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

Percentage Accurate: 41.8% → 73.8%

Time: 2.3s

Alternatives: 8

Speedup: 2.5×

Specification

\[im > 0\]

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* 1/2 (sqrt (* 2 (- (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[(1/2 * N[Sqrt[N[(2 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* 1/2 (sqrt (* 2 (- (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[(1/2 * N[Sqrt[N[(2 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -3199999999999999787627735359118791747552630218171077842525632124659835956804589448297552398506401745056568866397772249313852982157676790821687394304:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq \frac{-8924236958169793}{13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \leq 349999999999999977082398796899540174385819355252213370164883576067233347140147216384:\\ \;\;\;\;\sqrt{im + im} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{\frac{im}{re} \cdot im}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     -3199999999999999787627735359118791747552630218171077842525632124659835956804589448297552398506401745056568866397772249313852982157676790821687394304)
  (* 1/2 (sqrt (* -4 re)))
  (if (<=
       re
       -8924236958169793/13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304)
    (* 1/2 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re))))
    (if (<=
         re
         349999999999999977082398796899540174385819355252213370164883576067233347140147216384)
      (* (sqrt (+ im im)) 1/2)
      (* 1/2 (sqrt (* (/ im re) im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+147) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -6.5e-142) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 3.5e+83) {
		tmp = sqrt((im + im)) * 0.5;
	} else {
		tmp = 0.5 * sqrt(((im / re) * im));
	}
	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 <= (-3.2d+147)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= (-6.5d-142)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    else if (re <= 3.5d+83) then
        tmp = sqrt((im + im)) * 0.5d0
    else
        tmp = 0.5d0 * sqrt(((im / re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+147) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= -6.5e-142) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	} else if (re <= 3.5e+83) {
		tmp = Math.sqrt((im + im)) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt(((im / re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+147:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= -6.5e-142:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	elif re <= 3.5e+83:
		tmp = math.sqrt((im + im)) * 0.5
	else:
		tmp = 0.5 * math.sqrt(((im / re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+147)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -6.5e-142)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))));
	elseif (re <= 3.5e+83)
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e+147)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= -6.5e-142)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	elseif (re <= 3.5e+83)
		tmp = sqrt((im + im)) * 0.5;
	else
		tmp = 0.5 * sqrt(((im / re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3199999999999999787627735359118791747552630218171077842525632124659835956804589448297552398506401745056568866397772249313852982157676790821687394304], N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -8924236958169793/13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304], N[(1/2 * N[Sqrt[N[(2 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 349999999999999977082398796899540174385819355252213370164883576067233347140147216384], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision], N[(1/2 * N[Sqrt[N[(N[(im / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.1999999999999998e147

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -3.1999999999999998e147 < re < -6.5000000000000003e-142

   1. Initial program 41.8%

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

   if -6.5000000000000003e-142 < re < 3.4999999999999998e83

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 51.5%

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.5%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 51.5%

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

   6. Applied rewrites 51.5%

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

   if 3.4999999999999998e83 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 14.5%

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

   4. Applied rewrites 14.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 17.3%

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

   6. Applied rewrites 17.3%

      \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{im}{re} \cdot \color{blue}{im}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -19000000000000001018918648858612766219271529294930202244204011192320:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 339999999999999984815804307206829625576824593371778690788974276089217024:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{im}{\frac{re}{im}}} \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     -19000000000000001018918648858612766219271529294930202244204011192320)
  (* 1/2 (sqrt (* -4 re)))
  (if (<=
       re
       339999999999999984815804307206829625576824593371778690788974276089217024)
    (* (sqrt (* (- im re) 2)) 1/2)
    (* (sqrt (/ im (/ re im))) 1/2))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt((im / (re / im))) * 0.5;
	}
	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 <= (-1.9d+67)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 3.4d+71) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt((im / (re / im))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt((im / (re / im))) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+67:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 3.4e+71:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt((im / (re / im))) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+67)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 3.4e+71)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(im / Float64(re / im))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+67)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 3.4e+71)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = sqrt((im / (re / im))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -19000000000000001018918648858612766219271529294930202244204011192320], N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 339999999999999984815804307206829625576824593371778690788974276089217024], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision], N[(N[Sqrt[N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e67

