_divideComplex, imaginary part

Percentage Accurate: 62.2% → 84.0%

Time: 4.7s

Alternatives: 14

Speedup: 1.7×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re \cdot y.im, \frac{-1}{y.re} \cdot y.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y.im}{t\_0}, x.re, \frac{x.im}{t\_0} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.6e+41)
     (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
     (if (<= y.re -1.06e-135)
       (/ (fma (* x.re y.im) (* (/ -1.0 y.re) y.re) (* x.im y.re)) t_0)
       (if (<= y.re 9e-84)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.5e+117)
           (fma (/ (- y.im) t_0) x.re (* (/ x.im t_0) y.re))
           (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = fma((x_46_re * y_46_im), ((-1.0 / y_46_re) * y_46_re), (x_46_im * y_46_re)) / t_0;
	} else if (y_46_re <= 9e-84) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.5e+117) {
		tmp = fma((-y_46_im / t_0), x_46_re, ((x_46_im / t_0) * y_46_re));
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = Float64(fma(Float64(x_46_re * y_46_im), Float64(Float64(-1.0 / y_46_re) * y_46_re), Float64(x_46_im * y_46_re)) / t_0);
	elseif (y_46_re <= 9e-84)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.5e+117)
		tmp = fma(Float64(Float64(-y_46_im) / t_0), x_46_re, Float64(Float64(x_46_im / t_0) * y_46_re));
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], N[(N[(N[(x$46$re * y$46$im), $MachinePrecision] * N[(N[(-1.0 / y$46$re), $MachinePrecision] * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 9e-84], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+117], N[(N[((-y$46$im) / t$95$0), $MachinePrecision] * x$46$re + N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   5. Applied rewrites 62.1%

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

   if -1.06000000000000004e-135 < y.re < 9.00000000000000031e-84

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 9.00000000000000031e-84 < y.re < 1.5e117

   1. Initial program 62.2%

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

   3. Applied rewrites 62.0%

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

   if 1.5e117 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 55.1

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

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re \cdot y.im, \frac{-1}{y.re} \cdot y.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{t\_0} \cdot y.re - \frac{x.re}{t\_0} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.6e+41)
     (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
     (if (<= y.re -1.06e-135)
       (/ (fma (* x.re y.im) (* (/ -1.0 y.re) y.re) (* x.im y.re)) t_0)
       (if (<= y.re 1.35e-83)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 3.7e+118)
           (- (* (/ x.im t_0) y.re) (* (/ x.re t_0) y.im))
           (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = fma((x_46_re * y_46_im), ((-1.0 / y_46_re) * y_46_re), (x_46_im * y_46_re)) / t_0;
	} else if (y_46_re <= 1.35e-83) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.7e+118) {
		tmp = ((x_46_im / t_0) * y_46_re) - ((x_46_re / t_0) * y_46_im);
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = Float64(fma(Float64(x_46_re * y_46_im), Float64(Float64(-1.0 / y_46_re) * y_46_re), Float64(x_46_im * y_46_re)) / t_0);
	elseif (y_46_re <= 1.35e-83)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.7e+118)
		tmp = Float64(Float64(Float64(x_46_im / t_0) * y_46_re) - Float64(Float64(x_46_re / t_0) * y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], N[(N[(N[(x$46$re * y$46$im), $MachinePrecision] * N[(N[(-1.0 / y$46$re), $MachinePrecision] * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.35e-83], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.7e+118], N[(N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[(x$46$re / t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   5. Applied rewrites 62.1%

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

   if -1.06000000000000004e-135 < y.re < 1.34999999999999996e-83

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 1.34999999999999996e-83 < y.re < 3.69999999999999987e118

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 59.5%

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

   if 3.69999999999999987e118 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 55.1

