_divideComplex, imaginary part

Percentage Accurate: 61.2% → 84.4%

Time: 3.7s

Alternatives: 13

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: 61.2% 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: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_2 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y.im}{t\_0}, x.re, \frac{x.im}{t\_0} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{y.re}{t\_1} \cdot x.im - \frac{y.im}{t\_1} \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma y.re y.re (* y.im y.im)))
        (t_2 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -2.7e+84)
     t_2
     (if (<= y.re -7.2e-99)
       (fma (/ (- y.im) t_0) x.re (* (/ x.im t_0) y.re))
       (if (<= y.re 5.2e-155)
         (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
         (if (<= y.re 4.5e+83)
           (- (* (/ y.re t_1) x.im) (* (/ y.im t_1) x.re))
           t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_2 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.7e+84) {
		tmp = t_2;
	} else if (y_46_re <= -7.2e-99) {
		tmp = fma((-y_46_im / t_0), x_46_re, ((x_46_im / t_0) * y_46_re));
	} else if (y_46_re <= 5.2e-155) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_re <= 4.5e+83) {
		tmp = ((y_46_re / t_1) * x_46_im) - ((y_46_im / t_1) * x_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	t_2 = 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 <= -2.7e+84)
		tmp = t_2;
	elseif (y_46_re <= -7.2e-99)
		tmp = fma(Float64(Float64(-y_46_im) / t_0), x_46_re, Float64(Float64(x_46_im / t_0) * y_46_re));
	elseif (y_46_re <= 5.2e-155)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 4.5e+83)
		tmp = Float64(Float64(Float64(y_46_re / t_1) * x_46_im) - Float64(Float64(y_46_im / t_1) * x_46_re));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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, -2.7e+84], t$95$2, If[LessEqual[y$46$re, -7.2e-99], 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], If[LessEqual[y$46$re, 5.2e-155], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.5e+83], N[(N[(N[(y$46$re / t$95$1), $MachinePrecision] * x$46$im), $MachinePrecision] - N[(N[(y$46$im / t$95$1), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.7e84 or 4.4999999999999999e83 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -2.7e84 < y.re < -7.2000000000000001e-99

   1. Initial program 61.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. div-add N/A

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

      7. frac-2neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-neg-frac2 N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < y.re < 5.20000000000000016e-155

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 5.20000000000000016e-155 < y.re < 4.4999999999999999e83

   1. Initial program 61.2%

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

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

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

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-in N/A

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

      18. mult-flip N/A

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

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

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

   5. Applied rewrites 64.4%

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

Alternative 2: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_1 := \frac{y.re}{t\_0} \cdot x.im - \frac{y.im}{t\_0} \cdot x.re\\ t_2 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+87}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im)))
        (t_1 (- (* (/ y.re t_0) x.im) (* (/ y.im t_0) x.re)))
        (t_2 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -1.3e+87)
     t_2
     (if (<= y.re -7.2e-99)
       t_1
       (if (<= y.re 5.2e-155)
         (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
         (if (<= y.re 4.5e+83) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = ((y_46_re / t_0) * x_46_im) - ((y_46_im / t_0) * x_46_re);
	double t_2 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.3e+87) {
		tmp = t_2;
	} else if (y_46_re <= -7.2e-99) {
		tmp = t_1;
	} else if (y_46_re <= 5.2e-155) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_re <= 4.5e+83) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(Float64(y_46_re / t_0) * x_46_im) - Float64(Float64(y_46_im / t_0) * x_46_re))
	t_2 = 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 <= -1.3e+87)
		tmp = t_2;
	elseif (y_46_re <= -7.2e-99)
		tmp = t_1;
	elseif (y_46_re <= 5.2e-155)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 4.5e+83)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision] - N[(N[(y$46$im / t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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, -1.3e+87], t$95$2, If[LessEqual[y$46$re, -7.2e-99], t$95$1, If[LessEqual[y$46$re, 5.2e-155], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.5e+83], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.29999999999999999e87 or 4.4999999999999999e83 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -1.29999999999999999e87 < y.re < -7.2000000000000001e-99 or 5.20000000000000016e-155 < y.re < 4.4999999999999999e83

