math.sqrt on complex, real part

Percentage Accurate: 41.7% → 90.2%

Time: 10.6s

Alternatives: 10

Speedup: 1.4×

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: 90.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(im$95$m * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 12.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 12.4%

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

   5. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.5

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

   7. Simplified 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   9. Applied egg-rr 57.3%

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

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

   1. Initial program 45.2%

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

      1. lower-hypot.f64 93.0

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

   4. Applied egg-rr 93.0%

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

4. Final simplification 88.7%

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

Alternative 2: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\left(im\_m \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im\_m}, 2 \cdot \left(re + im\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(re, \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re \cdot re}, 1\right)}, re\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.9e-64)
   (* (* im_m (sqrt (/ -0.5 re))) (* (sqrt 2.0) 0.5))
   (if (<= re 1.7e-80)
     (* 0.5 (sqrt (fma re (/ re im_m) (* 2.0 (+ re im_m)))))
     (*
      0.5
      (sqrt (* 2.0 (fma re (sqrt (fma im_m (/ im_m (* re re)) 1.0)) re)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.9e-64) {
		tmp = (im_m * sqrt((-0.5 / re))) * (sqrt(2.0) * 0.5);
	} else if (re <= 1.7e-80) {
		tmp = 0.5 * sqrt(fma(re, (re / im_m), (2.0 * (re + im_m))));
	} else {
		tmp = 0.5 * sqrt((2.0 * fma(re, sqrt(fma(im_m, (im_m / (re * re)), 1.0)), re)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.9e-64)
		tmp = Float64(Float64(im_m * sqrt(Float64(-0.5 / re))) * Float64(sqrt(2.0) * 0.5));
	elseif (re <= 1.7e-80)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(re / im_m), Float64(2.0 * Float64(re + im_m)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * fma(re, sqrt(fma(im_m, Float64(im_m / Float64(re * re)), 1.0)), re))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.9e-64], N[(N[(im$95$m * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.7e-80], N[(0.5 * N[Sqrt[N[(re * N[(re / im$95$m), $MachinePrecision] + N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * N[Sqrt[N[(im$95$m * N[(im$95$m / N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.8999999999999997e-64

   1. Initial program 11.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 11.4%

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

   5. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.6

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

   7. Simplified 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   9. Applied egg-rr 45.8%

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

   if -3.8999999999999997e-64 < re < 1.7e-80

   1. Initial program 53.5%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l/ N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.8

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

   5. Simplified 41.8%

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

   7. Applied egg-rr 41.7%

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

   if 1.7e-80 < re

   1. Initial program 50.2%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-hypot.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 50.2

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

   6. Applied egg-rr 50.2%

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

   7. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 48.2

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

   9. Simplified 48.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. unpow1 N/A

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

       7. sqrt-pow1 N/A

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

       8. metadata-eval N/A

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

       9. pow1/2 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-+.f64 N/A

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

   11. Applied egg-rr 88.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\left(im \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im}, 2 \cdot \left(re + im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \mathsf{fma}\left(re, \sqrt{\mathsf{fma}\left(im, \frac{im}{re \cdot re}, 1\right)}, re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\left(im\_m \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im\_m}, 2 \cdot \left(re + im\_m\right)\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re}, re \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.9e-64)
   (* (* im_m (sqrt (/ -0.5 re))) (* (sqrt 2.0) 0.5))
   (if (<= re 1.25e-113)
     (* 0.5 (sqrt (fma re (/ re im_m) (* 2.0 (+ re im_m)))))
     (if (<= re 1.15e+57)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im_m im_m (* re re)))))))
       (* 0.5 (sqrt (fma im_m (/ im_m re) (* re 4.0))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.9e-64) {
		tmp = (im_m * sqrt((-0.5 / re))) * (sqrt(2.0) * 0.5);
	} else if (re <= 1.25e-113) {
		tmp = 0.5 * sqrt(fma(re, (re / im_m), (2.0 * (re + im_m))));
	} else if (re <= 1.15e+57) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im_m, im_m, (re * re))))));
	} else {
		tmp = 0.5 * sqrt(fma(im_m, (im_m / re), (re * 4.0)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.9e-64)
		tmp = Float64(Float64(im_m * sqrt(Float64(-0.5 / re))) * Float64(sqrt(2.0) * 0.5));
	elseif (re <= 1.25e-113)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(re / im_m), Float64(2.0 * Float64(re + im_m)))));
	elseif (re <= 1.15e+57)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im_m, im_m, Float64(re * re)))))));
	else
		tmp = Float64(0.5 * sqrt(fma(im_m, Float64(im_m / re), Float64(re * 4.0))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.9e-64], N[(N[(im$95$m * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e-113], N[(0.5 * N[Sqrt[N[(re * N[(re / im$95$m), $MachinePrecision] + N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+57], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im$95$m * N[(im$95$m / re), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.8999999999999997e-64

