math.sqrt on complex, real part

Percentage Accurate: 40.8% → 84.8%

Time: 4.7s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

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

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

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: 84.8% accurate, 0.4× speedup

Math

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

FPCore

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

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 * (pow(im, 2.0) / re))) * 0.5;
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) + re)));
	}
	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 * (Math.pow(im, 2.0) / re))) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) + re)));
	}
	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 * (math.pow(im, 2.0) / re))) * 0.5
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) + re)))
	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((im ^ 2.0) / re))) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) + re))));
	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 ^ 2.0) / re))) * 0.5;
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) + re)));
	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[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 40.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.7%

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

   4. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.7%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sum-to-mult-rev N/A

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

      12. lower-+.f64 29.8%

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

   6. Applied rewrites 29.8%

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

   7. Taylor expanded in re around -inf

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \cdot 0.5 \]
   8. 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. lower-pow.f64 15.4%

         \[\leadsto \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \cdot 0.5 \]

   9. Applied rewrites 15.4%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \cdot 0.5 \]

   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 40.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. add-flip N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

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

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

      14. distribute-lft-neg-out N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. fp-cancel-sub-sign-inv N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. sqr-neg-rev N/A

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

      20. sqr-abs-rev N/A

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

   3. Applied rewrites 79.4%

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

Alternative 2: 79.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] + re), $MachinePrecision]), $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. 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. sqrt-fabs-rev N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-+.f64 N/A

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

   9. add-flip N/A

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

   10. sub-flip N/A

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

   11. lift-*.f64 N/A

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

   12. sqr-abs-rev N/A

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

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

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

   14. distribute-lft-neg-out N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. fp-cancel-sub-sign-inv N/A

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

   17. lift-*.f64 N/A

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

   18. remove-double-neg N/A

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

   19. sqr-neg-rev N/A

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

   20. sqr-abs-rev N/A

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

3. Applied rewrites 79.4%

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

Alternative 3: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\left|im\right| + \left|im\right|} \cdot 0.5\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(\left|im\right|, \left|im\right|, re \cdot re\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-125)
   (* (sqrt (+ (fabs im) (fabs im))) 0.5)
   (if (<= re 6.2e+82)
     (* (sqrt (* (+ (sqrt (fma (fabs im) (fabs im) (* re re))) re) 2.0)) 0.5)
     (* 0.5 (sqrt (* 4.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-125) {
		tmp = sqrt((fabs(im) + fabs(im))) * 0.5;
	} else if (re <= 6.2e+82) {
		tmp = sqrt(((sqrt(fma(fabs(im), fabs(im), (re * re))) + re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * sqrt((4.0 * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e-125)
		tmp = Float64(sqrt(Float64(abs(im) + abs(im))) * 0.5);
	elseif (re <= 6.2e+82)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(abs(im), abs(im), Float64(re * re))) + re) * 2.0)) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-125], N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 6.2e+82], N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] * N[Abs[im], $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 6.20000000000000026e-125

   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 im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.7%

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

   4. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.7%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sum-to-mult-rev N/A

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

      12. lower-+.f64 29.8%

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

   6. Applied rewrites 29.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 26.0%

      \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. count-2-rev N/A

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

       4. lower-+.f64 26.0%

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

   11. Applied rewrites 26.0%

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

   if 6.20000000000000026e-125 < re < 6.20000000000000065e82

   1. Initial program 40.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 40.8%

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

   3. Applied rewrites 40.8%

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

   if 6.20000000000000065e82 < re

   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 inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 27.0%

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

   4. Applied rewrites 27.0%

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

Alternative 4: 64.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left|im\right| + \left|im\right|} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.3e-24)
   (* (sqrt (+ (fabs im) (fabs im))) 0.5)
   (* 0.5 (sqrt (* 4.0 re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.3e-24) {
		tmp = sqrt((fabs(im) + fabs(im))) * 0.5;
	} else {
		tmp = 0.5 * sqrt((4.0 * 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) :: tmp
    if (re <= 2.3d-24) then
        tmp = sqrt((abs(im) + abs(im))) * 0.5d0
    else
        tmp = 0.5d0 * sqrt((4.0d0 * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.3e-24) {
		tmp = Math.sqrt((Math.abs(im) + Math.abs(im))) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt((4.0 * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.3e-24:
		tmp = math.sqrt((math.fabs(im) + math.fabs(im))) * 0.5
	else:
		tmp = 0.5 * math.sqrt((4.0 * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.3e-24)
		tmp = Float64(sqrt(Float64(abs(im) + abs(im))) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(4.0 * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.3e-24)
		tmp = sqrt((abs(im) + abs(im))) * 0.5;
	else
		tmp = 0.5 * sqrt((4.0 * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.3e-24], N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.3000000000000001e-24

   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 im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 28.7%

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

   4. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.7%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sum-to-mult-rev N/A

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

      12. lower-+.f64 29.8%

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

   6. Applied rewrites 29.8%

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

   7. Taylor expanded in re around 0

      \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 26.0%

      \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. count-2-rev N/A

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

       4. lower-+.f64 26.0%

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

   11. Applied rewrites 26.0%

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

   if 2.3000000000000001e-24 < re

   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 inf

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re}} \]
   3. Step-by-step derivation

      1. lower-*.f64 27.0%

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

   4. Applied rewrites 27.0%

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

Alternative 5: 51.6% accurate, 2.0× speedup

Math

\[\sqrt{\left|im\right| + \left|im\right|} \cdot 0.5 \]

FPCore

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

C

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

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((abs(im) + abs(im))) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Sqrt[N[(N[Abs[im], $MachinePrecision] + N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $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 im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 28.7%

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

4. Applied rewrites 28.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 28.7%

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 28.7%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. sum-to-mult-rev N/A

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

   12. lower-+.f64 29.8%

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

6. Applied rewrites 29.8%

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

7. Taylor expanded in re around 0

   \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
8. Step-by-step derivation

9. Applied rewrites 26.0%

   \[\leadsto \sqrt{im \cdot 2} \cdot 0.5 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. count-2-rev N/A

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

    4. lower-+.f64 26.0%

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

11. Applied rewrites 26.0%

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

Developer Target 1: 48.2% accurate, 0.6× speedup

Math

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

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 2025183 
(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)))))