_divideComplex, real part

Percentage Accurate: 61.7% → 80.9%

Time: 6.1s

Alternatives: 10

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

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(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

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(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.6e-18)
   (/ (+ (* (/ x.im y.re) y.im) x.re) y.re)
   (if (<= y.re 1.35e-136)
     (/ (fma (/ x.re y.im) y.re x.im) y.im)
     (if (<= y.re 3.8e+124)
       (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
       (/ (fma (/ x.im y.re) y.im x.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.6e-18) {
		tmp = (((x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re;
	} else if (y_46_re <= 1.35e-136) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 3.8e+124) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -6.6e-18)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re);
	elseif (y_46_re <= 1.35e-136)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 3.8e+124)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.6e-18], N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.35e-136], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.8e+124], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.6000000000000003e-18

   1. Initial program 43.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 84.8

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.9%

      \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re} \]

   if -6.6000000000000003e-18 < y.re < 1.3499999999999999e-136

   1. Initial program 68.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im} + x.im}}{y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im} + x.im}{y.im} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}} + x.im}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re}{y.im} \cdot y.re} + x.im}{y.im} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}}{y.im} \]

      10. lower-/.f64 89.5

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.re}{y.im}}, y.re, x.im\right)}{y.im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}} \]

   if 1.3499999999999999e-136 < y.re < 3.7999999999999998e124

   1. Initial program 84.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.re \cdot y.re} + y.im \cdot y.im} \]

      3. lower-fma.f64 84.2

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   4. Applied rewrites 84.2%

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   if 3.7999999999999998e124 < y.re

   1. Initial program 35.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 85.9

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-252}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re x.re (* y.im x.im))))
   (if (<= y.re -3.1e-22)
     (/ x.re y.re)
     (if (<= y.re 3.9e-252)
       (/ x.im y.im)
       (if (<= y.re 1.1e-87)
         (/ t_0 (* y.im y.im))
         (if (<= y.re 1.3e+122) (/ t_0 (* y.re y.re)) (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, x_46_re, (y_46_im * x_46_im));
	double tmp;
	if (y_46_re <= -3.1e-22) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.9e-252) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1.1e-87) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_re <= 1.3e+122) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_re <= -3.1e-22)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.9e-252)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1.1e-87)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 1.3e+122)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e-22], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.9e-252], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.1e-87], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e+122], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.10000000000000013e-22 or 1.30000000000000004e122 < y.re

   1. Initial program 40.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 78.0

         \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   5. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.10000000000000013e-22 < y.re < 3.8999999999999999e-252

   1. Initial program 62.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 72.7

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if 3.8999999999999999e-252 < y.re < 1.09999999999999994e-87

   1. Initial program 88.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

      2. lower-*.f64 77.3

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

   5. Applied rewrites 77.3%

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{y.im \cdot y.im} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      4. lower-fma.f64 77.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}}{y.im \cdot y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{x.im \cdot y.im}\right)}{y.im \cdot y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

      7. lower-*.f64 77.3

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

   7. Applied rewrites 77.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}}{y.im \cdot y.im} \]

   if 1.09999999999999994e-87 < y.re < 1.30000000000000004e122

   1. Initial program 84.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

      2. lower-*.f64 29.4

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

   5. Applied rewrites 29.4%

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{y.im \cdot y.im} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      4. lower-fma.f64 29.4

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}}{y.im \cdot y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{x.im \cdot y.im}\right)}{y.im \cdot y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

      7. lower-*.f64 29.4

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

   7. Applied rewrites 29.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}}{y.im \cdot y.im} \]

   8. Taylor expanded in y.re around inf

      \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\color{blue}{{y.re}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\color{blue}{y.re \cdot y.re}} \]

      2. lower-*.f64 59.8

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\color{blue}{y.re \cdot y.re}} \]