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -1.9000000000000001e67 < re < 3.3999999999999998e71

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.0%

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

   4. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.0%

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

   6. Applied rewrites 54.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 54.1%

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

   8. Applied rewrites 54.1%

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

   if 3.3999999999999998e71 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 14.5%

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

   4. Applied rewrites 14.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.5%

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 14.5%

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

   6. Applied rewrites 14.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 17.3%

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

   8. Applied rewrites 17.3%

      \[\leadsto \sqrt{\frac{im}{\color{blue}{\frac{re}{im}}}} \cdot \frac{1}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -19000000000000001018918648858612766219271529294930202244204011192320:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 339999999999999984815804307206829625576824593371778690788974276089217024:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{im \cdot \left(2 + -2 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{im}{\frac{re}{im}}} \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     -19000000000000001018918648858612766219271529294930202244204011192320)
  (* 1/2 (sqrt (* -4 re)))
  (if (<=
       re
       339999999999999984815804307206829625576824593371778690788974276089217024)
    (* 1/2 (sqrt (* im (+ 2 (* -2 (/ re im))))))
    (* (sqrt (/ im (/ re im))) 1/2))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = 0.5 * sqrt((im * (2.0 + (-2.0 * (re / im)))));
	} else {
		tmp = sqrt((im / (re / im))) * 0.5;
	}
	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 <= (-1.9d+67)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 3.4d+71) then
        tmp = 0.5d0 * sqrt((im * (2.0d0 + ((-2.0d0) * (re / im)))))
    else
        tmp = sqrt((im / (re / im))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = 0.5 * Math.sqrt((im * (2.0 + (-2.0 * (re / im)))));
	} else {
		tmp = Math.sqrt((im / (re / im))) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+67:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 3.4e+71:
		tmp = 0.5 * math.sqrt((im * (2.0 + (-2.0 * (re / im)))))
	else:
		tmp = math.sqrt((im / (re / im))) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+67)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 3.4e+71)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(2.0 + Float64(-2.0 * Float64(re / im))))));
	else
		tmp = Float64(sqrt(Float64(im / Float64(re / im))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+67)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 3.4e+71)
		tmp = 0.5 * sqrt((im * (2.0 + (-2.0 * (re / im)))));
	else
		tmp = sqrt((im / (re / im))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -19000000000000001018918648858612766219271529294930202244204011192320], N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 339999999999999984815804307206829625576824593371778690788974276089217024], N[(1/2 * N[Sqrt[N[(im * N[(2 + N[(-2 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e67

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -1.9000000000000001e67 < re < 3.3999999999999998e71

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.0%

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

   4. Applied rewrites 53.0%

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

   if 3.3999999999999998e71 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 14.5%

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

   4. Applied rewrites 14.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.5%

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 14.5%

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

   6. Applied rewrites 14.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 17.3%

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

   8. Applied rewrites 17.3%

      \[\leadsto \sqrt{\frac{im}{\color{blue}{\frac{re}{im}}}} \cdot \frac{1}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -19000000000000001018918648858612766219271529294930202244204011192320:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 339999999999999984815804307206829625576824593371778690788974276089217024:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{\frac{im}{re} \cdot im}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     -19000000000000001018918648858612766219271529294930202244204011192320)
  (* 1/2 (sqrt (* -4 re)))
  (if (<=
       re
       339999999999999984815804307206829625576824593371778690788974276089217024)
    (* (sqrt (* (- im re) 2)) 1/2)
    (* 1/2 (sqrt (* (/ im re) im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * sqrt(((im / re) * im));
	}
	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 <= (-1.9d+67)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 3.4d+71) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else
        tmp = 0.5d0 * sqrt(((im / re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.9e+67) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 3.4e+71) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt(((im / re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.9e+67:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 3.4e+71:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	else:
		tmp = 0.5 * math.sqrt(((im / re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.9e+67)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 3.4e+71)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(im / re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.9e+67)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 3.4e+71)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	else
		tmp = 0.5 * sqrt(((im / re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -19000000000000001018918648858612766219271529294930202244204011192320], N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 339999999999999984815804307206829625576824593371778690788974276089217024], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision], N[(1/2 * N[Sqrt[N[(N[(im / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e67

