math.sqrt on complex, real part

Percentage Accurate: 41.9% → 73.2%

Time: 7.7s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\_m\right) \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{1}{\frac{1 - \mathsf{ratio\_of\_squares}\left(re, im\_m\right) \cdot 0.5}{im\_m}} + re\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im\_m \cdot im\_m} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.6e+22)
   (* 0.5 (sqrt (* (- im_m) (/ im_m re))))
   (if (<= re 1.9e-18)
     (*
      0.5
      (sqrt
       (*
        2.0
        (+ (/ 1.0 (/ (- 1.0 (* (ratio-of-squares re im_m) 0.5)) im_m)) re))))
     (if (<= re 2e+123)
       (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im_m im_m))) re))))
       (sqrt re)))))

Derivation

1. Split input into 4 regimes

2. if re < -3.6e22

   1. Initial program 7.7%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if -3.6e22 < re < 1.8999999999999999e-18

   1. Initial program 58.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 57.9

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

   4. Applied rewrites 57.9%

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

   5. Taylor expanded in im around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lower-ratio-of-squares.f64 47.1

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\frac{1}{\frac{1 - \mathsf{ratio\_of\_squares}\left(re, im\right) \cdot 0.5}{im}} + re\right)} \]

   7. Applied rewrites 47.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\frac{1}{\color{blue}{\frac{1 - \mathsf{ratio\_of\_squares}\left(re, im\right) \cdot 0.5}{im}}} + re\right)} \]

   if 1.8999999999999999e-18 < re < 1.99999999999999996e123

   1. Initial program 75.2%

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

   if 1.99999999999999996e123 < re

   1. Initial program 18.2%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 14.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 14.1

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in re around inf

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

      1. count-2-rev N/A

         \[\leadsto \sqrt{re} \]

      2. lower-sqrt.f64 86.7

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 86.7%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{1}{\frac{1 - \mathsf{ratio\_of\_squares}\left(re, im\right) \cdot 0.5}{im}} + re\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\_m\right) \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im\_m \cdot im\_m} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.6e+22)
   (* 0.5 (sqrt (* (- im_m) (/ im_m re))))
   (if (<= re 1.9e-18)
     (* 0.5 (sqrt (* 2.0 (+ im_m re))))
     (if (<= re 2e+123)
       (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im_m im_m))) re))))
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e+22) {
		tmp = 0.5 * sqrt((-im_m * (im_m / re)));
	} else if (re <= 1.9e-18) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else if (re <= 2e+123) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im_m * im_m))) + re)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.6d+22)) then
        tmp = 0.5d0 * sqrt((-im_m * (im_m / re)))
    else if (re <= 1.9d-18) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + re)))
    else if (re <= 2d+123) then
        tmp = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im_m * im_m))) + re)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e+22) {
		tmp = 0.5 * Math.sqrt((-im_m * (im_m / re)));
	} else if (re <= 1.9e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + re)));
	} else if (re <= 2e+123) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im_m * im_m))) + re)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.6e+22:
		tmp = 0.5 * math.sqrt((-im_m * (im_m / re)))
	elif re <= 1.9e-18:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + re)))
	elif re <= 2e+123:
		tmp = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im_m * im_m))) + re)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.6e+22)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im_m) * Float64(im_m / re))));
	elseif (re <= 1.9e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	elseif (re <= 2e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im_m * im_m))) + re))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.6e+22)
		tmp = 0.5 * sqrt((-im_m * (im_m / re)));
	elseif (re <= 1.9e-18)
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	elseif (re <= 2e+123)
		tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im_m * im_m))) + re)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.6e+22], N[(0.5 * N[Sqrt[N[((-im$95$m) * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.6e22

   1. Initial program 7.7%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if -3.6e22 < re < 1.8999999999999999e-18

   1. Initial program 58.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 46.7%

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

   if 1.8999999999999999e-18 < re < 1.99999999999999996e123

   1. Initial program 75.2%

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

   if 1.99999999999999996e123 < re

   1. Initial program 18.2%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 14.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 14.1

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in re around inf

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

      1. count-2-rev N/A

         \[\leadsto \sqrt{re} \]