   10. Applied rewrites 59.8%

       \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\color{blue}{y.re \cdot y.re}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-252}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+132}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.1e-22)
   (/ x.re y.re)
   (if (<= y.re 3.9e-252)
     (/ x.im y.im)
     (if (<= y.re 2.5e-53)
       (/ (fma y.re x.re (* y.im x.im)) (* y.im y.im))
       (if (<= y.re 1.35e+132)
         (* x.re (/ y.re (fma y.re y.re (* y.im y.im))))
         (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e-22) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.9e-252) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 2.5e-53) {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 1.35e+132) {
		tmp = x_46_re * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.1e-22)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.9e-252)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 2.5e-53)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 1.35e+132)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.1e-22], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.9e-252], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.5e-53], N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+132], N[(x$46$re * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.10000000000000013e-22 or 1.35e132 < y.re

   1. Initial program 40.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 78.7

         \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.10000000000000013e-22 < y.re < 3.8999999999999999e-252

   1. Initial program 62.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 72.7

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if 3.8999999999999999e-252 < y.re < 2.5e-53

   1. Initial program 87.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

      2. lower-*.f64 70.8

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{y.im \cdot y.im}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{y.im \cdot y.im} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re} + x.im \cdot y.im}{y.im \cdot y.im} \]

      4. lower-fma.f64 70.8

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}}{y.im \cdot y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{x.im \cdot y.im}\right)}{y.im \cdot y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

      7. lower-*.f64 70.8

         \[\leadsto \frac{\mathsf{fma}\left(y.re, x.re, \color{blue}{y.im \cdot x.im}\right)}{y.im \cdot y.im} \]

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}}{y.im \cdot y.im} \]

   if 2.5e-53 < y.re < 1.35e132

   1. Initial program 82.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2} + {y.re}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y.re \cdot \frac{x.re}{{y.im}^{2} + {y.re}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}} \cdot y.re} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}} \cdot y.re} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}}} \cdot y.re \]

      6. unpow2 N/A

         \[\leadsto \frac{x.re}{{y.im}^{2} + \color{blue}{y.re \cdot y.re}} \cdot y.re \]

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

         \[\leadsto \frac{x.re}{\color{blue}{{y.im}^{2} - \left(\mathsf{neg}\left(y.re\right)\right) \cdot y.re}} \cdot y.re \]

      8. mul-1-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \left(\color{blue}{1} \cdot y.re\right) \cdot y.re} \cdot y.re \]

      13. associate-*r* N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \color{blue}{1 \cdot \left(y.re \cdot y.re\right)}} \cdot y.re \]

      14. unpow2 N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + 1 \cdot \color{blue}{{y.re}^{2}}} \cdot y.re \]

      15. *-lft-identity N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \color{blue}{{y.re}^{2}}} \cdot y.re \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot y.re \]

      17. unpow2 N/A

          \[\leadsto \frac{x.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot y.re \]

      18. lower-*.f64 55.3

          \[\leadsto \frac{x.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot y.re \]

   5. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.4%

      \[\leadsto x.re \cdot \color{blue}{\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.6e-18)
   (/ (+ (* (/ x.im y.re) y.im) x.re) y.re)
   (if (<= y.re 1.35e-136)
     (/ (fma (/ x.re y.im) y.re x.im) y.im)
     (if (<= y.re 3.8e+124)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (/ (fma (/ x.im y.re) y.im x.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.6e-18) {
		tmp = (((x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re;
	} else if (y_46_re <= 1.35e-136) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 3.8e+124) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -6.6e-18)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re);
	elseif (y_46_re <= 1.35e-136)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 3.8e+124)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.6e-18], N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.35e-136], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.8e+124], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.6000000000000003e-18

   1. Initial program 43.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 84.8

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.9%

      \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re} \]

   if -6.6000000000000003e-18 < y.re < 1.3499999999999999e-136

   1. Initial program 68.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im} + x.im}}{y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im} + x.im}{y.im} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}} + x.im}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re}{y.im} \cdot y.re} + x.im}{y.im} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}}{y.im} \]

