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

Percentage Accurate: 41.1% → 88.2%

Time: 7.3s

Alternatives: 8

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: 41.1% 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: 88.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.6e+28)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.6e+28) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7.6e+28) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7.6e+28:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7.6e+28)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7.6e+28)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7.6e+28], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.5999999999999998e28

   1. Initial program 56.4%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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%

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

   if 7.5999999999999998e28 < re

   1. Initial program 7.0%

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

   3. Taylor expanded in re around 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 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 87.4%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{{re}^{-1}} \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 0.000155)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* (sqrt (pow re -1.0)) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 0.000155) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (sqrt(pow(re, -1.0)) * im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d-98)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 0.000155d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (sqrt((re ** (-1.0d0))) * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 0.000155) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (Math.sqrt(Math.pow(re, -1.0)) * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-98:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 0.000155:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (math.sqrt(math.pow(re, -1.0)) * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 0.000155)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(sqrt((re ^ -1.0)) * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e-98)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 0.000155)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (sqrt((re ^ -1.0)) * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000155], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e-98

   1. Initial program 49.3%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -4.8000000000000001e-98 < re < 1.55e-4

   1. Initial program 61.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

   if 1.55e-4 < re

   1. Initial program 10.5%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 33.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   6. Applied rewrites 33.7%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(\mathsf{hypot}\left(im, re\right) - re\right) \cdot 2\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 85.3

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

   9. Applied rewrites 85.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{{re}^{-1}} \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -2.95 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 28:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.32e+122)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re -2.95e-105)
     (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
     (if (<= re 28.0)
       (* 0.5 (sqrt (+ (fma (- (/ re im) 2.0) re im) im)))
       (* 0.5 (/ im (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.32e+122) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= -2.95e-105) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else if (re <= 28.0) {
		tmp = 0.5 * sqrt((fma(((re / im) - 2.0), re, im) + im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.32e+122)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= -2.95e-105)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	elseif (re <= 28.0)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.32e+122], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -2.95e-105], 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, 28.0], N[(0.5 * N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.31999999999999992e122

   1. Initial program 14.0%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   if -1.31999999999999992e122 < re < -2.9499999999999999e-105

   1. Initial program 80.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 80.1%

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

   if -2.9499999999999999e-105 < re < 28

   1. Initial program 61.0%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 89.6%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   9. Applied rewrites 81.3%

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

   if 28 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 86.9%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq -2.95 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{elif}\;re \leq 28:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 28:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 28.0)
     (* 0.5 (sqrt (+ (fma (- (/ re im) 2.0) re im) im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 28.0) {
		tmp = 0.5 * sqrt((fma(((re / im) - 2.0), re, im) + im));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 28.0)
		tmp = Float64(0.5 * sqrt(Float64(fma(Float64(Float64(re / im) - 2.0), re, im) + im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 28.0], N[(0.5 * N[Sqrt[N[(N[(N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision] * re + im), $MachinePrecision] + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e-98

   1. Initial program 49.3%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -4.8000000000000001e-98 < re < 28

   1. Initial program 61.7%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 89.8%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.8

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.8%

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

   if 28 < re

   1. Initial program 8.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 86.9%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 28:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} - 2, re, im\right) + im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 0.000155)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 0.000155) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d-98)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 0.000155d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 0.000155) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-98:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 0.000155:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 0.000155)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e-98)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 0.000155)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.000155], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e-98

   1. Initial program 49.3%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -4.8000000000000001e-98 < re < 1.55e-4

   1. Initial program 61.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

   if 1.55e-4 < re

   1. Initial program 10.5%

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

   3. Taylor expanded in re around 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 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 85.4%

      \[\leadsto 0.5 \cdot \frac{1 \cdot \left(im \cdot 1\right)}{\color{blue}{\sqrt{re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-98)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 1.65e+171) (* 0.5 (sqrt (+ im im))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 1.65e+171) {
		tmp = 0.5 * sqrt((im + im));
	} 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 <= (-4.8d-98)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 1.65d+171) then
        tmp = 0.5d0 * sqrt((im + im))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-98) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 1.65e+171) {
		tmp = 0.5 * Math.sqrt((im + im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-98:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 1.65e+171:
		tmp = 0.5 * math.sqrt((im + im))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-98)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 1.65e+171)
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e-98)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 1.65e+171)
		tmp = 0.5 * sqrt((im + im));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-98], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.65e+171], N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e-98

   1. Initial program 49.3%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 100.0%

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

   5. Taylor expanded in re around -inf

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

      1. lower-*.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -4.8000000000000001e-98 < re < 1.64999999999999996e171

   1. Initial program 53.3%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 79.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 70.1

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 70.1%

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

   if 1.64999999999999996e171 < re

   1. Initial program 2.5%

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

   4. Applied rewrites 22.5%

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

Alternative 7: 52.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{-216}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im + im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2e-216) 0.0 (* 0.5 (sqrt (+ im im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2e-216) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * sqrt((im + im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2d-216) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * sqrt((im + im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2e-216) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * Math.sqrt((im + im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2e-216:
		tmp = 0.0
	else:
		tmp = 0.5 * math.sqrt((im + im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2e-216)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * sqrt(Float64(im + im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2e-216)
		tmp = 0.0;
	else
		tmp = 0.5 * sqrt((im + im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2e-216], 0.0, N[(0.5 * N[Sqrt[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.0000000000000001e-216

   1. Initial program 35.2%

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

   4. Applied rewrites 21.8%

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

   if 2.0000000000000001e-216 < im

   1. Initial program 47.8%

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

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

      4. sqr-abs-rev N/A

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

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

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

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

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

      12. lift-*.f64 N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. remove-double-neg N/A

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

      19. remove-double-neg N/A

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

      20. remove-double-neg N/A

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

      21. unpow1 N/A

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

      22. metadata-eval N/A

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

      23. sqrt-pow1 N/A

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

      24. pow2 N/A

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

      25. 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 82.9%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 57.4%

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

Alternative 8: 6.4% 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 45.6%

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

4. Applied rewrites 6.3%

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

Reproduce

herbie shell --seed 2024347 
(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)))))