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -1.9000000000000001e67 < re < 3.3999999999999998e71

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.0%

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

   4. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.0%

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

   6. Applied rewrites 54.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 54.1%

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

   8. Applied rewrites 54.1%

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

   if 3.3999999999999998e71 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 14.5%

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

   4. Applied rewrites 14.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 17.3%

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

   6. Applied rewrites 17.3%

      \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{im}{re} \cdot \color{blue}{im}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{if}\;re \leq -19000000000000001018918648858612766219271529294930202244204011192320:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq \frac{-207784017051299}{76957043352332967211482500195592995713046365762627825523336510555167425334955489475418488779072100860950445293568}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2} \cdot \frac{1}{2}\\ \mathbf{elif}\;re \leq \frac{-8924236958169793}{13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 1/2 (sqrt (* -4 re)))))
  (if (<=
       re
       -19000000000000001018918648858612766219271529294930202244204011192320)
    t_0
    (if (<=
         re
         -207784017051299/76957043352332967211482500195592995713046365762627825523336510555167425334955489475418488779072100860950445293568)
      (* (sqrt (* (- im re) 2)) 1/2)
      (if (<=
           re
           -8924236958169793/13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304)
        t_0
        (* (sqrt (+ im im)) 1/2))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((-4.0 * re));
	double tmp;
	if (re <= -1.9e+67) {
		tmp = t_0;
	} else if (re <= -2.7e-99) {
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	} else if (re <= -6.5e-142) {
		tmp = t_0;
	} else {
		tmp = sqrt((im + im)) * 0.5;
	}
	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 = 0.5d0 * sqrt(((-4.0d0) * re))
    if (re <= (-1.9d+67)) then
        tmp = t_0
    else if (re <= (-2.7d-99)) then
        tmp = sqrt(((im - re) * 2.0d0)) * 0.5d0
    else if (re <= (-6.5d-142)) then
        tmp = t_0
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((-4.0 * re));
	double tmp;
	if (re <= -1.9e+67) {
		tmp = t_0;
	} else if (re <= -2.7e-99) {
		tmp = Math.sqrt(((im - re) * 2.0)) * 0.5;
	} else if (re <= -6.5e-142) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((-4.0 * re))
	tmp = 0
	if re <= -1.9e+67:
		tmp = t_0
	elif re <= -2.7e-99:
		tmp = math.sqrt(((im - re) * 2.0)) * 0.5
	elif re <= -6.5e-142:
		tmp = t_0
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(-4.0 * re)))
	tmp = 0.0
	if (re <= -1.9e+67)
		tmp = t_0;
	elseif (re <= -2.7e-99)
		tmp = Float64(sqrt(Float64(Float64(im - re) * 2.0)) * 0.5);
	elseif (re <= -6.5e-142)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((-4.0 * re));
	tmp = 0.0;
	if (re <= -1.9e+67)
		tmp = t_0;
	elseif (re <= -2.7e-99)
		tmp = sqrt(((im - re) * 2.0)) * 0.5;
	elseif (re <= -6.5e-142)
		tmp = t_0;
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -19000000000000001018918648858612766219271529294930202244204011192320], t$95$0, If[LessEqual[re, -207784017051299/76957043352332967211482500195592995713046365762627825523336510555167425334955489475418488779072100860950445293568], N[(N[Sqrt[N[(N[(im - re), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision], If[LessEqual[re, -8924236958169793/13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304], t$95$0, N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9000000000000001e67 or -2.6999999999999999e-99 < re < -6.5000000000000003e-142

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -1.9000000000000001e67 < re < -2.6999999999999999e-99