   1. Initial program 61.2%

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

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

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

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-in N/A

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

      18. mult-flip N/A

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

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

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

   5. Applied rewrites 64.4%

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

   if -7.2000000000000001e-99 < y.re < 5.20000000000000016e-155

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{1}{\frac{y.im}{x.im}}, t\_1\right)\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{t\_0} \cdot y.re - \frac{x.re}{t\_0} \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))) (t_1 (/ x.re (- y.im))))
   (if (<= y.im -1.15e+111)
     (fma (/ y.re y.im) (/ 1.0 (/ y.im x.im)) t_1)
     (if (<= y.im -2.3e-151)
       (/ (- (* x.im y.re) (* x.re y.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 2.15e-29)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 2.35e+101)
           (- (* (/ x.im t_0) y.re) (* (/ x.re t_0) y.im))
           (fma (/ y.re y.im) (/ x.im y.im) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -1.15e+111) {
		tmp = fma((y_46_re / y_46_im), (1.0 / (y_46_im / x_46_im)), t_1);
	} else if (y_46_im <= -2.3e-151) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 2.15e-29) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.35e+101) {
		tmp = ((x_46_im / t_0) * y_46_re) - ((x_46_re / t_0) * y_46_im);
	} else {
		tmp = fma((y_46_re / y_46_im), (x_46_im / y_46_im), t_1);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.15e+111)
		tmp = fma(Float64(y_46_re / y_46_im), Float64(1.0 / Float64(y_46_im / x_46_im)), t_1);
	elseif (y_46_im <= -2.3e-151)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.15e-29)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.35e+101)
		tmp = Float64(Float64(Float64(x_46_im / t_0) * y_46_re) - Float64(Float64(x_46_re / t_0) * y_46_im));
	else
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_im / y_46_im), t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+111], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(1.0 / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[y$46$im, -2.3e-151], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.15e-29], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.35e+101], 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[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.15000000000000001e111

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity 52.9

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

   6. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-unsound-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

   if -1.15000000000000001e111 < y.im < -2.29999999999999996e-151

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

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

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -2.29999999999999996e-151 < y.im < 2.1499999999999999e-29

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

   if 2.1499999999999999e-29 < y.im < 2.34999999999999985e101

   1. Initial program 61.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 58.4%

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

   if 2.34999999999999985e101 < y.im

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re \cdot x.im - y.im \cdot x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -3.4e+86)
     t_0
     (if (<= y.re -7.2e-99)
       (/ (- (* x.im y.re) (* x.re y.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 2.45e-115)
         (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
         (if (<= y.re 1.85e+50)
           (/
            1.0
            (/ (fma y.im y.im (* y.re y.re)) (- (* y.re x.im) (* y.im x.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_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -3.4e+86) {
		tmp = t_0;
	} else if (y_46_re <= -7.2e-99) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 2.45e-115) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_re <= 1.85e+50) {
		tmp = 1.0 / (fma(y_46_im, y_46_im, (y_46_re * y_46_re)) / ((y_46_re * x_46_im) - (y_46_im * x_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_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.4e+86)
		tmp = t_0;
	elseif (y_46_re <= -7.2e-99)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.45e-115)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 1.85e+50)
		tmp = Float64(1.0 / Float64(fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)) / Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_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[(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, -3.4e+86], t$95$0, If[LessEqual[y$46$re, -7.2e-99], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.45e-115], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+50], N[(1.0 / N[(N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.3999999999999998e86 or 1.85e50 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -3.3999999999999998e86 < y.re < -7.2000000000000001e-99

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

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

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < y.re < 2.44999999999999994e-115

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 2.44999999999999994e-115 < y.re < 1.85e50

