_divideComplex, imaginary part

Percentage Accurate: 62.0% → 84.0%

Time: 3.6s

Alternatives: 9

Speedup: 1.7×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{x.im + y.im \cdot \mathsf{fma}\left(-1, \frac{x.re}{y.re}, -1 \cdot \frac{x.im \cdot y.im}{{y.re}^{2}}\right)}{y.re}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ t_2 := \frac{1}{\mathsf{fma}\left(y.im, \frac{y.im}{t\_1}, \frac{y.re \cdot y.re}{t\_1}\right)}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (+
           x.im
           (*
            y.im
            (fma
             -1.0
             (/ x.re y.re)
             (* -1.0 (/ (* x.im y.im) (pow y.re 2.0))))))
          y.re))
        (t_1 (- (* x.im y.re) (* x.re y.im)))
        (t_2 (/ 1.0 (fma y.im (/ y.im t_1) (/ (* y.re y.re) t_1)))))
   (if (<= y.re -5e+146)
     t_0
     (if (<= y.re -1e-107)
       t_2
       (if (<= y.re 2.2e-98)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 2.25e+112) t_2 t_0))))))

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(-1.0, (x_46_re / y_46_re), (-1.0 * ((x_46_im * y_46_im) / pow(y_46_re, 2.0)))))) / y_46_re;
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_2 = 1.0 / fma(y_46_im, (y_46_im / t_1), ((y_46_re * y_46_re) / t_1));
	double tmp;
	if (y_46_re <= -5e+146) {
		tmp = t_0;
	} else if (y_46_re <= -1e-107) {
		tmp = t_2;
	} else if (y_46_re <= 2.2e-98) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.25e+112) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(y_46_im * fma(-1.0, Float64(x_46_re / y_46_re), Float64(-1.0 * Float64(Float64(x_46_im * y_46_im) / (y_46_re ^ 2.0)))))) / y_46_re)
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_2 = Float64(1.0 / fma(y_46_im, Float64(y_46_im / t_1), Float64(Float64(y_46_re * y_46_re) / t_1)))
	tmp = 0.0
	if (y_46_re <= -5e+146)
		tmp = t_0;
	elseif (y_46_re <= -1e-107)
		tmp = t_2;
	elseif (y_46_re <= 2.2e-98)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.25e+112)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(y$46$im * N[(-1.0 * N[(x$46$re / y$46$re), $MachinePrecision] + N[(-1.0 * N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(y$46$im * N[(y$46$im / t$95$1), $MachinePrecision] + N[(N[(y$46$re * y$46$re), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5e+146], t$95$0, If[LessEqual[y$46$re, -1e-107], t$95$2, If[LessEqual[y$46$re, 2.2e-98], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.25e+112], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.9999999999999999e146 or 2.24999999999999995e112 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 39.9%

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

   4. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 39.9%

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{y.re \cdot \color{blue}{x.im}}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y.re \cdot y.re + y.im \cdot y.im\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y.re \cdot \color{blue}{x.im}}{\mathsf{Rewrite=>}\left(+-commutative, \left(y.im \cdot y.im + y.re \cdot y.re\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y.re \cdot \color{blue}{x.im}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(y.im \cdot y.im\right)\right) + y.re \cdot y.re} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{y.re \cdot \color{blue}{x.im}}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\right)\right)} \]

   6. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. /-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. add-to-fraction N/A

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

      10. mult-flip N/A

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

      11. lift-/.f64 N/A

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

      12. remove-double-div N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in y.im around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-pow.f64 46.1%

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

   11. Applied rewrites 46.1%

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

   if -4.9999999999999999e146 < y.re < -1e-107 or 2.19999999999999996e-98 < y.re < 2.24999999999999995e112

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.8%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. distribute-lft-neg-in N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.8%

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

   3. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re \cdot x.im - y.im \cdot x.re}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. remove-double-neg N/A

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

      11. div-add N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sqr-neg-rev N/A

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

      17. lift-*.f64 N/A

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

      18. /-rgt-identity N/A

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

      19. associate-/l/ N/A

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

      20. *-lft-identity N/A

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

   5. Applied rewrites 67.9%

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

   if -1e-107 < y.re < 2.19999999999999996e-98

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

Alternative 2: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ t_2 := \frac{1}{\mathsf{fma}\left(y.im, \frac{y.im}{t\_1}, \frac{y.re \cdot y.re}{t\_1}\right)}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+109}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* -1.0 (/ (* x.re y.im) y.re))) y.re))
        (t_1 (- (* x.im y.re) (* x.re y.im)))
        (t_2 (/ 1.0 (fma y.im (/ y.im t_1) (/ (* y.re y.re) t_1)))))
   (if (<= y.re -4.8e+146)
     t_0
     (if (<= y.re -1e-107)
       t_2
       (if (<= y.re 2.2e-98)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 6.8e+109) t_2 t_0))))))

