math.sqrt on complex, real part

Percentage Accurate: 40.8% → 87.1%

Time: 4.2s

Alternatives: 7

Speedup: 2.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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: 40.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \sqrt{-0.5 \cdot \frac{1}{re}}\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\_m\right) + re\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -8e+112)
   (* 0.5 (* (* im_m (sqrt (* -0.5 (/ 1.0 re)))) (sqrt 2.0)))
   (* 0.5 (sqrt (* 2.0 (+ (hypot re im_m) re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -8e+112) {
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im_m) + re)));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -8e+112) {
		tmp = 0.5 * ((im_m * Math.sqrt((-0.5 * (1.0 / re)))) * Math.sqrt(2.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im_m) + re)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -8e+112:
		tmp = 0.5 * ((im_m * math.sqrt((-0.5 * (1.0 / re)))) * math.sqrt(2.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im_m) + re)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -8e+112)
		tmp = Float64(0.5 * Float64(Float64(im_m * sqrt(Float64(-0.5 * Float64(1.0 / re)))) * sqrt(2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im_m) + re))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -8e+112)
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im_m) + re)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -8e+112], N[(0.5 * N[(N[(im$95$m * N[Sqrt[N[(-0.5 * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -7.9999999999999994e112

   1. Initial program 5.7%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 16.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 16.6

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

   6. Applied rewrites 16.6%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 20.8%

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

   10. Taylor expanded in re around -inf

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

       1. associate-*l* N/A

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

       2. lower-*.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. metadata-eval N/A

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

       5. sqrt-unprod N/A

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

       6. metadata-eval N/A

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

       7. lower-sqrt.f64 N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 83.5

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

   12. Applied rewrites 83.5%

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

   if -7.9999999999999994e112 < re

   1. Initial program 47.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lower-hypot.f64 87.8

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

   3. Applied rewrites 87.8%

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

Alternative 2: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+56}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \sqrt{-0.5 \cdot \frac{1}{re}}\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.3e+56)
   (* 0.5 (* (* im_m (sqrt (* -0.5 (/ 1.0 re)))) (sqrt 2.0)))
   (if (<= re 8.5e-20)
     (* 0.5 (sqrt (fma (+ (/ re im_m) 2.0) re (+ im_m im_m))))
     (* 0.5 (sqrt (* 4.0 re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.3e+56) {
		tmp = 0.5 * ((im_m * sqrt((-0.5 * (1.0 / re)))) * sqrt(2.0));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * sqrt(fma(((re / im_m) + 2.0), re, (im_m + im_m)));
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.3e+56)
		tmp = Float64(0.5 * Float64(Float64(im_m * sqrt(Float64(-0.5 * Float64(1.0 / re)))) * sqrt(2.0)));
	elseif (re <= 8.5e-20)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m + im_m))));
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.3e+56], N[(0.5 * N[(N[(im$95$m * N[Sqrt[N[(-0.5 * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e-20], N[(0.5 * N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.30000000000000002e56

   1. Initial program 9.1%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 22.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 22.0

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

   6. Applied rewrites 22.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 26.0%

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

   10. Taylor expanded in re around -inf

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

       1. associate-*l* N/A

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

       2. lower-*.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. metadata-eval N/A

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

       5. sqrt-unprod N/A

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

       6. metadata-eval N/A

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

       7. lower-sqrt.f64 N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 78.4

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

   12. Applied rewrites 78.4%

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

   if -3.30000000000000002e56 < re < 8.5000000000000005e-20

   1. Initial program 52.7%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 76.3

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

   4. Applied rewrites 76.3%

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

   if 8.5000000000000005e-20 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around inf

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

      1. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{-im\_m \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im\_m} + 2, re, im\_m + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.6e+113)
   (* 0.5 (sqrt (- (* im_m (/ im_m re)))))
   (if (<= re 8.5e-20)
     (* 0.5 (sqrt (fma (+ (/ re im_m) 2.0) re (+ im_m im_m))))
     (* 0.5 (sqrt (* 4.0 re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e+113) {
		tmp = 0.5 * sqrt(-(im_m * (im_m / re)));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * sqrt(fma(((re / im_m) + 2.0), re, (im_m + im_m)));
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.6e+113)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m * Float64(im_m / re)))));
	elseif (re <= 8.5e-20)
		tmp = Float64(0.5 * sqrt(fma(Float64(Float64(re / im_m) + 2.0), re, Float64(im_m + im_m))));
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.6e+113], N[(0.5 * N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e-20], N[(0.5 * N[Sqrt[N[(N[(N[(re / im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * re + N[(im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.59999999999999992e113

