_divideComplex, real part

Percentage Accurate: 60.0% → 82.7%

Time: 6.4s

Alternatives: 10

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.0% 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: 82.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(x.re, \frac{y.re}{t\_0}, y.im \cdot \frac{x.im}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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.re y.im) y.re x.im) y.im)))
   (if (<= y.im -1.75e+27)
     t_1
     (if (<= y.im -2.8e-178)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 2.45e-136)
         (/ (fma (/ x.im y.re) y.im x.re) y.re)
         (if (<= y.im 7.2e+96)
           (fma x.re (/ y.re t_0) (* y.im (/ x.im 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 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.75e+27) {
		tmp = t_1;
	} else if (y_46_im <= -2.8e-178) {
		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 if (y_46_im <= 2.45e-136) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 7.2e+96) {
		tmp = fma(x_46_re, (y_46_re / t_0), (y_46_im * (x_46_im / 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 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.75e+27)
		tmp = t_1;
	elseif (y_46_im <= -2.8e-178)
		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)));
	elseif (y_46_im <= 2.45e-136)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 7.2e+96)
		tmp = fma(x_46_re, Float64(y_46_re / t_0), Float64(y_46_im * Float64(x_46_im / 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[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.75e+27], t$95$1, If[LessEqual[y$46$im, -2.8e-178], 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], If[LessEqual[y$46$im, 2.45e-136], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+96], 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], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7500000000000001e27 or 7.20000000000000026e96 < y.im

   1. Initial program 41.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -1.7500000000000001e27 < y.im < -2.80000000000000019e-178

   1. Initial program 87.6%

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

   if -2.80000000000000019e-178 < y.im < 2.45e-136

   1. Initial program 68.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if 2.45e-136 < y.im < 7.20000000000000026e96

   1. Initial program 80.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. 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 89.5

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

      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 89.5

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

   4. Applied rewrites 89.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(x.re, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.im \cdot \frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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.re y.im) y.re x.im) y.im)))
   (if (<= y.im -1.75e+27)
     t_1
     (if (<= y.im -2.8e-178)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 4.6e-126)
         (/ (fma (/ x.im y.re) y.im x.re) y.re)
         (if (<= y.im 4.2e+98)
           (fma x.im (/ y.im t_0) (* y.re (/ x.re 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 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.75e+27) {
		tmp = t_1;
	} else if (y_46_im <= -2.8e-178) {
		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 if (y_46_im <= 4.6e-126) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 4.2e+98) {
		tmp = fma(x_46_im, (y_46_im / t_0), (y_46_re * (x_46_re / 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 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.75e+27)
		tmp = t_1;
	elseif (y_46_im <= -2.8e-178)
		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)));
	elseif (y_46_im <= 4.6e-126)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 4.2e+98)
		tmp = fma(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / 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[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.75e+27], t$95$1, If[LessEqual[y$46$im, -2.8e-178], 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], If[LessEqual[y$46$im, 4.6e-126], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+98], 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], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7500000000000001e27 or 4.20000000000000008e98 < y.im

   1. Initial program 41.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -1.7500000000000001e27 < y.im < -2.80000000000000019e-178

   1. Initial program 87.6%

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

   if -2.80000000000000019e-178 < y.im < 4.60000000000000021e-126

   1. Initial program 69.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 95.2

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

   5. Applied rewrites 95.2%

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

   if 4.60000000000000021e-126 < y.im < 4.20000000000000008e98

   1. Initial program 80.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.7

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

      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 87.7

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

   4. Applied rewrites 87.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.re \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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.re y.im) y.re x.im) y.im)))
   (if (<= y.im -1.75e+27)
     t_1
     (if (<= y.im -2.8e-178)
       t_0
       (if (<= y.im 5.2e-137)
         (/ (fma (/ x.im y.re) y.im x.re) y.re)
         (if (<= y.im 6e+54) 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_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.75e+27) {
		tmp = t_1;
	} else if (y_46_im <= -2.8e-178) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-137) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 6e+54) {
		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(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.75e+27)
		tmp = t_1;
	elseif (y_46_im <= -2.8e-178)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-137)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 6e+54)
		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[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.75e+27], t$95$1, If[LessEqual[y$46$im, -2.8e-178], t$95$0, If[LessEqual[y$46$im, 5.2e-137], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6e+54], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.7500000000000001e27 or 5.9999999999999998e54 < y.im

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -1.7500000000000001e27 < y.im < -2.80000000000000019e-178 or 5.1999999999999999e-137 < y.im < 5.9999999999999998e54

   1. Initial program 88.1%

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

   if -2.80000000000000019e-178 < y.im < 5.1999999999999999e-137

   1. Initial program 68.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -7.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7.7 \cdot 10^{-208}:\\ \;\;\;\;x.im \cdot \frac{y.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{t\_0} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

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.im -7.9e+93)
     (/ x.im y.im)
     (if (<= y.im -7.7e-208)
       (* x.im (/ y.im t_0))
       (if (<= y.im 1.75e-82)
         (/ x.re y.re)
         (if (<= y.im 9e+83) (* (/ x.im t_0) y.im) (/ x.im y.im)))))))