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 53.0%

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

   4. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.0%

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

   6. Applied rewrites 54.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 54.1%

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

   8. Applied rewrites 54.1%

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

   if -6.5000000000000003e-142 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 51.5%

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.5%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 51.5%

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

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{im + im} \cdot \frac{1}{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -61000000000000001154395395843627917390008246875717559787641110528:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<=
     re
     -61000000000000001154395395843627917390008246875717559787641110528)
  (* 1/2 (sqrt (* -4 re)))
  (* (sqrt (+ im im)) 1/2)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.1e+64) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = sqrt((im + im)) * 0.5;
	}
	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 <= (-6.1d+64)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = sqrt((im + im)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.1e+64) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.1e+64:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.1e+64)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.1e+64)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -61000000000000001154395395843627917390008246875717559787641110528], N[(1/2 * N[Sqrt[N[(-4 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.1000000000000001e64

   1. Initial program 41.8%

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

   2. Taylor expanded in re around -inf

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

      1. lower-*.f64 27.6%

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

   4. Applied rewrites 27.6%

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

   if -6.1000000000000001e64 < re

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 51.5%

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.5%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 51.5%

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

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{im + im} \cdot \frac{1}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{im + im}\\ \mathbf{if}\;im \leq \frac{921786315201311}{1707011694817242694164442058424641996069058130512872489061441999811593532881313810309486643423117898430190057111918909554147533223454557460573019149396692491800360340355587726966548041193424390330615044130786970107312831497593974090537952608256}:\\ \;\;\;\;t\_0 \cdot 0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sqrt (+ im im))))
  (if (<=
       im
       921786315201311/1707011694817242694164442058424641996069058130512872489061441999811593532881313810309486643423117898430190057111918909554147533223454557460573019149396692491800360340355587726966548041193424390330615044130786970107312831497593974090537952608256)
    (* t_0 0)
    (* t_0 1/2))))

C

double code(double re, double im) {
	double t_0 = sqrt((im + im));
	double tmp;
	if (im <= 5.4e-229) {
		tmp = t_0 * 0.0;
	} else {
		tmp = t_0 * 0.5;
	}
	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((im + im))
    if (im <= 5.4d-229) then
        tmp = t_0 * 0.0d0
    else
        tmp = t_0 * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((im + im));
	double tmp;
	if (im <= 5.4e-229) {
		tmp = t_0 * 0.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((im + im))
	tmp = 0
	if im <= 5.4e-229:
		tmp = t_0 * 0.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(im + im))
	tmp = 0.0
	if (im <= 5.4e-229)
		tmp = Float64(t_0 * 0.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((im + im));
	tmp = 0.0;
	if (im <= 5.4e-229)
		tmp = t_0 * 0.0;
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[im, 921786315201311/1707011694817242694164442058424641996069058130512872489061441999811593532881313810309486643423117898430190057111918909554147533223454557460573019149396692491800360340355587726966548041193424390330615044130786970107312831497593974090537952608256], N[(t$95$0 * 0), $MachinePrecision], N[(t$95$0 * 1/2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 5.3999999999999997e-229

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 51.5%

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.5%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 51.5%

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in undef-var around zero

      \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 6.0%

      \[\leadsto \sqrt{im + im} \cdot \color{blue}{0} \]

   if 5.3999999999999997e-229 < im

   1. Initial program 41.8%

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

   2. Taylor expanded in im around inf

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

      1. lower-*.f64 51.5%

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.5%

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 51.5%

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

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\sqrt{im + im} \cdot \frac{1}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.5% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (sqrt (+ im im)) 1/2))

C

double code(double re, double im) {
	return sqrt((im + 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((im + im)) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]

Derivation

1. Initial program 41.8%

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

2. Taylor expanded in im around inf

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

   1. lower-*.f64 51.5%

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

4. Applied rewrites 51.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 51.5%

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 51.5%

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

6. Applied rewrites 51.5%

   \[\leadsto \color{blue}{\sqrt{im + im} \cdot \frac{1}{2}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025274 -o generate:evaluate
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0)
  (* 1/2 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))