Log[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[Log[(-re)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Exp[N[(N[Log[N[(N[(N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 8.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 8.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 8.2

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-hypot.f64 8.2

          \[\leadsto 0.5 \cdot e^{\log \left(\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2\right) \cdot 0.5} \]

   4. Applied rewrites 8.2%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2\right) \cdot 0.5}} \]

   5. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 61.7

         \[\leadsto 0.5 \cdot e^{\left(\log \left(im \cdot im\right) + \log \color{blue}{\left(\frac{-1}{re}\right)}\right) \cdot 0.5} \]

   7. Applied rewrites 61.7%

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

   9. Applied rewrites 38.9%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 43.1%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 40.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 40.9

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-hypot.f64 81.7

          \[\leadsto 0.5 \cdot e^{\log \left(\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2\right) \cdot 0.5} \]

   4. Applied rewrites 81.7%

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

4. Final simplification 74.6%

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

Alternative 2: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\frac{\left(\left(-im\_m\right) \cdot im\_m\right) \cdot 2}{re - \mathsf{hypot}\left(im\_m, re\right)}} \cdot 0.5\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(\mathsf{hypot}\left(im\_m, re\right) + re\right) \cdot 2\right) \cdot 0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e+199)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re -1.1e-23)
     (* (sqrt (/ (* (* (- im_m) im_m) 2.0) (- re (hypot im_m re)))) 0.5)
     (if (<= re 5e-252)
       (* (sqrt (fma (+ (/ re im_m) 2.0) re (* im_m 2.0))) 0.5)
       (* (exp (* (log (* (+ (hypot im_m re) re) 2.0)) 0.5)) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e+199) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= -1.1e-23) {
		tmp = sqrt((((-im_m * im_m) * 2.0) / (re - hypot(im_m, re)))) * 0.5;
	} else if (re <= 5e-252) {
		tmp = sqrt(fma(((re / im_m) + 2.0), re, (im_m * 2.0))) * 0.5;
	} else {
		tmp = exp((log(((hypot(im_m, re) + re) * 2.0)) * 0.5)) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e+199)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= -1.1e-23)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(-im_m) * im_m) * 2.0) / Float64(re - hypot(im_m, re)))) * 0.5);
	elseif (re <= 5e-252)
		tmp = Float64(sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m * 2.0))) * 0.5);
	else
		tmp = Float64(exp(Float64(log(Float64(Float64(hypot(im_m, re) + re) * 2.0)) * 0.5)) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5e+199], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -1.1e-23], N[(N[Sqrt[N[(N[(N[((-im$95$m) * im$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(re - N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 5e-252], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Exp[N[(N[Log[N[(N[(N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.9999999999999998e199

   1. Initial program 2.8%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -4.9999999999999998e199 < re < -1.1e-23

   1. Initial program 19.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 18.7%

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

   5. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 66.7

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

   7. Applied rewrites 66.7%

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

   if -1.1e-23 < re < 5.00000000000000008e-252

   1. Initial program 40.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative 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 N/A

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

      7. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

   if 5.00000000000000008e-252 < re

   1. Initial program 53.6%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 51.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 51.1

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

      10. lift-sqrt.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-hypot.f64 92.8

          \[\leadsto 0.5 \cdot e^{\log \left(\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2\right) \cdot 0.5} \]

   4. Applied rewrites 92.8%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2\right) \cdot 0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\frac{\left(\left(-im\right) \cdot im\right) \cdot 2}{re - \mathsf{hypot}\left(im, re\right)}} \cdot 0.5\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, im \cdot 2\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2\right) \cdot 0.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\left(\left(-im\_m\right) \cdot im\_m\right) \cdot 2}{re - \mathsf{hypot}\left(im\_m, re\right)}} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m}, 2, 2\right) \cdot im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e+199)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re -3.5e-26)
     (* (sqrt (/ (* (* (- im_m) im_m) 2.0) (- re (hypot im_m re)))) 0.5)
     (if (<= re 400000000000.0)
       (* (sqrt (* (fma (/ re im_m) 2.0 2.0) im_m)) 0.5)
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e+199) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= -3.5e-26) {
		tmp = sqrt((((-im_m * im_m) * 2.0) / (re - hypot(im_m, re)))) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = sqrt((fma((re / im_m), 2.0, 2.0) * im_m)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e+199)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= -3.5e-26)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(-im_m) * im_m) * 2.0) / Float64(re - hypot(im_m, re)))) * 0.5);
	elseif (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(fma(Float64(re / im_m), 2.0, 2.0) * im_m)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5e+199], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, -3.5e-26], N[(N[Sqrt[N[(N[(N[((-im$95$m) * im$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(re - N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.9999999999999998e199

   1. Initial program 2.8%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -4.9999999999999998e199 < re < -3.49999999999999985e-26