      10. lower-/.f64 89.5

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.re}{y.im}}, y.re, x.im\right)}{y.im} \]

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}} \]

   if 1.3499999999999999e-136 < y.re < 3.7999999999999998e124

   1. Initial program 84.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{x.re \cdot y.re + x.im \cdot y.im}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x.im \cdot y.im + x.re \cdot y.re}}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x.im \cdot y.im} + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.im \cdot x.im} + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \]

      5. lower-fma.f64 84.1

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, \color{blue}{x.re \cdot y.re}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, \color{blue}{y.re \cdot x.re}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

      8. lower-*.f64 84.1

         \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, \color{blue}{y.re \cdot x.re}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\color{blue}{y.im \cdot y.im} + y.re \cdot y.re} \]

      12. lower-fma.f64 84.1

          \[\leadsto \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\color{blue}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \]

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \]

   if 3.7999999999999998e124 < y.re

   1. Initial program 35.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 85.9

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+132}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.1e-22)
   (/ x.re y.re)
   (if (<= y.re 2.25e-128)
     (/ x.im y.im)
     (if (<= y.re 1.35e+132)
       (* x.re (/ y.re (fma y.re y.re (* y.im y.im))))
       (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e-22) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.25e-128) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1.35e+132) {
		tmp = x_46_re * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.1e-22)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.25e-128)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1.35e+132)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.1e-22], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.25e-128], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+132], N[(x$46$re * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.10000000000000013e-22 or 1.35e132 < y.re

   1. Initial program 40.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 78.7

         \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.10000000000000013e-22 < y.re < 2.25e-128

   1. Initial program 69.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.9

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if 2.25e-128 < y.re < 1.35e132

   1. Initial program 83.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2} + {y.re}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y.re \cdot \frac{x.re}{{y.im}^{2} + {y.re}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}} \cdot y.re} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}} \cdot y.re} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re}{{y.im}^{2} + {y.re}^{2}}} \cdot y.re \]

      6. unpow2 N/A

         \[\leadsto \frac{x.re}{{y.im}^{2} + \color{blue}{y.re \cdot y.re}} \cdot y.re \]

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

         \[\leadsto \frac{x.re}{\color{blue}{{y.im}^{2} - \left(\mathsf{neg}\left(y.re\right)\right) \cdot y.re}} \cdot y.re \]

      8. mul-1-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \left(\color{blue}{1} \cdot y.re\right) \cdot y.re} \cdot y.re \]

      13. associate-*r* N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \color{blue}{1 \cdot \left(y.re \cdot y.re\right)}} \cdot y.re \]

      14. unpow2 N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + 1 \cdot \color{blue}{{y.re}^{2}}} \cdot y.re \]

      15. *-lft-identity N/A

          \[\leadsto \frac{x.re}{y.im \cdot y.im + \color{blue}{{y.re}^{2}}} \cdot y.re \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot y.re \]

      17. unpow2 N/A

          \[\leadsto \frac{x.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot y.re \]

      18. lower-*.f64 51.1

          \[\leadsto \frac{x.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot y.re \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.5%

      \[\leadsto x.re \cdot \color{blue}{\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-18} \lor \neg \left(y.re \leq 0.205\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.6e-18) (not (<= y.re 0.205)))
   (/ (fma (/ x.im y.re) y.im x.re) y.re)
   (/ (fma (/ x.re y.im) y.re x.im) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.6e-18) || !(y_46_re <= 0.205)) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.6e-18) || !(y_46_re <= 0.205))
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6.6e-18], N[Not[LessEqual[y$46$re, 0.205]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.6000000000000003e-18 or 0.204999999999999988 < y.re

   1. Initial program 50.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 80.7

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]

   if -6.6000000000000003e-18 < y.re < 0.204999999999999988

   1. Initial program 72.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im} + x.im}}{y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im} + x.im}{y.im} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}} + x.im}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re}{y.im} \cdot y.re} + x.im}{y.im} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}}{y.im} \]