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

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re \cdot y.im, \frac{-1}{y.re} \cdot y.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;y.re \cdot \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.6e+41)
     (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
     (if (<= y.re -1.06e-135)
       (/ (fma (* x.re y.im) (* (/ -1.0 y.re) y.re) (* x.im y.re)) t_0)
       (if (<= y.re 1.9e-83)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.5e+117)
           (* y.re (/ (- x.im (* y.im (/ x.re y.re))) t_0))
           (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = fma((x_46_re * y_46_im), ((-1.0 / y_46_re) * y_46_re), (x_46_im * y_46_re)) / t_0;
	} else if (y_46_re <= 1.9e-83) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.5e+117) {
		tmp = y_46_re * ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / t_0);
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = Float64(fma(Float64(x_46_re * y_46_im), Float64(Float64(-1.0 / y_46_re) * y_46_re), Float64(x_46_im * y_46_re)) / t_0);
	elseif (y_46_re <= 1.9e-83)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.5e+117)
		tmp = Float64(y_46_re * Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / t_0));
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], N[(N[(N[(x$46$re * y$46$im), $MachinePrecision] * N[(N[(-1.0 / y$46$re), $MachinePrecision] * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-83], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+117], N[(y$46$re * N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   5. Applied rewrites 62.1%

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

   if -1.06000000000000004e-135 < y.re < 1.89999999999999988e-83

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 1.89999999999999988e-83 < y.re < 1.5e117

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 55.1%

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

   if 1.5e117 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 55.1

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

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+117}:\\ \;\;\;\;y.re \cdot \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.6e+41)
     (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
     (if (<= y.re -1.06e-135)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.re 1.9e-83)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.5e+117)
           (* y.re (/ (- x.im (* y.im (/ x.re y.re))) t_0))
           (/ (- x.im (* (/ y.im y.re) x.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_re <= 1.9e-83) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.5e+117) {
		tmp = y_46_re * ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / t_0);
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_re <= 1.9e-83)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.5e+117)
		tmp = Float64(y_46_re * Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / t_0));
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-83], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+117], N[(y$46$re * N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   if -1.06000000000000004e-135 < y.re < 1.89999999999999988e-83

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 1.89999999999999988e-83 < y.re < 1.5e117

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 55.1%

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

   if 1.5e117 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 55.1

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

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.55 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y.im}{y.re}, \frac{x.re}{y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e+41)
   (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
   (if (<= y.re -1.06e-135)
     (/ (- (* y.re x.im) (* y.im x.re)) (fma y.im y.im (* y.re y.re)))
     (if (<= y.re 2.5e-140)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (if (<= y.re 3.55e+50)
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (fma (/ (- y.im) y.re) (/ x.re y.re) (/ 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 <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_re <= 2.5e-140) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.55e+50) {
		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));
	} else {
		tmp = fma((-y_46_im / y_46_re), (x_46_re / y_46_re), (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 <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 2.5e-140)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.55e+50)
		tmp = 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)));
	else
		tmp = fma(Float64(Float64(-y_46_im) / y_46_re), Float64(x_46_re / y_46_re), 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, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e-140], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.55e+50], 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], N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   if -1.06000000000000004e-135 < y.re < 2.50000000000000007e-140

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 2.50000000000000007e-140 < y.re < 3.54999999999999996e50

   1. Initial program 62.2%

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

   if 3.54999999999999996e50 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. div-add N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. associate-/l/ N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. times-frac N/A

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

      18. distribute-neg-frac N/A

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

      19. distribute-neg-frac2 N/A

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

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lift-/.f64 N/A

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

   6. Applied rewrites 53.7%

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

Alternative 6: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.55 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y.im}{y.re}, \frac{x.re}{y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (fma y.im y.im (* y.re y.re)))))
   (if (<= y.re -2.6e+41)
     (/ (- x.im (* (/ x.re y.re) y.im)) y.re)
     (if (<= y.re -1.06e-135)
       t_0
       (if (<= y.re 2.5e-140)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 3.55e+50)
           t_0
           (fma (/ (- y.im) y.re) (/ x.re y.re) (/ x.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.6e+41) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} else if (y_46_re <= -1.06e-135) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-140) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.55e+50) {
		tmp = t_0;
	} else {
		tmp = fma((-y_46_im / y_46_re), (x_46_re / y_46_re), (x_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	elseif (y_46_re <= -1.06e-135)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-140)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.55e+50)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-y_46_im) / y_46_re), Float64(x_46_re / y_46_re), 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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+41], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-135], t$95$0, If[LessEqual[y$46$re, 2.5e-140], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.55e+50], t$95$0, N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.6000000000000001e41

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -2.6000000000000001e41 < y.re < -1.06000000000000004e-135 or 2.50000000000000007e-140 < y.re < 3.54999999999999996e50

