_divideComplex, real part

Percentage Accurate: 62.0% → 84.5%

Time: 7.9s

Alternatives: 11

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: 62.0% 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: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1.12 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(x.re, \frac{y.re}{t\_0}, y.im \cdot \frac{x.im}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -1.12e+163)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (if (<= y.re -5.2e-104)
       (fma x.re (/ y.re t_0) (* y.im (/ x.im t_0)))
       (if (<= y.re 2.35e-104)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.5e+119)
           (fma x.im (/ y.im t_0) (* y.re (/ x.re t_0)))
           (fma (/ y.im y.re) (/ x.im 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_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -1.12e+163) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -5.2e-104) {
		tmp = fma(x_46_re, (y_46_re / t_0), (y_46_im * (x_46_im / t_0)));
	} else if (y_46_re <= 2.35e-104) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.5e+119) {
		tmp = fma(x_46_im, (y_46_im / t_0), (y_46_re * (x_46_re / t_0)));
	} else {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (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_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.12e+163)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -5.2e-104)
		tmp = fma(x_46_re, Float64(y_46_re / t_0), Float64(y_46_im * Float64(x_46_im / t_0)));
	elseif (y_46_re <= 2.35e-104)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.5e+119)
		tmp = fma(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / t_0)));
	else
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), 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$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.12e+163], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-104], N[(x$46$re * N[(y$46$re / t$95$0), $MachinePrecision] + N[(y$46$im * N[(x$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.35e-104], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+119], N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.11999999999999996e163

   1. Initial program 29.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if -1.11999999999999996e163 < y.re < -5.20000000000000005e-104

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 83.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 83.6

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

   4. Applied rewrites 83.6%

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

   if -5.20000000000000005e-104 < y.re < 2.35e-104

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

   if 2.35e-104 < y.re < 1.50000000000000001e119

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 85.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 85.6

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

   4. Applied rewrites 85.6%

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

   if 1.50000000000000001e119 < y.re

   1. Initial program 31.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{y.im}{y.re}, \color{blue}{\frac{x.im}{y.re}}, \frac{x.re}{y.re}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.12 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(x.re, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.im \cdot \frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.re \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ t_2 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.72 \cdot 10^{+118}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma x.im (/ y.im t_0) (* y.re (/ x.re t_0))))
        (t_2 (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))))
   (if (<= y.re -1.72e+118)
     t_2
     (if (<= y.re -5.2e-104)
       t_1
       (if (<= y.re 2.35e-104)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.5e+119) t_1 t_2))))))

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_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(x_46_im, (y_46_im / t_0), (y_46_re * (x_46_re / t_0)));
	double t_2 = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.72e+118) {
		tmp = t_2;
	} else if (y_46_re <= -5.2e-104) {
		tmp = t_1;
	} else if (y_46_re <= 2.35e-104) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.5e+119) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / t_0)))
	t_2 = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.72e+118)
		tmp = t_2;
	elseif (y_46_re <= -5.2e-104)
		tmp = t_1;
	elseif (y_46_re <= 2.35e-104)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.5e+119)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.72e+118], t$95$2, If[LessEqual[y$46$re, -5.2e-104], t$95$1, If[LessEqual[y$46$re, 2.35e-104], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+119], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.71999999999999999e118 or 1.50000000000000001e119 < y.re

   1. Initial program 33.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 92.8%

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

   if -1.71999999999999999e118 < y.re < -5.20000000000000005e-104 or 2.35e-104 < y.re < 1.50000000000000001e119

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 86.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if -5.20000000000000005e-104 < y.re < 2.35e-104

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.72 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.re \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.re \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \frac{\mathsf{fma}\left(\frac{y.im}{x.re}, x.im, y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))))
   (if (<= y.re -1.65e+118)
     t_0
     (if (<= y.re -1.75e-100)
       (* x.re (/ (fma (/ y.im x.re) x.im y.re) (fma y.im y.im (* y.re y.re))))
       (if (<= y.re 2.35e-104)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.7e+96)
           (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
           t_0))))))