   1. Initial program 5.6%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5

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

   6. Applied rewrites 61.5%

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

   if -3.59999999999999992e113 < re < 8.5000000000000005e-20

   1. Initial program 50.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 73.5

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

   4. Applied rewrites 73.5%

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

   if 8.5000000000000005e-20 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around inf

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

      1. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \sqrt{-im\_m \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.1e+113)
   (* 0.5 (sqrt (- (* im_m (/ im_m re)))))
   (if (<= re 8.5e-20)
     (* 0.5 (sqrt (* 2.0 (+ im_m re))))
     (* 0.5 (sqrt (* 4.0 re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.1e+113) {
		tmp = 0.5 * sqrt(-(im_m * (im_m / re)));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.1d+113)) then
        tmp = 0.5d0 * sqrt(-(im_m * (im_m / re)))
    else if (re <= 8.5d-20) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + re)))
    else
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.1e+113) {
		tmp = 0.5 * Math.sqrt(-(im_m * (im_m / re)));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + re)));
	} else {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.1e+113:
		tmp = 0.5 * math.sqrt(-(im_m * (im_m / re)))
	elif re <= 8.5e-20:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + re)))
	else:
		tmp = 0.5 * math.sqrt((4.0 * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.1e+113)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(im_m * Float64(im_m / re)))));
	elseif (re <= 8.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.1e+113)
		tmp = 0.5 * sqrt(-(im_m * (im_m / re)));
	elseif (re <= 8.5e-20)
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	else
		tmp = 0.5 * sqrt((4.0 * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.1e+113], N[(0.5 * N[Sqrt[(-N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e-20], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.09999999999999991e113

   1. Initial program 5.6%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5

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

   6. Applied rewrites 61.5%

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

   if -3.09999999999999991e113 < re < 8.5000000000000005e-20

   1. Initial program 50.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 73.3%

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

   if 8.5000000000000005e-20 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around inf

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

      1. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+210}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) + re\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.45e+210)
   (* 0.5 (sqrt (* 2.0 (+ (- re) re))))
   (if (<= re 8.5e-20)
     (* 0.5 (sqrt (+ im_m im_m)))
     (* 0.5 (sqrt (* 4.0 re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.45e+210) {
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.45d+210)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-re + re)))
    else if (re <= 8.5d-20) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.45e+210) {
		tmp = 0.5 * Math.sqrt((2.0 * (-re + re)));
	} else if (re <= 8.5e-20) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.45e+210:
		tmp = 0.5 * math.sqrt((2.0 * (-re + re)))
	elif re <= 8.5e-20:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = 0.5 * math.sqrt((4.0 * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.45e+210)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-re) + re))));
	elseif (re <= 8.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.45e+210)
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	elseif (re <= 8.5e-20)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = 0.5 * sqrt((4.0 * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.45e+210], N[(0.5 * N[Sqrt[N[(2.0 * N[((-re) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e-20], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.44999999999999996e210

   1. Initial program 2.6%

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

   2. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.8

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

   4. Applied rewrites 21.8%

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

   if -1.44999999999999996e210 < re < 8.5000000000000005e-20

   1. Initial program 45.2%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 68.1

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

   4. Applied rewrites 68.1%

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

   if 8.5000000000000005e-20 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around inf

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

      1. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 8.5e-20) (* 0.5 (sqrt (+ im_m im_m))) (* 0.5 (sqrt (* 4.0 re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 8.5e-20) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 8.5d-20) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = 0.5d0 * sqrt((4.0d0 * 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 <= 8.5e-20) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 8.5e-20:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = 0.5 * math.sqrt((4.0 * re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 8.5e-20)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 8.5e-20)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = 0.5 * sqrt((4.0 * re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 8.5e-20], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.5000000000000005e-20

   1. Initial program 40.4%

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

   2. Taylor expanded in re around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 61.8

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

   4. Applied rewrites 61.8%

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

   if 8.5000000000000005e-20 < re

   1. Initial program 42.0%

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

   2. Taylor expanded in re around inf

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

      1. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \sqrt{im\_m + im\_m} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (sqrt (+ im_m im_m))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * sqrt((im_m + im_m));
}

Fortran

im_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.5d0 * sqrt((im_m + im_m))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5 * Math.sqrt((im_m + im_m));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * math.sqrt((im_m + im_m))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * sqrt(Float64(im_m + im_m)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5 * sqrt((im_m + im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.8%

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

2. Taylor expanded in re around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 52.9

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

4. Applied rewrites 52.9%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im + im}} \]
5. Add Preprocessing

Developer Target 1: 47.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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 2025112 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (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)))))