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.2

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

      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 62.2

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

      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. remove-double-neg N/A

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

      16. sqr-neg-rev N/A

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

      17. sqr-abs-rev N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 62.2

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

   3. Applied rewrites 62.2%

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

   if -1.06000000000000004e-135 < y.re < 2.50000000000000007e-140

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

   if 3.54999999999999996e50 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. div-add N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. associate-/l/ N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. times-frac N/A

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

      18. distribute-neg-frac N/A

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

      19. distribute-neg-frac2 N/A

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

      20. distribute-frac-neg N/A

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

      21. frac-2neg N/A

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

      22. lift-/.f64 N/A

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

   6. Applied rewrites 53.7%

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

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-y.im}{y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -1.7e+41)
     t_0
     (if (<= y.im 9e+81) (/ (fma (/ (- y.im) y.re) x.re x.im) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+41) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = fma((-y_46_im / y_46_re), x_46_re, x_46_im) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.7e+41)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(fma(Float64(Float64(-y_46_im) / y_46_re), x_46_re, x_46_im) / y_46_re);
	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[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+41], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.69999999999999999e41 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 53.0

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

   6. Applied rewrites 53.0%

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

   if -1.69999999999999999e41 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. frac-2neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. associate-/l* N/A

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

      11. distribute-neg-frac2 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-rgt-identity 55.1

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

   6. Applied rewrites 55.1%

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

Alternative 8: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -1.7e+41)
     t_0
     (if (<= y.im 9e+81) (/ (- x.im (* (/ y.im y.re) x.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+41) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.7e+41)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	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[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+41], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.69999999999999999e41 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 53.0

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

   6. Applied rewrites 53.0%

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

   if -1.69999999999999999e41 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 55.1

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

   8. Applied rewrites 55.1%

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

Alternative 9: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.22 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -1.22e-28)
     t_0
     (if (<= y.re 5.2e-79) (/ (- (/ (* x.im y.re) 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 - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.22e-28) {
		tmp = t_0;
	} else if (y_46_re <= 5.2e-79) {
		tmp = (((x_46_im * y_46_re) / 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 - ((x_46re / y_46re) * y_46im)) / y_46re
    if (y_46re <= (-1.22d-28)) then
        tmp = t_0
    else if (y_46re <= 5.2d-79) then
        tmp = (((x_46im * y_46re) / 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 - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.22e-28) {
		tmp = t_0;
	} else if (y_46_re <= 5.2e-79) {
		tmp = (((x_46_im * y_46_re) / 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 - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	tmp = 0
	if y_46_re <= -1.22e-28:
		tmp = t_0
	elif y_46_re <= 5.2e-79:
		tmp = (((x_46_im * y_46_re) / 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(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.22e-28)
		tmp = t_0;
	elseif (y_46_re <= 5.2e-79)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / 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 - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.22e-28)
		tmp = t_0;
	elseif (y_46_re <= 5.2e-79)
		tmp = (((x_46_im * y_46_re) / 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[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.22e-28], t$95$0, If[LessEqual[y$46$re, 5.2e-79], N[(N[(N[(N[(x$46$im * y$46$re), $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 < -1.22e-28 or 5.19999999999999987e-79 < y.re

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if -1.22e-28 < y.re < 5.19999999999999987e-79

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-rgt-identity 51.8

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

   6. Applied rewrites 51.8%

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

Alternative 10: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re}{y.im} \cdot x.im - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* (/ y.re y.im) x.im) x.re) y.im)))
   (if (<= y.im -1.7e+41)
     t_0
     (if (<= y.im 9e+81) (/ (- x.im (* (/ y.im y.re) x.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+41) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} 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 = (((y_46re / y_46im) * x_46im) - x_46re) / y_46im
    if (y_46im <= (-1.7d+41)) then
        tmp = t_0
    else if (y_46im <= 9d+81) then
        tmp = (x_46im - ((y_46im / y_46re) * x_46re)) / y_46re
    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 = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+41) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.7e+41:
		tmp = t_0
	elif y_46_im <= 9e+81:
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re
	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(Float64(Float64(y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.7e+41)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	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 = (((y_46_re / y_46_im) * x_46_im) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.7e+41)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	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[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+41], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.69999999999999999e41 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

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

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-rgt-identity 53.0

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

   6. Applied rewrites 53.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. sub-flip-reverse N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ 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. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

   if -1.69999999999999999e41 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 53.3

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

      22. *-rgt-identity 54.5

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

   6. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. remove-double-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*l* N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.re} \]