   1. Initial program 11.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 11.4%

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

   5. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.6

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

   7. Simplified 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   9. Applied egg-rr 45.8%

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

   if -3.8999999999999997e-64 < re < 1.2499999999999999e-113

   1. Initial program 49.6%

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

   3. Taylor expanded in re around 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. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l/ N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 42.9

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

   5. Simplified 42.9%

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

   7. Applied egg-rr 42.8%

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

   if 1.2499999999999999e-113 < re < 1.1499999999999999e57

   1. Initial program 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 79.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.3

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

   4. Applied egg-rr 79.3%

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

   if 1.1499999999999999e57 < re

   1. Initial program 39.4%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-hypot.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 39.4

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

   6. Applied egg-rr 39.4%

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

   7. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 39.4

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

   9. Simplified 39.4%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re} + 4 \cdot re}} \cdot \frac{1}{2} \]

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 89.0

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

   12. Simplified 89.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\left(im \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(\sqrt{2} \cdot 0.5\right)\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im}, 2 \cdot \left(re + im\right)\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im\_m \cdot im\_m}{-re}}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im\_m}, 2 \cdot \left(re + im\_m\right)\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re}, re \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -6.8e+21)
   (* 0.5 (sqrt (/ (* im_m im_m) (- re))))
   (if (<= re 1.25e-113)
     (* 0.5 (sqrt (fma re (/ re im_m) (* 2.0 (+ re im_m)))))
     (if (<= re 1.15e+57)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im_m im_m (* re re)))))))
       (* 0.5 (sqrt (fma im_m (/ im_m re) (* re 4.0))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -6.8e+21) {
		tmp = 0.5 * sqrt(((im_m * im_m) / -re));
	} else if (re <= 1.25e-113) {
		tmp = 0.5 * sqrt(fma(re, (re / im_m), (2.0 * (re + im_m))));
	} else if (re <= 1.15e+57) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im_m, im_m, (re * re))))));
	} else {
		tmp = 0.5 * sqrt(fma(im_m, (im_m / re), (re * 4.0)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -6.8e+21)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im_m * im_m) / Float64(-re))));
	elseif (re <= 1.25e-113)
		tmp = Float64(0.5 * sqrt(fma(re, Float64(re / im_m), Float64(2.0 * Float64(re + im_m)))));
	elseif (re <= 1.15e+57)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im_m, im_m, Float64(re * re)))))));
	else
		tmp = Float64(0.5 * sqrt(fma(im_m, Float64(im_m / re), Float64(re * 4.0))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -6.8e+21], N[(0.5 * N[Sqrt[N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.25e-113], N[(0.5 * N[Sqrt[N[(re * N[(re / im$95$m), $MachinePrecision] + N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+57], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im$95$m * N[(im$95$m / re), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -6.8e21

   1. Initial program 12.6%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.5

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

   5. Simplified 55.5%

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

   if -6.8e21 < re < 1.2499999999999999e-113

   1. Initial program 44.8%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l/ N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.6

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

   5. Simplified 41.6%

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

   7. Applied egg-rr 41.6%

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

   if 1.2499999999999999e-113 < re < 1.1499999999999999e57

   1. Initial program 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 79.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 79.3

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

   4. Applied egg-rr 79.3%

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

   if 1.1499999999999999e57 < re

   1. Initial program 39.4%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-hypot.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 39.4