      2. lower-sqrt.f64 86.7

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 86.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\_m\right) \cdot \frac{im\_m}{re}}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im\_m + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.6e+22)
   (* 0.5 (sqrt (* (- im_m) (/ im_m re))))
   (if (<= re 8.2e-17) (* 0.5 (sqrt (* 2.0 (+ im_m re)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e+22) {
		tmp = 0.5 * sqrt((-im_m * (im_m / re)));
	} else if (re <= 8.2e-17) {
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.6d+22)) then
        tmp = 0.5d0 * sqrt((-im_m * (im_m / re)))
    else if (re <= 8.2d-17) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im_m + re)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e+22) {
		tmp = 0.5 * Math.sqrt((-im_m * (im_m / re)));
	} else if (re <= 8.2e-17) {
		tmp = 0.5 * Math.sqrt((2.0 * (im_m + re)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.6e+22:
		tmp = 0.5 * math.sqrt((-im_m * (im_m / re)))
	elif re <= 8.2e-17:
		tmp = 0.5 * math.sqrt((2.0 * (im_m + re)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.6e+22)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im_m) * Float64(im_m / re))));
	elseif (re <= 8.2e-17)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im_m + re))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.6e+22)
		tmp = 0.5 * sqrt((-im_m * (im_m / re)));
	elseif (re <= 8.2e-17)
		tmp = 0.5 * sqrt((2.0 * (im_m + re)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.6e+22], N[(0.5 * N[Sqrt[N[((-im$95$m) * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.2e-17], N[(0.5 * N[Sqrt[N[(2.0 * N[(im$95$m + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.6e22

   1. Initial program 7.7%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.9

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

   7. Applied rewrites 47.9%

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

   if -3.6e22 < re < 8.2000000000000001e-17

   1. Initial program 58.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 46.7%

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

   if 8.2000000000000001e-17 < re

   1. Initial program 42.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 15.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 15.5

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in re around inf

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

      1. count-2-rev N/A

         \[\leadsto \sqrt{re} \]

      2. lower-sqrt.f64 75.8

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 75.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{+211}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) + re\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.9e+211)
   (* 0.5 (sqrt (* 2.0 (+ (- re) re))))
   (if (<= re 6.5e-17) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.9e+211) {
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	} else if (re <= 6.5e-17) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.9d+211)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-re + re)))
    else if (re <= 6.5d-17) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.9e+211) {
		tmp = 0.5 * Math.sqrt((2.0 * (-re + re)));
	} else if (re <= 6.5e-17) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.9e+211:
		tmp = 0.5 * math.sqrt((2.0 * (-re + re)))
	elif re <= 6.5e-17:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.9e+211)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-re) + re))));
	elseif (re <= 6.5e-17)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.9e+211)
		tmp = 0.5 * sqrt((2.0 * (-re + re)));
	elseif (re <= 6.5e-17)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.9e+211], N[(0.5 * N[Sqrt[N[(2.0 * N[((-re) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e-17], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.90000000000000023e211

   1. Initial program 2.3%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.1

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

   5. Applied rewrites 35.1%

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

   if -3.90000000000000023e211 < re < 6.4999999999999996e-17

   1. Initial program 45.1%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 38.6

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

   7. Applied rewrites 38.6%

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

   if 6.4999999999999996e-17 < re

   1. Initial program 42.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 15.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 15.5

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in re around inf

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

      1. count-2-rev N/A

         \[\leadsto \sqrt{re} \]

      2. lower-sqrt.f64 75.8

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 75.8%

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

Alternative 5: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m + im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 6.5e-17) (* 0.5 (sqrt (+ im_m im_m))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 6.5e-17) {
		tmp = 0.5 * sqrt((im_m + im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 6.5d-17) then
        tmp = 0.5d0 * sqrt((im_m + im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 6.5e-17) {
		tmp = 0.5 * Math.sqrt((im_m + im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 6.5e-17:
		tmp = 0.5 * math.sqrt((im_m + im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 6.5e-17)
		tmp = Float64(0.5 * sqrt(Float64(im_m + im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 6.5e-17)
		tmp = 0.5 * sqrt((im_m + im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 6.5e-17], N[(0.5 * N[Sqrt[N[(im$95$m + im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.4999999999999996e-17

   1. Initial program 40.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 34.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 34.1

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

   7. Applied rewrites 34.1%

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

   if 6.4999999999999996e-17 < re

   1. Initial program 42.8%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 15.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 15.5

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in re around inf

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

      1. count-2-rev N/A

         \[\leadsto \sqrt{re} \]

      2. lower-sqrt.f64 75.8

         \[\leadsto \sqrt{re} \]

   10. Applied rewrites 75.8%

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

Alternative 6: 26.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around 0

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

5. Applied rewrites 28.9%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 28.9

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

7. Applied rewrites 28.9%

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

8. Taylor expanded in re around inf

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

   1. count-2-rev N/A

      \[\leadsto \sqrt{re} \]

   2. lower-sqrt.f64 27.0

      \[\leadsto \sqrt{re} \]

10. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

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

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