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 + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_2 = 1.0 / fma(y_46_im, (y_46_im / t_1), ((y_46_re * y_46_re) / t_1));
	double tmp;
	if (y_46_re <= -4.8e+146) {
		tmp = t_0;
	} else if (y_46_re <= -1e-107) {
		tmp = t_2;
	} else if (y_46_re <= 2.2e-98) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.8e+109) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(-1.0 * Float64(Float64(x_46_re * y_46_im) / y_46_re))) / y_46_re)
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_2 = Float64(1.0 / fma(y_46_im, Float64(y_46_im / t_1), Float64(Float64(y_46_re * y_46_re) / t_1)))
	tmp = 0.0
	if (y_46_re <= -4.8e+146)
		tmp = t_0;
	elseif (y_46_re <= -1e-107)
		tmp = t_2;
	elseif (y_46_re <= 2.2e-98)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6.8e+109)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(-1.0 * N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(y$46$im * N[(y$46$im / t$95$1), $MachinePrecision] + N[(N[(y$46$re * y$46$re), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.8e+146], t$95$0, If[LessEqual[y$46$re, -1e-107], t$95$2, If[LessEqual[y$46$re, 2.2e-98], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.8e+109], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.8000000000000004e146 or 6.80000000000000013e109 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower-*.f64 51.8%

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 51.8%

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

   if -4.8000000000000004e146 < y.re < -1e-107 or 2.19999999999999996e-98 < y.re < 6.80000000000000013e109

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.8%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. distribute-lft-neg-in N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.8%

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

   3. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re \cdot x.im - y.im \cdot x.re}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. remove-double-neg N/A

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

      11. div-add N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sqr-neg-rev N/A

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

      17. lift-*.f64 N/A

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

      18. /-rgt-identity N/A

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

      19. associate-/l/ N/A

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

      20. *-lft-identity N/A

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

   5. Applied rewrites 67.9%

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

   if -1e-107 < y.re < 2.19999999999999996e-98

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

Alternative 3: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right) \cdot \frac{y.re}{x.im \cdot y.re - x.re \cdot y.im}}\\ t_1 := \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (*
           (fma y.im (/ y.im y.re) y.re)
           (/ y.re (- (* x.im y.re) (* x.re y.im))))))
        (t_1 (/ (+ x.im (* -1.0 (/ (* x.re y.im) y.re))) y.re)))
   (if (<= y.re -5e+146)
     t_1
     (if (<= y.re -8e-17)
       t_0
       (if (<= y.re 4.2e-95)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 6.8e+109) 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 = 1.0 / (fma(y_46_im, (y_46_im / y_46_re), y_46_re) * (y_46_re / ((x_46_im * y_46_re) - (x_46_re * y_46_im))));
	double t_1 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -5e+146) {
		tmp = t_1;
	} else if (y_46_re <= -8e-17) {
		tmp = t_0;
	} else if (y_46_re <= 4.2e-95) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.8e+109) {
		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(1.0 / Float64(fma(y_46_im, Float64(y_46_im / y_46_re), y_46_re) * Float64(y_46_re / Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)))))
	t_1 = Float64(Float64(x_46_im + Float64(-1.0 * Float64(Float64(x_46_re * y_46_im) / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -5e+146)
		tmp = t_1;
	elseif (y_46_re <= -8e-17)
		tmp = t_0;
	elseif (y_46_re <= 4.2e-95)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6.8e+109)
		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[(1.0 / N[(N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + y$46$re), $MachinePrecision] * N[(y$46$re / N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(-1.0 * N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -5e+146], t$95$1, If[LessEqual[y$46$re, -8e-17], t$95$0, If[LessEqual[y$46$re, 4.2e-95], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.8e+109], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.9999999999999999e146 or 6.80000000000000013e109 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower-*.f64 51.8%

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 51.8%

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

   if -4.9999999999999999e146 < y.re < -8.00000000000000057e-17 or 4.2e-95 < y.re < 6.80000000000000013e109

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.8%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. distribute-lft-neg-in N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.8%

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

   3. Applied rewrites 61.8%

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

   5. Applied rewrites 63.2%

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

   if -8.00000000000000057e-17 < y.re < 4.2e-95

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

Alternative 4: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* -1.0 (/ (* x.re y.im) y.re))) y.re)))
   (if (<= y.re -8.5e+22)
     t_0
     (if (<= y.re 2.5e-98)
       (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
       (if (<= y.re 6.4e+109)
         (/ (- (* x.im y.re) (* x.re y.im)) (fma y.re y.re (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-98) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.4e+109) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(-1.0 * Float64(Float64(x_46_re * y_46_im) / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -8.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-98)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6.4e+109)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(-1.0 * N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -8.5e+22], t$95$0, If[LessEqual[y$46$re, 2.5e-98], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.4e+109], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.49999999999999979e22 or 6.4000000000000002e109 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower-*.f64 51.8%