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_im <= -7.9e+93) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -7.7e-208) {
		tmp = x_46_im * (y_46_im / t_0);
	} else if (y_46_im <= 1.75e-82) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 9e+83) {
		tmp = (x_46_im / t_0) * y_46_im;
	} 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)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -7.9e+93)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -7.7e-208)
		tmp = Float64(x_46_im * Float64(y_46_im / t_0));
	elseif (y_46_im <= 1.75e-82)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 9e+83)
		tmp = Float64(Float64(x_46_im / t_0) * y_46_im);
	else
		tmp = Float64(x_46_im / y_46_im);
	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$46$im, -7.9e+93], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.7e-208], N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.75e-82], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 9e+83], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.8999999999999999e93 or 8.9999999999999999e83 < y.im

   1. Initial program 39.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -7.8999999999999999e93 < y.im < -7.69999999999999972e-208

   1. Initial program 83.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 58.7%

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

   if -7.69999999999999972e-208 < y.im < 1.7499999999999999e-82

   1. Initial program 71.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

   if 1.7499999999999999e-82 < y.im < 8.9999999999999999e83

   1. Initial program 80.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7.7 \cdot 10^{-208}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.im \leq -7.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7.7 \cdot 10^{-208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))))
   (if (<= y.im -7.9e+93)
     (/ x.im y.im)
     (if (<= y.im -7.7e-208)
       t_0
       (if (<= y.im 1.75e-82)
         (/ x.re y.re)
         (if (<= y.im 2.7e+84) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_im <= -7.9e+93) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -7.7e-208) {
		tmp = t_0;
	} else if (y_46_im <= 1.75e-82) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.7e+84) {
		tmp = t_0;
	} 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)
	t_0 = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_im <= -7.9e+93)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -7.7e-208)
		tmp = t_0;
	elseif (y_46_im <= 1.75e-82)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 2.7e+84)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.9e+93], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.7e-208], t$95$0, If[LessEqual[y$46$im, 1.75e-82], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.7e+84], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.8999999999999999e93 or 2.7e84 < y.im

   1. Initial program 39.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -7.8999999999999999e93 < y.im < -7.69999999999999972e-208 or 1.7499999999999999e-82 < y.im < 2.7e84

   1. Initial program 82.5%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 63.0%

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

   if -7.69999999999999972e-208 < y.im < 1.7499999999999999e-82

   1. Initial program 71.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7.7 \cdot 10^{-208}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 920000000000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.2e+91) (not (<= y.re 920000000000.0)))
   (/ (fma (/ x.im y.re) y.im x.re) y.re)
   (/ (+ (* (/ 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+91) || !(y_46_re <= 920000000000.0)) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else {
		tmp = (((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+91) || !(y_46_re <= 920000000000.0))
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(Float64(Float64(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+91], N[Not[LessEqual[y$46$re, 920000000000.0]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.19999999999999999e91 or 9.2e11 < y.re

   1. Initial program 52.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -2.19999999999999999e91 < y.re < 9.2e11

   1. Initial program 71.7%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. mul-1-neg N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   7. 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 79.3

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

   8. Applied rewrites 79.3%

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

   10. Applied rewrites 79.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 920000000000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.re + x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 920000000000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.2e+91) (not (<= y.re 920000000000.0)))
   (/ (fma (/ x.im y.re) y.im 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+91) || !(y_46_re <= 920000000000.0)) {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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+91) || !(y_46_re <= 920000000000.0))
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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+91], N[Not[LessEqual[y$46$re, 920000000000.0]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(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 < -2.19999999999999999e91 or 9.2e11 < y.re

   1. Initial program 52.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -2.19999999999999999e91 < y.re < 9.2e11

   1. Initial program 71.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 71.3

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

      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 71.3

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

   4. Applied rewrites 71.3%

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

   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 79.3

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

   7. Applied rewrites 79.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 920000000000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+94} \lor \neg \left(y.re \leq 7.8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.2e+94) (not (<= y.re 7.8e+22)))
   (/ 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 <= -6.2e+94) || !(y_46_re <= 7.8e+22)) {
		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 <= -6.2e+94) || !(y_46_re <= 7.8e+22))
		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, -6.2e+94], N[Not[LessEqual[y$46$re, 7.8e+22]], $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 < -6.19999999999999983e94 or 7.80000000000000042e22 < y.re

   1. Initial program 52.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   if -6.19999999999999983e94 < y.re < 7.80000000000000042e22

   1. Initial program 71.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 71.5

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

      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 71.5

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

   4. Applied rewrites 71.5%

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

   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 78.9

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

   7. Applied rewrites 78.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+94} \lor \neg \left(y.re \leq 7.8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 7.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.2e+91) (not (<= y.re 7.5e+20)))
   (/ 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 <= -2.2e+91) || !(y_46_re <= 7.5e+20)) {
		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 <= (-2.2d+91)) .or. (.not. (y_46re <= 7.5d+20))) 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 <= -2.2e+91) || !(y_46_re <= 7.5e+20)) {
		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 <= -2.2e+91) or not (y_46_re <= 7.5e+20):
		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 <= -2.2e+91) || !(y_46_re <= 7.5e+20))
		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 <= -2.2e+91) || ~((y_46_re <= 7.5e+20)))
		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, -2.2e+91], N[Not[LessEqual[y$46$re, 7.5e+20]], $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 < -2.19999999999999999e91 or 7.5e20 < y.re

   1. Initial program 52.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   if -2.19999999999999999e91 < y.re < 7.5e20

   1. Initial program 71.8%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 60.5

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

   5. Applied rewrites 60.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+91} \lor \neg \left(y.re \leq 7.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.3% 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 64.9%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 45.3

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

5. Applied rewrites 45.3%

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

6. Final simplification 45.3%

   \[\leadsto \frac{x.im}{y.im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024362 
(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))))