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

   6. Applied egg-rr 39.4%

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

   7. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 39.4

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

   9. Simplified 39.4%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re} + 4 \cdot re}} \cdot \frac{1}{2} \]

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 89.0

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

   12. Simplified 89.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(re, \frac{re}{im}, 2 \cdot \left(re + im\right)\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im\_m \cdot im\_m}{-re}}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re}, re \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.8)
   (* 0.5 (sqrt (/ (* im_m im_m) (- re))))
   (if (<= re 1.45e+102)
     (* 0.5 (sqrt (* 2.0 (+ re im_m))))
     (* 0.5 (sqrt (fma im_m (/ im_m re) (* re 4.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5.8) {
		tmp = 0.5 * sqrt(((im_m * im_m) / -re));
	} else if (re <= 1.45e+102) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * sqrt(fma(im_m, (im_m / re), (re * 4.0)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5.8)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im_m * im_m) / Float64(-re))));
	elseif (re <= 1.45e+102)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = Float64(0.5 * sqrt(fma(im_m, Float64(im_m / re), Float64(re * 4.0))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5.8], N[(0.5 * N[Sqrt[N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.45e+102], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im$95$m * N[(im$95$m / re), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.79999999999999982

   1. Initial program 12.3%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.4

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

   5. Simplified 55.4%

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

   if -5.79999999999999982 < re < 1.4500000000000001e102

   1. Initial program 55.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 38.7

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

   5. Simplified 38.7%

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

   if 1.4500000000000001e102 < re

   1. Initial program 36.2%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-hypot.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 36.2

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

   6. Applied egg-rr 36.2%

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

   7. Taylor expanded in re around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 36.2

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

   9. Simplified 36.2%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{{im}^{2}}{re} + 4 \cdot re}} \cdot \frac{1}{2} \]

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 95.2

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

   12. Simplified 95.2%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im\_m \cdot im\_m}{-re}}\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.8)
   (* 0.5 (sqrt (/ (* im_m im_m) (- re))))
   (if (<= re 1.95e+56) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5.8) {
		tmp = 0.5 * sqrt(((im_m * im_m) / -re));
	} else if (re <= 1.95e+56) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-5.8d0)) then
        tmp = 0.5d0 * sqrt(((im_m * im_m) / -re))
    else if (re <= 1.95d+56) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -5.8) {
		tmp = 0.5 * Math.sqrt(((im_m * im_m) / -re));
	} else if (re <= 1.95e+56) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5.8:
		tmp = 0.5 * math.sqrt(((im_m * im_m) / -re))
	elif re <= 1.95e+56:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5.8)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im_m * im_m) / Float64(-re))));
	elseif (re <= 1.95e+56)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -5.8)
		tmp = 0.5 * sqrt(((im_m * im_m) / -re));
	elseif (re <= 1.95e+56)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5.8], N[(0.5 * N[Sqrt[N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.95e+56], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.79999999999999982

   1. Initial program 12.3%

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

   3. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.4

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

   5. Simplified 55.4%

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

   if -5.79999999999999982 < re < 1.94999999999999997e56

   1. Initial program 55.3%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 39.5

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

   5. Simplified 39.5%

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

   if 1.94999999999999997e56 < re

   1. Initial program 39.4%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   6. 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. lower-*.f64 N/A

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

      7. lower-sqrt.f64 88.5

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

   7. Simplified 88.5%

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

      1. lift-sqrt.f64 N/A

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

      2. *-lft-identity 88.5

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

   9. Applied egg-rr 88.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+165}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.2e+165)
   0.0
   (if (<= re 1.95e+56) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.2e+165) {
		tmp = 0.0;
	} else if (re <= 1.95e+56) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-2.2d+165)) then
        tmp = 0.0d0
    else if (re <= 1.95d+56) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -2.2e+165) {
		tmp = 0.0;
	} else if (re <= 1.95e+56) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.2e+165:
		tmp = 0.0
	elif re <= 1.95e+56:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.2e+165)
		tmp = 0.0;
	elseif (re <= 1.95e+56)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -2.2e+165)
		tmp = 0.0;
	elseif (re <= 1.95e+56)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.2e+165], 0.0, If[LessEqual[re, 1.95e+56], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1999999999999999e165

   1. Initial program 2.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
   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 37.2

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

   5. Simplified 37.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. associate-*l* N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-neg.f64 N/A

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

      17. unsub-neg N/A

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

      18. +-inverses N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left({\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2}\right) \]

      19. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left(\color{blue}{0} \cdot \frac{1}{2}\right) \]

      20. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{0} \]

      21. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{{0}^{\frac{1}{2}}} \]