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_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.65e+118) {
		tmp = t_0;
	} else if (y_46_re <= -1.75e-100) {
		tmp = x_46_re * (fma((y_46_im / x_46_re), x_46_im, y_46_re) / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_re <= 2.35e-104) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.7e+96) {
		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));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.65e+118)
		tmp = t_0;
	elseif (y_46_re <= -1.75e-100)
		tmp = Float64(x_46_re * Float64(fma(Float64(y_46_im / x_46_re), x_46_im, y_46_re) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 2.35e-104)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.7e+96)
		tmp = 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)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.65e+118], t$95$0, If[LessEqual[y$46$re, -1.75e-100], N[(x$46$re * N[(N[(N[(y$46$im / x$46$re), $MachinePrecision] * x$46$im + y$46$re), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.35e-104], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+96], 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], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.65e118 or 1.7e96 < y.re

   1. Initial program 33.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.1%

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

   if -1.65e118 < y.re < -1.75e-100

   1. Initial program 72.9%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/r* N/A

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

      2. div-add-rev N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 72.9%

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

   7. Applied rewrites 82.3%

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

   if -1.75e-100 < y.re < 2.35e-104

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

   if 2.35e-104 < y.re < 1.7e96

   1. Initial program 77.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \frac{\mathsf{fma}\left(\frac{y.im}{x.re}, x.im, y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -2.08 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))))
   (if (<= y.re -2.08e+65)
     t_1
     (if (<= y.re -1.6e-100)
       t_0
       (if (<= y.re 2.35e-104)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.7e+96) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -2.08e+65) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-100) {
		tmp = t_0;
	} else if (y_46_re <= 2.35e-104) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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)))
	t_1 = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.08e+65)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-100)
		tmp = t_0;
	elseif (y_46_re <= 2.35e-104)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.7e+96)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.08e+65], t$95$1, If[LessEqual[y$46$re, -1.6e-100], t$95$0, If[LessEqual[y$46$re, 2.35e-104], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+96], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.07999999999999997e65 or 1.7e96 < y.re

   1. Initial program 36.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 91.0%

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

   if -2.07999999999999997e65 < y.re < -1.60000000000000008e-100 or 2.35e-104 < y.re < 1.7e96

   1. Initial program 75.5%

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

   if -1.60000000000000008e-100 < y.re < 2.35e-104

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.08 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.08 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -2.08e+65)
     t_1
     (if (<= y.re -1.6e-100)
       t_0
       (if (<= y.re 2.35e-104)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.7e+96) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2.08e+65) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-100) {
		tmp = t_0;
	} else if (y_46_re <= 2.35e-104) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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)))
	t_1 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.08e+65)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-100)
		tmp = t_0;
	elseif (y_46_re <= 2.35e-104)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.7e+96)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.08e+65], t$95$1, If[LessEqual[y$46$re, -1.6e-100], t$95$0, If[LessEqual[y$46$re, 2.35e-104], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+96], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.07999999999999997e65 or 1.7e96 < y.re

   1. Initial program 36.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 91.0%

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

   if -2.07999999999999997e65 < y.re < -1.60000000000000008e-100 or 2.35e-104 < y.re < 1.7e96

   1. Initial program 75.5%

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

   if -1.60000000000000008e-100 < y.re < 2.35e-104

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.08 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+50} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.4e+50) (not (<= y.re 6.2e-26)))
   (/ (fma x.im (/ y.im y.re) x.re) y.re)
   (/ (fma (/ y.re y.im) x.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 <= -2.4e+50) || !(y_46_re <= 6.2e-26)) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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 <= -2.4e+50) || !(y_46_re <= 6.2e-26))
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_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, -2.4e+50], N[Not[LessEqual[y$46$re, 6.2e-26]], $MachinePrecision]], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.4000000000000002e50 or 6.19999999999999966e-26 < y.re

   1. Initial program 45.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   if -2.4000000000000002e50 < y.re < 6.19999999999999966e-26

