math.sqrt on complex, real part

Percentage Accurate: 41.3% → 84.6%

Time: 3.6s

Alternatives: 5

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 41.3% 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: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leq 0:\\ \;\;\;\;\sqrt{-1 \cdot \frac{im \cdot im}{re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))) 0.0)
   (* (sqrt (* -1.0 (/ (* im im) re))) 0.5)
   (* (sqrt (* (+ (hypot im re) re) 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0) {
		tmp = sqrt((-1.0 * ((im * im) / re))) * 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 ((0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)))) <= 0.0) {
		tmp = Math.sqrt((-1.0 * ((im * im) / re))) * 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 (0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))) <= 0.0:
		tmp = math.sqrt((-1.0 * ((im * im) / re))) * 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(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re)))) <= 0.0)
		tmp = Float64(sqrt(Float64(-1.0 * Float64(Float64(im * im) / re))) * 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 ((0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)))) <= 0.0)
		tmp = sqrt((-1.0 * ((im * im) / re))) * 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[(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], 0.0], N[(N[Sqrt[N[(-1.0 * N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $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 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 8.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 8.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 8.9

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

      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 8.9

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

   3. Applied rewrites 8.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 8.9

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.0

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

   8. Applied rewrites 52.0%

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

   if 0.0 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))

   1. Initial program 45.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 45.6

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

      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 45.6

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

      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 89.0

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

   3. Applied rewrites 89.0%

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

Alternative 2: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{-1 \cdot \frac{im \cdot im}{re}} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))))))
   (if (<= t_0 0.0)
     (* (sqrt (* -1.0 (/ (* im im) re))) 0.5)
     (if (<= t_0 4e+71)
       (* 0.5 (sqrt (* 2.0 (+ (sqrt (fma re re (* im im))) re))))
       (* (sqrt (* (+ im re) 2.0)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((-1.0 * ((im * im) / re))) * 0.5;
	} else if (t_0 <= 4e+71) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) + re)));
	} else {
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(-1.0 * Float64(Float64(im * im) / re))) * 0.5);
	elseif (t_0 <= 4e+71)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) + re))));
	else
		tmp = Float64(sqrt(Float64(Float64(im + re) * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(-1.0 * N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 4e+71], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 8.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 8.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 8.9

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

      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 8.9

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

   3. Applied rewrites 8.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 8.9

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.0

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

   8. Applied rewrites 52.0%

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

   if 0.0 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 4.0000000000000002e71

   1. Initial program 93.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 93.0

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

   3. Applied rewrites 93.0%

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

   if 4.0000000000000002e71 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))

   1. Initial program 7.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 7.4

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

      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 7.4

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

      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 80.4

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

   3. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 4.8

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 4.8

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 31.5

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

   8. Applied rewrites 31.5%

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

Alternative 3: 49.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{-1 \cdot \frac{im \cdot im}{re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e-26)
   (* (sqrt (* -1.0 (/ (* im im) re))) 0.5)
   (if (<= re 1.4e+64) (* (sqrt (* (+ im re) 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e-26) {
		tmp = sqrt((-1.0 * ((im * im) / re))) * 0.5;
	} else if (re <= 1.4e+64) {
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(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) :: tmp
    if (re <= (-3.8d-26)) then
        tmp = sqrt(((-1.0d0) * ((im * im) / re))) * 0.5d0
    else if (re <= 1.4d+64) then
        tmp = sqrt(((im + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.8e-26) {
		tmp = Math.sqrt((-1.0 * ((im * im) / re))) * 0.5;
	} else if (re <= 1.4e+64) {
		tmp = Math.sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8e-26:
		tmp = math.sqrt((-1.0 * ((im * im) / re))) * 0.5
	elif re <= 1.4e+64:
		tmp = math.sqrt(((im + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8e-26)
		tmp = Float64(sqrt(Float64(-1.0 * Float64(Float64(im * im) / re))) * 0.5);
	elseif (re <= 1.4e+64)
		tmp = Float64(sqrt(Float64(Float64(im + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.8e-26)
		tmp = sqrt((-1.0 * ((im * im) / re))) * 0.5;
	elseif (re <= 1.4e+64)
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8e-26], N[(N[Sqrt[N[(-1.0 * N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.4e+64], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.80000000000000015e-26

   1. Initial program 13.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 13.0

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

      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 13.0

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

      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 42.9

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

   3. Applied rewrites 42.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 11.8

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 43.5

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

   8. Applied rewrites 43.5%

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

   if -3.80000000000000015e-26 < re < 1.40000000000000012e64

   1. Initial program 58.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 58.9

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

      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 58.9

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

      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 90.9

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

   3. Applied rewrites 90.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 44.1

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if 1.40000000000000012e64 < re

   1. Initial program 31.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 31.8

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

      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 31.8

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

      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 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in re around inf

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

      1. lower-sqrt.f64 81.9

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 81.9%

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

Alternative 4: 43.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.1 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.1e-67) (* (sqrt (* im 2.0)) 0.5) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.1e-67) {
		tmp = sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = sqrt(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) :: tmp
    if (re <= 4.1d-67) then
        tmp = sqrt((im * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.1e-67) {
		tmp = Math.sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.1e-67:
		tmp = math.sqrt((im * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.1e-67)
		tmp = Float64(sqrt(Float64(im * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.1e-67)
		tmp = sqrt((im * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.1e-67], N[(N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.0999999999999997e-67

   1. Initial program 38.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 38.6

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

      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 38.6

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

      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 70.9

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

   3. Applied rewrites 70.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 34.7

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

      14. lift-hypot.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-hypot.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.6%

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

   if 4.0999999999999997e-67 < re

   1. Initial program 47.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. 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 47.6

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

      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 47.6

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

      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 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-hypot.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in re around inf

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

      1. lower-sqrt.f64 72.2

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 72.2%

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

Alternative 5: 25.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

double code(double re, double im) {
	return sqrt(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 = sqrt(re)
end function

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 41.3%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. 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 41.3

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

   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 41.3

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

   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 79.5

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

3. Applied rewrites 79.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-hypot.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-hypot.f64 N/A

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

   15. lower-sqrt.f64 79.0

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

5. Applied rewrites 79.0%

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

   1. lift-sqrt.f64 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lower-/.f64 78.8

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

7. Applied rewrites 78.8%

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

8. Taylor expanded in re around inf

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

   1. lower-sqrt.f64 25.8

      \[\leadsto \sqrt{re} \]

10. Applied rewrites 25.8%

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

Developer Target 1: 48.5% 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 2025106 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

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

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