      16. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(x.re \cdot -1\right)\right)}{y.re} \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)}{y.re} \]

      18. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(x.re \cdot 1\right)}{y.re} \]

      19. *-rgt-identity 55.1

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.6e+44)
     t_0
     (if (<= y.im 9e+81) (/ (- x.im (* (/ y.im y.re) x.re)) y.re) 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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.6e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} 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_46re / y_46im
    if (y_46im <= (-2.6d+44)) then
        tmp = t_0
    else if (y_46im <= 9d+81) then
        tmp = (x_46im - ((y_46im / y_46re) * x_46re)) / y_46re
    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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.6e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} 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_re / y_46_im
	tmp = 0
	if y_46_im <= -2.6e+44:
		tmp = t_0
	elif y_46_im <= 9e+81:
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re
	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_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.6e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	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_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.6e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.6e+44], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5999999999999999e44 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.5

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot x.re}{y.im} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x.re\right)}{y.im} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(x.re \cdot -1\right)\right)}{y.im} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \frac{-x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)}{y.im} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-x.re \cdot 1}{y.im} \]

      11. *-rgt-identity 42.5

          \[\leadsto \frac{-x.re}{y.im} \]

   9. Applied rewrites 42.5%

      \[\leadsto \frac{-x.re}{\color{blue}{y.im}} \]

   if -2.5999999999999999e44 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   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 53.3

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)}{y.re} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower--.f64 53.3

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re} \]

      9. associate-/l* N/A

         \[\leadsto \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re} \]

      10. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      12. frac-2neg N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot 1\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      14. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x.re \cdot -1\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      16. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      17. frac-2neg-rev N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(-1 \cdot x.re\right)}{y.re} \cdot y.im}{y.re} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(-1 \cdot x.re\right)}{y.re} \cdot y.im}{y.re} \]

      19. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot -1\right)}{y.re} \cdot y.im}{y.re} \]

      20. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)}{y.re} \cdot y.im}{y.re} \]

      21. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{x.re \cdot 1}{y.re} \cdot y.im}{y.re} \]

      22. *-rgt-identity 54.5

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

   6. Applied rewrites 54.5%

      \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      4. mult-flip N/A

         \[\leadsto \frac{x.im - \left(x.re \cdot y.im\right) \cdot \frac{1}{y.re}}{y.re} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.re \cdot y.im\right)\right)\right)\right) \cdot \frac{1}{y.re}}{y.re} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.re\right)\right) \cdot y.im\right)\right) \cdot \frac{1}{y.re}}{y.re} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(\left(-1 \cdot x.re\right) \cdot y.im\right)\right) \cdot \frac{1}{y.re}}{y.re} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(\left(-1 \cdot x.re\right) \cdot y.im\right)\right) \cdot \frac{1}{y.re}}{y.re} \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \frac{x.im - \left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right) \cdot y.im\right) \cdot \frac{1}{y.re}}{y.re} \]

      10. associate-*l* N/A

          \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right) \cdot \left(y.im \cdot \frac{1}{y.re}\right)}{y.re} \]

      11. mult-flip N/A

          \[\leadsto \frac{x.im - \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right) \cdot \frac{y.im}{y.re}}{y.re} \]

      12. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.re} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.re} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.re} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.re} \]

      16. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(\mathsf{neg}\left(x.re \cdot -1\right)\right)}{y.re} \]

      17. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)}{y.re} \]

      18. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot \left(x.re \cdot 1\right)}{y.re} \]

      19. *-rgt-identity 55.1

          \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]

   8. Applied rewrites 55.1%

      \[\leadsto \frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.6e+44)
     t_0
     (if (<= y.im 9e+81) (/ (- x.im (* (/ x.re y.re) y.im)) y.re) 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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.6e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} 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_46re / y_46im
    if (y_46im <= (-2.6d+44)) then
        tmp = t_0
    else if (y_46im <= 9d+81) then
        tmp = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.6e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	} 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_re / y_46_im
	tmp = 0
	if y_46_im <= -2.6e+44:
		tmp = t_0
	elif y_46_im <= 9e+81:
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	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_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.6e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	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_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.6e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.6e+44], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5999999999999999e44 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.5

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot x.re}{y.im} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x.re\right)}{y.im} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(x.re \cdot -1\right)\right)}{y.im} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \frac{-x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)}{y.im} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-x.re \cdot 1}{y.im} \]