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 62.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 80.8

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

   7. Applied rewrites 80.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+50} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+50} \lor \neg \left(y.re \leq 9.1 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.2e+50) (not (<= y.re 9.1e-26)))
   (/ x.re y.re)
   (/ (fma (/ y.re y.im) x.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 <= -2.2e+50) || !(y_46_re <= 9.1e-26)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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 <= -2.2e+50) || !(y_46_re <= 9.1e-26))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_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, -2.2e+50], N[Not[LessEqual[y$46$re, 9.1e-26]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.20000000000000017e50 or 9.0999999999999995e-26 < y.re

   1. Initial program 45.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   if -2.20000000000000017e50 < y.re < 9.0999999999999995e-26

   1. Initial program 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 62.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. 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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 80.8

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

   7. Applied rewrites 80.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+50} \lor \neg \left(y.re \leq 9.1 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.4e+154)
   (/ x.re y.re)
   (if (<= y.re -7.2e-69)
     (* (/ y.re (fma y.im y.im (* y.re y.re))) x.re)
     (if (<= y.re 6.2e-26) (/ 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_re <= -1.4e+154) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -7.2e-69) {
		tmp = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_re;
	} else if (y_46_re <= 6.2e-26) {
		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_re <= -1.4e+154)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -7.2e-69)
		tmp = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_re);
	elseif (y_46_re <= 6.2e-26)
		tmp = Float64(x_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, -1.4e+154], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -7.2e-69], N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-26], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.4e154 or 6.19999999999999966e-26 < y.re

   1. Initial program 41.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

   if -1.4e154 < y.re < -7.20000000000000035e-69

   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

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2} + {y.re}^{2}}} \]
   9. 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 63.6

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

   10. Applied rewrites 63.6%

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

   if -7.20000000000000035e-69 < y.re < 6.19999999999999966e-26

   1. Initial program 62.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.8e+153)
   (/ x.re y.re)
   (if (<= y.re -7.2e-69)
     (* (/ x.re (fma y.im y.im (* y.re y.re))) y.re)
     (if (<= y.re 6.2e-26) (/ 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_re <= -3.8e+153) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -7.2e-69) {
		tmp = (x_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_re;
	} else if (y_46_re <= 6.2e-26) {
		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_re <= -3.8e+153)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -7.2e-69)
		tmp = Float64(Float64(x_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_re);
	elseif (y_46_re <= 6.2e-26)
		tmp = Float64(x_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, -3.8e+153], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -7.2e-69], N[(N[(x$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-26], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.79999999999999966e153 or 6.19999999999999966e-26 < y.re

   1. Initial program 41.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.79999999999999966e153 < y.re < -7.20000000000000035e-69

   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 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 63.5

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

   5. Applied rewrites 63.5%

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

   if -7.20000000000000035e-69 < y.re < 6.19999999999999966e-26

   1. Initial program 62.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-64} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.7e-64) (not (<= y.re 6.2e-26)))
   (/ x.re 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 <= -2.7e-64) || !(y_46_re <= 6.2e-26)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_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(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_46re <= (-2.7d-64)) .or. (.not. (y_46re <= 6.2d-26))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    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_re <= -2.7e-64) || !(y_46_re <= 6.2e-26)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.7e-64) or not (y_46_re <= 6.2e-26):
		tmp = x_46_re / y_46_re
	else:
		tmp = 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 <= -2.7e-64) || !(y_46_re <= 6.2e-26))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	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_re <= -2.7e-64) || ~((y_46_re <= 6.2e-26)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.7e-64], N[Not[LessEqual[y$46$re, 6.2e-26]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.69999999999999986e-64 or 6.19999999999999966e-26 < y.re

   1. Initial program 49.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   if -2.69999999999999986e-64 < y.re < 6.19999999999999966e-26

   1. Initial program 62.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-64} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.9% 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 54.9%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 41.6

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

5. Applied rewrites 41.6%

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

Reproduce

herbie shell --seed 2024357 
(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))))