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

Percentage Accurate: 40.9% → 89.9%

Time: 7.2s

Alternatives: 8

Speedup: 1.7×

Specification

\[im > 0\]

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* im im) (* re re))) re) 0.0)
   (* (* (pow re -0.5) im) 0.5)
   (* (sqrt (* (- (hypot im re) re) 2.0)) 0.5)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[(N[Power[re, -0.5], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 92.8%

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

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

   1. Initial program 52.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 52.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.4

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

   4. Applied rewrites 90.4%

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

4. Final simplification 90.8%

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

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{im \cdot im + re \cdot re} - re\\ t_1 := \sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left({re}^{-0.5} \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* im im) (* re re))) re))
        (t_1 (* (sqrt (* (- im re) 2.0)) 0.5)))
   (if (<= t_0 0.0)
     (* (* (pow re -0.5) im) 0.5)
     (if (<= t_0 2e-230)
       t_1
       (if (<= t_0 1e+148)
         (* (sqrt (* (- (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
         t_1)))))

C

double code(double re, double im) {
	double t_0 = sqrt(((im * im) + (re * re))) - re;
	double t_1 = sqrt(((im - re) * 2.0)) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (pow(re, -0.5) * im) * 0.5;
	} else if (t_0 <= 2e-230) {
		tmp = t_1;
	} else if (t_0 <= 1e+148) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) - re) * 2.0)) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(im * im) + Float64(re * re))) - re)
	t_1 = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64((re ^ -0.5) * im) * 0.5);
	elseif (t_0 <= 2e-230)
		tmp = t_1;
	elseif (t_0 <= 1e+148)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) - re) * 2.0)) * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[re, -0.5], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-230], t$95$1, If[LessEqual[t$95$0, 1e+148], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 92.8%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 2.00000000000000009e-230 or 1e148 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 6.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 2.00000000000000009e-230 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 1e148

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{im \cdot im + re \cdot re} - re \leq 0:\\ \;\;\;\;\left({re}^{-0.5} \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;\sqrt{im \cdot im + re \cdot re} - re \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;\sqrt{im \cdot im + re \cdot re} - re \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{im \cdot im + re \cdot re} - re\\ t_1 := \sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ (* im im) (* re re))) re))
        (t_1 (* (sqrt (* (- im re) 2.0)) 0.5)))
   (if (<= t_0 0.0)
     (* (* (sqrt (/ 1.0 re)) im) 0.5)
     (if (<= t_0 2e-230)
       t_1
       (if (<= t_0 1e+148)
         (* (sqrt (* (- (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
         t_1)))))

C

double code(double re, double im) {
	double t_0 = sqrt(((im * im) + (re * re))) - re;
	double t_1 = sqrt(((im - re) * 2.0)) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	} else if (t_0 <= 2e-230) {
		tmp = t_1;
	} else if (t_0 <= 1e+148) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) - re) * 2.0)) * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(im * im) + Float64(re * re))) - re)
	t_1 = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * im) * 0.5);
	elseif (t_0 <= 2e-230)
		tmp = t_1;
	elseif (t_0 <= 1e+148)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) - re) * 2.0)) * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-230], t$95$1, If[LessEqual[t$95$0, 1e+148], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.5%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 92.7%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 2.00000000000000009e-230 or 1e148 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 6.1%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 2.00000000000000009e-230 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 1e148

   1. Initial program 95.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 95.1

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

   4. Applied rewrites 95.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{im \cdot im + re \cdot re} - re \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \mathbf{elif}\;\sqrt{im \cdot im + re \cdot re} - re \leq 2 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;\sqrt{im \cdot im + re \cdot re} - re \leq 10^{+148}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+34)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 2.6e-52)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (* (* (sqrt (/ 1.0 re)) im) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+34) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 2.6e-52) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.2d+34)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 2.6d-52) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = (sqrt((1.0d0 / re)) * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+34) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 2.6e-52) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (Math.sqrt((1.0 / re)) * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+34:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 2.6e-52:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = (math.sqrt((1.0 / re)) * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+34)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 2.6e-52)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / re)) * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e+34)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 2.6e-52)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = (sqrt((1.0 / re)) * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+34], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.6e-52], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1999999999999998e34

   1. Initial program 55.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -3.1999999999999998e34 < re < 2.5999999999999999e-52

   1. Initial program 53.2%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2.5999999999999999e-52 < re

   1. Initial program 19.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 72.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{re}} \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+34)
   (* (sqrt (* -4.0 re)) 0.5)
   (if (<= re 2.6e-52)
     (* (sqrt (* (- im re) 2.0)) 0.5)
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+34) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 2.6e-52) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.2d+34)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else if (re <= 2.6d-52) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = (im / sqrt(re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+34) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else if (re <= 2.6e-52) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+34:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	elif re <= 2.6e-52:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+34)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	elseif (re <= 2.6e-52)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e+34)
		tmp = sqrt((-4.0 * re)) * 0.5;
	elseif (re <= 2.6e-52)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+34], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.6e-52], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.1999999999999998e34

   1. Initial program 55.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -3.1999999999999998e34 < re < 2.5999999999999999e-52

   1. Initial program 53.2%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2.5999999999999999e-52 < re

   1. Initial program 19.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 72.2

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.2

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

   7. Applied rewrites 72.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.5e-183)
   (* (sqrt (* -4.0 re)) 0.5)
   (* (sqrt (* (- im re) 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.5e-183) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.5d-183) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.5e-183) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.5e-183:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.5e-183)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.5e-183)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.5e-183], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.49999999999999971e-183

   1. Initial program 33.9%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

   if 4.49999999999999971e-183 < im

   1. Initial program 48.9%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 71.1

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

   5. Applied rewrites 71.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -40000000000000:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -40000000000000.0)
   (* (sqrt (* -4.0 re)) 0.5)
   (* (sqrt (* 2.0 im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -40000000000000.0) {
		tmp = sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = sqrt((2.0 * im)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-40000000000000.0d0)) then
        tmp = sqrt(((-4.0d0) * re)) * 0.5d0
    else
        tmp = sqrt((2.0d0 * im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -40000000000000.0) {
		tmp = Math.sqrt((-4.0 * re)) * 0.5;
	} else {
		tmp = Math.sqrt((2.0 * im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -40000000000000.0:
		tmp = math.sqrt((-4.0 * re)) * 0.5
	else:
		tmp = math.sqrt((2.0 * im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -40000000000000.0)
		tmp = Float64(sqrt(Float64(-4.0 * re)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(2.0 * im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -40000000000000.0)
		tmp = sqrt((-4.0 * re)) * 0.5;
	else
		tmp = sqrt((2.0 * im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -40000000000000.0], N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4e13

   1. Initial program 60.1%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -4e13 < re

   1. Initial program 40.1%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -40000000000000:\\ \;\;\;\;\sqrt{-4 \cdot re} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot im} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{-4 \cdot re} \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sqrt (* -4.0 re)) 0.5))

C

double code(double re, double im) {
	return sqrt((-4.0 * re)) * 0.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(((-4.0d0) * re)) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return Math.sqrt((-4.0 * re)) * 0.5;
}

Python

def code(re, im):
	return math.sqrt((-4.0 * re)) * 0.5

Julia

function code(re, im)
	return Float64(sqrt(Float64(-4.0 * re)) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((-4.0 * re)) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 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 -inf

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

   1. lower-*.f64 24.1

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

5. Applied rewrites 24.1%

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

6. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))