   7. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{0} \]

   if -2.1999999999999999e165 < re < 1.94999999999999997e56

   1. Initial program 48.2%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 33.9

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

   5. Simplified 33.9%

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

   if 1.94999999999999997e56 < re

   1. Initial program 39.4%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   6. 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. lower-*.f64 N/A

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

      7. lower-sqrt.f64 88.5

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

   7. Simplified 88.5%

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

      1. lift-sqrt.f64 N/A

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

      2. *-lft-identity 88.5

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

   9. Applied egg-rr 88.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+165}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+165}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.2e+165)
   0.0
   (if (<= re 4.8e+18) (* 0.5 (sqrt (* 2.0 im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.2e+165) {
		tmp = 0.0;
	} else if (re <= 4.8e+18) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} 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 <= (-2.2d+165)) then
        tmp = 0.0d0
    else if (re <= 4.8d+18) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    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 <= -2.2e+165) {
		tmp = 0.0;
	} else if (re <= 4.8e+18) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.2e+165:
		tmp = 0.0
	elif re <= 4.8e+18:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.2e+165)
		tmp = 0.0;
	elseif (re <= 4.8e+18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	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 <= -2.2e+165)
		tmp = 0.0;
	elseif (re <= 4.8e+18)
		tmp = 0.5 * sqrt((2.0 * im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.2e+165], 0.0, If[LessEqual[re, 4.8e+18], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1999999999999999e165

   1. Initial program 2.4%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
   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 37.2

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

   5. Simplified 37.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. associate-*l* N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-neg.f64 N/A

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

      17. unsub-neg N/A

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

      18. +-inverses N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left({\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2}\right) \]

      19. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left(\color{blue}{0} \cdot \frac{1}{2}\right) \]

      20. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{0} \]

      21. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{{0}^{\frac{1}{2}}} \]

   7. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{0} \]

   if -2.1999999999999999e165 < re < 4.8e18

   1. Initial program 46.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.5

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

   5. Simplified 31.5%

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

   if 4.8e18 < re

   1. Initial program 43.5%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   6. 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. lower-*.f64 N/A

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

      7. lower-sqrt.f64 84.0

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

   7. Simplified 84.0%

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

      1. lift-sqrt.f64 N/A

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

      2. *-lft-identity 84.0

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

   9. Applied egg-rr 84.0%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+165}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (if (<= re -5e-310) 0.0 (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.0;
	} 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 <= (-5d-310)) then
        tmp = 0.0d0
    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 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.0
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = 0.0;
	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 <= -5e-310)
		tmp = 0.0;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5e-310], 0.0, N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 24.8%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
   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 13.8

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

   5. Simplified 13.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. associate-*l* N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-neg.f64 N/A

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

      17. unsub-neg N/A

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

      18. +-inverses N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left({\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2}\right) \]

      19. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \left(\color{blue}{0} \cdot \frac{1}{2}\right) \]

      20. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{0} \]

      21. metadata-eval N/A

          \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{{0}^{\frac{1}{2}}} \]

   7. Applied egg-rr 13.8%

      \[\leadsto \color{blue}{0} \]

   if -4.999999999999985e-310 < re

   1. Initial program 53.3%

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

      1. lower-hypot.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   6. 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. lower-*.f64 N/A

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

      7. lower-sqrt.f64 55.7

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

   7. Simplified 55.7%

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

      1. lift-sqrt.f64 N/A

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

      2. *-lft-identity 55.7

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

   9. Applied egg-rr 55.7%

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

Alternative 10: 6.2% accurate, 47.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.0

Derivation

1. Initial program 41.2%

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

3. Taylor expanded in re around -inf

   \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
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 7.5

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

5. Simplified 7.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. lift-sqrt.f64 N/A

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

   10. pow1/2 N/A

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

   11. associate-*l* N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow1/2 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-neg.f64 N/A

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

   17. unsub-neg N/A

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

   18. +-inverses N/A

       \[\leadsto {2}^{\frac{1}{2}} \cdot \left({\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2}\right) \]

   19. metadata-eval N/A

       \[\leadsto {2}^{\frac{1}{2}} \cdot \left(\color{blue}{0} \cdot \frac{1}{2}\right) \]

   20. metadata-eval N/A

       \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{0} \]

   21. metadata-eval N/A

       \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{{0}^{\frac{1}{2}}} \]

7. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 49.0% 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 2024215 
(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)))))