      10. lower-/.f64 83.7

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.re}{y.im}}, y.re, x.im\right)}{y.im} \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-18} \lor \neg \left(y.re \leq 0.205\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+135} \lor \neg \left(y.im \leq 2.65 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1e+135) (not (<= y.im 2.65e-6)))
   (/ x.im y.im)
   (/ (fma (/ x.im y.re) y.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1e+135) || !(y_46_im <= 2.65e-6)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1e+135) || !(y_46_im <= 2.65e-6))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1e+135], N[Not[LessEqual[y$46$im, 2.65e-6]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.99999999999999962e134 or 2.65e-6 < y.im

   1. Initial program 45.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.2

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -9.99999999999999962e134 < y.im < 2.65e-6

   1. Initial program 69.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 80.6

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+135} \lor \neg \left(y.im \leq 2.65 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.6e-18)
   (/ (+ (* (/ x.im y.re) y.im) x.re) y.re)
   (if (<= y.re 0.205)
     (/ (fma (/ x.re y.im) y.re x.im) y.im)
     (/ (fma (/ x.im y.re) y.im x.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.6e-18) {
		tmp = (((x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re;
	} else if (y_46_re <= 0.205) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -6.6e-18)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) + x_46_re) / y_46_re);
	elseif (y_46_re <= 0.205)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.6e-18], N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 0.205], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.6000000000000003e-18

   1. Initial program 43.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 84.8

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.9%

      \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + x.re}{y.re} \]

   if -6.6000000000000003e-18 < y.re < 0.204999999999999988

   1. Initial program 72.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re}{y.im} + x.im}}{y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.re}}{y.im} + x.im}{y.im} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}} + x.im}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x.re}{y.im} \cdot y.re} + x.im}{y.im} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}}{y.im} \]

      10. lower-/.f64 83.7

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.re}{y.im}}, y.re, x.im\right)}{y.im} \]

   5. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}} \]

   if 0.204999999999999988 < y.re

   1. Initial program 55.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}{y.re} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)}}{y.re} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)}}{y.re} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re} + -1 \cdot x.re\right)}}{y.re} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right) \cdot -1 + \left(-1 \cdot x.re\right) \cdot -1}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y.im \cdot x.im}}{y.re} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      13. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      14. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{x.im}{y.re} \cdot y.im} + \left(-1 \cdot x.re\right) \cdot -1}{y.re} \]

      15. *-commutative N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{-1 \cdot \left(-1 \cdot x.re\right)}}{y.re} \]

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}\right)\right)}{y.re} \]

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{x.im}{y.re} \cdot y.im + \color{blue}{x.re}}{y.re} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}{y.re} \]

      20. lower-/.f64 77.4

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x.im}{y.re}}, y.im, x.re\right)}{y.re} \]

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+131} \lor \neg \left(y.im \leq 1.25 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.6e+131) (not (<= y.im 1.25e+36)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.6e+131) || !(y_46_im <= 1.25e+36)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_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(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.6d+131)) .or. (.not. (y_46im <= 1.25d+36))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.6e+131) || !(y_46_im <= 1.25e+36)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.6e+131) or not (y_46_im <= 1.25e+36):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.6e+131) || !(y_46_im <= 1.25e+36))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.6e+131) || ~((y_46_im <= 1.25e+36)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.6e+131], N[Not[LessEqual[y$46$im, 1.25e+36]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.60000000000000031e131 or 1.24999999999999994e36 < y.im

   1. Initial program 42.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
   4. Step-by-step derivation

      1. lower-/.f64 71.7

         \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -3.60000000000000031e131 < y.im < 1.24999999999999994e36

   1. Initial program 70.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.9

         \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+131} \lor \neg \left(y.im \leq 1.25 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

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(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46im
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$im), $MachinePrecision]

Derivation

1. Initial program 60.1%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
2. Add Preprocessing

3. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
4. Step-by-step derivation

   1. lower-/.f64 38.8

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

5. Applied rewrites 38.8%

   \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))