   1. Initial program 18.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 66.2

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

   7. Applied rewrites 66.2%

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

   if -3.49999999999999985e-26 < re < 4e11

   1. Initial program 54.0%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.0

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

   5. Applied rewrites 47.0%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq -3.5 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\left(\left(-im\right) \cdot im\right) \cdot 2}{re - \mathsf{hypot}\left(im, re\right)}} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im}, 2, 2\right) \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{\frac{re}{im\_m}}} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m}, 2, 2\right) \cdot im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.4e-25)
   (* (sqrt (/ (- im_m) (/ re im_m))) 0.5)
   (if (<= re 400000000000.0)
     (* (sqrt (* (fma (/ re im_m) 2.0 2.0) im_m)) 0.5)
     (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.4e-25) {
		tmp = sqrt((-im_m / (re / im_m))) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = sqrt((fma((re / im_m), 2.0, 2.0) * im_m)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.4e-25)
		tmp = Float64(sqrt(Float64(Float64(-im_m) / Float64(re / im_m))) * 0.5);
	elseif (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(fma(Float64(re / im_m), 2.0, 2.0) * im_m)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.4e-25], N[(N[Sqrt[N[((-im$95$m) / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000002e-25

   1. Initial program 14.2%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.1%

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

   if -3.40000000000000002e-25 < re < 4e11

   1. Initial program 54.0%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.0

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

   5. Applied rewrites 47.0%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im}{\frac{re}{im}}} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im}, 2, 2\right) \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im\_m}, 2, 2\right) \cdot im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.4e-25)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 400000000000.0)
     (* (sqrt (* (fma (/ re im_m) 2.0 2.0) im_m)) 0.5)
     (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.4e-25) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = sqrt((fma((re / im_m), 2.0, 2.0) * im_m)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.4e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(fma(Float64(re / im_m), 2.0, 2.0) * im_m)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.4e-25], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] * 2.0 + 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000002e-25

   1. Initial program 14.2%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if -3.40000000000000002e-25 < re < 4e11

   1. Initial program 54.0%

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

   3. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 47.0

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

   5. Applied rewrites 47.0%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{re}{im}, 2, 2\right) \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\left(im\_m + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.4e-25)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 400000000000.0) (* (sqrt (* (+ im_m re) 2.0)) 0.5) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.4e-25) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.4d-25)) then
        tmp = sqrt(((-im_m / re) * im_m)) * 0.5d0
    else if (re <= 400000000000.0d0) then
        tmp = sqrt(((im_m + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.4e-25) {
		tmp = Math.sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = Math.sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.4e-25:
		tmp = math.sqrt(((-im_m / re) * im_m)) * 0.5
	elif re <= 400000000000.0:
		tmp = math.sqrt(((im_m + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.4e-25)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(Float64(im_m + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.4e-25)
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	elseif (re <= 400000000000.0)
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.4e-25], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(N[(im$95$m + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000002e-25

   1. Initial program 14.2%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 61.1

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

   5. Applied rewrites 61.1%

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

   if -3.40000000000000002e-25 < re < 4e11

   1. Initial program 54.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 47.0

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

   5. Applied rewrites 47.0%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{im\_m \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.9e+147)
   (* (sqrt (* (+ (- re) re) 2.0)) 0.5)
   (if (<= re 400000000000.0) (* (sqrt (* im_m 2.0)) 0.5) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.9e+147) {
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.9d+147)) then
        tmp = sqrt(((-re + re) * 2.0d0)) * 0.5d0
    else if (re <= 400000000000.0d0) then
        tmp = sqrt((im_m * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.9e+147) {
		tmp = Math.sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 400000000000.0) {
		tmp = Math.sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.9e+147:
		tmp = math.sqrt(((-re + re) * 2.0)) * 0.5
	elif re <= 400000000000.0:
		tmp = math.sqrt((im_m * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.9e+147)
		tmp = Float64(sqrt(Float64(Float64(Float64(-re) + re) * 2.0)) * 0.5);
	elseif (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(im_m * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.9e+147)
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	elseif (re <= 400000000000.0)
		tmp = sqrt((im_m * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.9e+147], N[(N[Sqrt[N[(N[((-re) + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.89999999999999985e147

   1. Initial program 5.3%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

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

   5. Applied rewrites 26.7%

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

   if -1.89999999999999985e147 < re < 4e11

   1. Initial program 44.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-*.f64 40.5

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

   5. Applied rewrites 40.5%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.9 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{im\_m \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 400000000000.0) (* (sqrt (* im_m 2.0)) 0.5) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 400000000000.0) {
		tmp = sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 400000000000.0d0) then
        tmp = sqrt((im_m * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 400000000000.0) {
		tmp = Math.sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 400000000000.0:
		tmp = math.sqrt((im_m * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 400000000000.0)
		tmp = Float64(sqrt(Float64(im_m * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 400000000000.0)
		tmp = sqrt((im_m * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 400000000000.0], N[(N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4e11

   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 0

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

      1. lower-*.f64 33.8

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

   5. Applied rewrites 33.8%

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

   if 4e11 < re

   1. Initial program 38.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 86.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 86.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 400000000000:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sqrt(re);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = sqrt(re)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sqrt(re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sqrt(re)

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sqrt(re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 37.4%

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

3. Taylor expanded in re around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{\sqrt{re}} \]

   7. lower-sqrt.f64 24.5

      \[\leadsto \color{blue}{\sqrt{re}} \]

5. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\sqrt{re}} \]
6. Add Preprocessing

Developer Target 1: 48.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024296 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))