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 51.8%

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

   if -8.49999999999999979e22 < y.re < 2.50000000000000009e-98

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

   if 2.50000000000000009e-98 < y.re < 6.4000000000000002e109

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

      6. distribute-lft-neg-in N/A

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

      7. distribute-rgt-neg-out N/A

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

      12. distribute-lft-neg-in N/A

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

Alternative 5: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* -1.0 (/ (* x.re y.im) y.re))) y.re)))
   (if (<= y.re -8.5e+22)
     t_0
     (if (<= y.re 5.5e+17) (/ (- (/ (* y.re x.im) y.im) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e+17) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + ((-1.0d0) * ((x_46re * y_46im) / y_46re))) / y_46re
    if (y_46re <= (-8.5d+22)) then
        tmp = t_0
    else if (y_46re <= 5.5d+17) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else
        tmp = t_0
    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 t_0 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e+17) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -8.5e+22:
		tmp = t_0
	elif y_46_re <= 5.5e+17:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(-1.0 * Float64(Float64(x_46_re * y_46_im) / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -8.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 5.5e+17)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (-1.0 * ((x_46_re * y_46_im) / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -8.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 5.5e+17)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(-1.0 * N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -8.5e+22], t$95$0, If[LessEqual[y$46$re, 5.5e+17], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.49999999999999979e22 or 5.5e17 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower-*.f64 51.8%

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 51.8%

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

   if -8.49999999999999979e22 < y.re < 5.5e17

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

Alternative 6: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1e+34)
   (/ x.im y.re)
   (if (<= y.re 5.5e+17)
     (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
     (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1e+34) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.5e+17) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1e+34)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 5.5e+17)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1e+34], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.5e+17], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.99999999999999946e33 or 5.5e17 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.8%

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

   4. Applied rewrites 42.8%

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

   if -9.99999999999999946e33 < y.re < 5.5e17

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      5. associate-/l* N/A

         \[\leadsto \frac{x.im \cdot \frac{y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.im + -1 \cdot x.re}{y.im} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -1 \cdot x.re\right)}{y.im} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -1 \cdot x.re\right)}{y.im} \]

      9. mul-1-neg N/A

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

      10. lift-neg.f64 54.7%

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

   10. Applied rewrites 54.7%

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

Alternative 7: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.5e+22)
   (/ x.im y.re)
   (if (<= y.re 5.5e+17)
     (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
     (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.5e+17) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-8.5d+22)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 5.5d+17) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.5e+17) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8.5e+22:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 5.5e+17:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.5e+22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 5.5e+17)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8.5e+22)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 5.5e+17)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.5e+22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.5e+17], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.49999999999999979e22 or 5.5e17 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.8%

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

   4. Applied rewrites 42.8%

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

   if -8.49999999999999979e22 < y.re < 5.5e17

   1. Initial program 62.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.0%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-abs-rev N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. sqr-neg-rev N/A

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

      18. sqr-abs-rev N/A

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

      19. lift-*.f64 N/A

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

      20. remove-double-neg N/A

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

      21. lower-fma.f64 62.0%

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

   3. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sub N/A

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

      5. lower-unsound-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. inv-pow N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-/.f64 62.0%

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

      4. lower-*.f64 52.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x.im \cdot y.re}{y.im} + -1 \cdot x.re}{y.im} \]

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 52.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.9%

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

   10. Applied rewrites 52.9%

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

Alternative 8: 65.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{+17}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.5e+22)
   (/ x.im y.re)
   (if (<= y.re 1.16e+17) (/ (- x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.16e+17) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-8.5d+22)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 1.16d+17) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.16e+17) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8.5e+22:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 1.16e+17:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.5e+22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.16e+17)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8.5e+22)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 1.16e+17)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.5e+22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.16e+17], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.49999999999999979e22 or 1.16e17 < y.re

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.8%

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

   4. Applied rewrites 42.8%

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

   if -8.49999999999999979e22 < y.re < 1.16e17

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 43.5%

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

   4. Applied rewrites 43.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{-x.re \cdot 1}{y.im} \]

      16. *-rgt-identity 43.5%

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

   6. Applied rewrites 43.5%

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

Alternative 9: 42.8% accurate, 4.9× speedup

Math

\[\frac{x.im}{y.re} \]

FPCore

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

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_re;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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_46re
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_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.8%

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

4. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2025179 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))