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

Percentage Accurate: 40.8% → 78.0%

Time: 5.6s

Alternatives: 7

Speedup: 1.9×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -9.6 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({re}^{-0.5} \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e+68)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -9.6e-95)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 5.6e-27)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (* (* (pow re -0.5) im) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+68) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -9.6e-95) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 5.6e-27) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (pow(re, -0.5) * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e+68)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -9.6e-95)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 5.6e-27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64((re ^ -0.5) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e+68], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -9.6e-95], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-27], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[re, -0.5], $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.84999999999999999e68

   1. Initial program 30.1%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -1.84999999999999999e68 < re < -9.6e-95

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

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

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. sqrt-pow1 N/A

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

      23. pow2 N/A

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

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

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

   4. Applied rewrites 94.0%

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

   if -9.6e-95 < re < 5.5999999999999999e-27

   1. Initial program 53.0%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 5.5999999999999999e-27 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 78.3

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

   5. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.3

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

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{im \cdot 1}{\sqrt{re}} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.0%

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

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.85 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -9.6 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.85e+68)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -9.6e-95)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 5.6e-27)
       (* 0.5 (sqrt (* 2.0 (- im re))))
       (* (/ im (sqrt re)) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.85e+68) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -9.6e-95) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 5.6e-27) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.85e+68)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -9.6e-95)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 5.6e-27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.85e+68], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -9.6e-95], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-27], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.84999999999999999e68

   1. Initial program 30.1%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -1.84999999999999999e68 < re < -9.6e-95

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

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

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. unpow1 N/A

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

      21. metadata-eval N/A

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

      22. sqrt-pow1 N/A

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

      23. pow2 N/A

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

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

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

   4. Applied rewrites 94.0%

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

   if -9.6e-95 < re < 5.5999999999999999e-27

   1. Initial program 53.0%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 5.5999999999999999e-27 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 78.3

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

   5. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.3

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

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{im \cdot 1}{\sqrt{re}} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.9%

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

Alternative 3: 75.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-9)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 5.6e-27)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-9) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 5.6e-27) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(re)) * 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 <= (-3.3d-9)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 5.6d-27) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im / sqrt(re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-9) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 5.6e-27) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e-9:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 5.6e-27:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e-9)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 5.6e-27)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e-9)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 5.6e-27)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-9], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e-27], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.30000000000000018e-9

   1. Initial program 45.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -3.30000000000000018e-9 < re < 5.5999999999999999e-27

   1. Initial program 57.7%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 5.5999999999999999e-27 < re

   1. Initial program 6.1%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 78.3

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

   5. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.3

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

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{im \cdot 1}{\sqrt{re}} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.9%

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

Alternative 4: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+180}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-9)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 8.5e+180) (* 0.5 (sqrt (* 2.0 (- im re)))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-9) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 8.5e+180) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.0;
	}
	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 <= (-3.3d-9)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 8.5d+180) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-9) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 8.5e+180) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e-9:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 8.5e+180:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e-9)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 8.5e+180)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e-9)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 8.5e+180)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-9], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+180], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -3.30000000000000018e-9

   1. Initial program 45.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -3.30000000000000018e-9 < re < 8.50000000000000077e180

   1. Initial program 48.3%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if 8.50000000000000077e180 < re

   1. Initial program 2.6%

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

   4. Applied rewrites 32.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e-9)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 8.5e+180) (* (sqrt (+ im im)) 0.5) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-9) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 8.5e+180) {
		tmp = sqrt((im + im)) * 0.5;
	} else {
		tmp = 0.0;
	}
	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 <= (-1.5d-9)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 8.5d+180) then
        tmp = sqrt((im + im)) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e-9) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 8.5e+180) {
		tmp = Math.sqrt((im + im)) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.5e-9:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 8.5e+180:
		tmp = math.sqrt((im + im)) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.5e-9)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 8.5e+180)
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e-9)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 8.5e+180)
		tmp = sqrt((im + im)) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.5e-9], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+180], N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -1.49999999999999999e-9

   1. Initial program 45.9%

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

   3. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in re around -inf

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

      1. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -1.49999999999999999e-9 < re < 8.50000000000000077e180

   1. Initial program 48.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.3

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

   7. Applied rewrites 72.9%

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

   9. Applied rewrites 72.9%

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

   if 8.50000000000000077e180 < re

   1. Initial program 2.6%

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

   4. Applied rewrites 32.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{-210}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im + im} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.9e-210) 0.0 (* (sqrt (+ im im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.9e-210) {
		tmp = 0.0;
	} 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) :: tmp
    if (im <= 1.9d-210) then
        tmp = 0.0d0
    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 (im <= 1.9e-210) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt((im + im)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.9e-210:
		tmp = 0.0
	else:
		tmp = math.sqrt((im + im)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.9e-210)
		tmp = 0.0;
	else
		tmp = Float64(sqrt(Float64(im + im)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.9e-210)
		tmp = 0.0;
	else
		tmp = sqrt((im + im)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.9e-210], 0.0, N[(N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.90000000000000002e-210

   1. Initial program 28.3%

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

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{0} \]

   if 1.90000000000000002e-210 < im

   1. Initial program 45.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.3

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

   7. Applied rewrites 59.7%

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

   9. Applied rewrites 59.7%

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

Alternative 7: 6.3% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 42.8%

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

4. Applied rewrites 6.4%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))