      11. *-rgt-identity 42.5

          \[\leadsto \frac{-x.re}{y.im} \]

   9. Applied rewrites 42.5%

      \[\leadsto \frac{-x.re}{\color{blue}{y.im}} \]

   if -2.5999999999999999e44 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   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 53.3

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

   4. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)}{y.re} \]

      4. sub-flip-reverse N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      5. lower--.f64 53.3

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re} \]

      8. *-commutative N/A

         \[\leadsto \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re} \]

      9. associate-/l* N/A

         \[\leadsto \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re} \]

      10. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

      12. frac-2neg N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot 1\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      14. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x.re \cdot -1\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      16. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{\mathsf{neg}\left(y.re\right)} \cdot y.im}{y.re} \]

      17. frac-2neg-rev N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(-1 \cdot x.re\right)}{y.re} \cdot y.im}{y.re} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(-1 \cdot x.re\right)}{y.re} \cdot y.im}{y.re} \]

      19. *-commutative N/A

          \[\leadsto \frac{x.im - \frac{\mathsf{neg}\left(x.re \cdot -1\right)}{y.re} \cdot y.im}{y.re} \]

      20. distribute-rgt-neg-out N/A

          \[\leadsto \frac{x.im - \frac{x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)}{y.re} \cdot y.im}{y.re} \]

      21. metadata-eval N/A

          \[\leadsto \frac{x.im - \frac{x.re \cdot 1}{y.re} \cdot y.im}{y.re} \]

      22. *-rgt-identity 54.5

          \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]

   6. Applied rewrites 54.5%

      \[\leadsto \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.35 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.35e+44) t_0 (if (<= y.im 9e+81) (/ x.im y.re) 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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.35e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = x_46_im / y_46_re;
	} 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_46re / y_46im
    if (y_46im <= (-2.35d+44)) then
        tmp = t_0
    else if (y_46im <= 9d+81) then
        tmp = x_46im / y_46re
    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_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.35e+44) {
		tmp = t_0;
	} else if (y_46_im <= 9e+81) {
		tmp = x_46_im / y_46_re;
	} 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_re / y_46_im
	tmp = 0
	if y_46_im <= -2.35e+44:
		tmp = t_0
	elif y_46_im <= 9e+81:
		tmp = x_46_im / y_46_re
	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_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.35e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = Float64(x_46_im / y_46_re);
	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_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.35e+44)
		tmp = t_0;
	elseif (y_46_im <= 9e+81)
		tmp = x_46_im / y_46_re;
	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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.35e+44], t$95$0, If[LessEqual[y$46$im, 9e+81], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.35000000000000009e44 or 9.00000000000000034e81 < y.im

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   3. 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 51.8

         \[\leadsto \frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im} \]

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x.re, \frac{x.im \cdot y.re}{y.im}\right)}{y.im}} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.5

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

   7. Applied rewrites 42.5%

      \[\leadsto \frac{-1 \cdot x.re}{y.im} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot x.re}{y.im} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot x.re}{y.im} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x.re\right)}{y.im} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)\right)}{y.im} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(-1 \cdot x.re\right)\right)}{y.im} \]

      8. *-commutative N/A

         \[\leadsto \frac{-\left(\mathsf{neg}\left(x.re \cdot -1\right)\right)}{y.im} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto \frac{-x.re \cdot \left(\mathsf{neg}\left(-1\right)\right)}{y.im} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-x.re \cdot 1}{y.im} \]

      11. *-rgt-identity 42.5

          \[\leadsto \frac{-x.re}{y.im} \]

   9. Applied rewrites 42.5%

      \[\leadsto \frac{-x.re}{\color{blue}{y.im}} \]

   if -2.35000000000000009e44 < y.im < 9.00000000000000034e81

   1. Initial program 62.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
   3. Step-by-step derivation

      1. lower-/.f64 43.4

         \[\leadsto \frac{x.im}{\color{blue}{y.re}} \]

   4. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.re} \end{array} \]

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 62.2%

   \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

2. Taylor expanded in y.re around inf

   \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
3. Step-by-step derivation

   1. lower-/.f64 43.4

      \[\leadsto \frac{x.im}{\color{blue}{y.re}} \]

4. Applied rewrites 43.4%

   \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))