   1. Initial program 61.2%

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

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

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

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re \cdot x.im - y.im \cdot x.re}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im y.re) (* x.re y.im)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -3.4e+86)
     t_1
     (if (<= y.re -7.2e-99)
       t_0
       (if (<= y.re 5.9e-117)
         (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
         (if (<= y.re 1.85e+50) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -3.4e+86) {
		tmp = t_1;
	} else if (y_46_re <= -7.2e-99) {
		tmp = t_0;
	} else if (y_46_re <= 5.9e-117) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_re <= 1.85e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	t_1 = 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 <= -3.4e+86)
		tmp = t_1;
	elseif (y_46_re <= -7.2e-99)
		tmp = t_0;
	elseif (y_46_re <= 5.9e-117)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 1.85e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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, -3.4e+86], t$95$1, If[LessEqual[y$46$re, -7.2e-99], t$95$0, If[LessEqual[y$46$re, 5.9e-117], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.3999999999999998e86 or 1.85e50 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -3.3999999999999998e86 < y.re < -7.2000000000000001e-99 or 5.9000000000000003e-117 < y.re < 1.85e50

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

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

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < y.re < 5.9000000000000003e-117

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 6: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14500:\\ \;\;\;\;\frac{x.im - \frac{1}{\frac{y.re}{x.re}} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -14500.0)
   (/ (- x.im (* (/ 1.0 (/ y.re x.re)) y.im)) y.re)
   (if (<= y.re 7.5e-8)
     (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
     (/ (- x.im (* (/ x.re y.re) y.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 <= -14500.0) {
		tmp = (x_46_im - ((1.0 / (y_46_re / x_46_re)) * y_46_im)) / y_46_re;
	} else if (y_46_re <= 7.5e-8) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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 <= -14500.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(1.0 / Float64(y_46_re / x_46_re)) * y_46_im)) / y_46_re);
	elseif (y_46_re <= 7.5e-8)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_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, -14500.0], N[(N[(x$46$im - N[(N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-8], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -14500

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-unsound-/.f64 53.0

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

   10. Applied rewrites 53.0%

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

   if -14500 < y.re < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 7.4999999999999997e-8 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 7: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14500:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.re \cdot \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -14500.0)
   (/ (fma y.im (* x.re (/ -1.0 y.re)) x.im) y.re)
   (if (<= y.re 7.5e-8)
     (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
     (/ (- x.im (* (/ x.re y.re) y.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 <= -14500.0) {
		tmp = fma(y_46_im, (x_46_re * (-1.0 / y_46_re)), x_46_im) / y_46_re;
	} else if (y_46_re <= 7.5e-8) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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 <= -14500.0)
		tmp = Float64(fma(y_46_im, Float64(x_46_re * Float64(-1.0 / y_46_re)), x_46_im) / y_46_re);
	elseif (y_46_re <= 7.5e-8)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_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, -14500.0], N[(N[(y$46$im * N[(x$46$re * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-8], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -14500

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. mult-flip N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. frac-2neg-rev N/A

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

      22. lower-/.f64 53.0

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

   6. Applied rewrites 53.0%

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

   if -14500 < y.re < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 7.4999999999999997e-8 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 8: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -14500.0)
     t_0
     (if (<= y.re 7.5e-8) (/ (fma (/ y.re y.im) x.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 <= -14500.0) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e-8) {
		tmp = fma((y_46_re / y_46_im), x_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 <= -14500.0)
		tmp = t_0;
	elseif (y_46_re <= 7.5e-8)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -14500.0], t$95$0, If[LessEqual[y$46$re, 7.5e-8], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -14500 or 7.4999999999999997e-8 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -14500 < y.re < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. lower-neg.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 9: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -14500.0)
     t_0
     (if (<= y.re 7.5e-8) (/ (- (/ (* 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 <= -14500.0) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e-8) {
		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 <= (-14500.0d0)) then
        tmp = t_0
    else if (y_46re <= 7.5d-8) 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 <= -14500.0) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e-8) {
		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 <= -14500.0:
		tmp = t_0
	elif y_46_re <= 7.5e-8:
		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 <= -14500.0)
		tmp = t_0;
	elseif (y_46_re <= 7.5e-8)
		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 <= -14500.0)
		tmp = t_0;
	elseif (y_46_re <= 7.5e-8)
		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, -14500.0], t$95$0, If[LessEqual[y$46$re, 7.5e-8], 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 < -14500 or 7.4999999999999997e-8 < y.re

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -14500 < y.re < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 10: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.75 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -3e+44)
     t_0
     (if (<= y.im 2.75e+52) (/ (- x.im (/ (* x.re y.im) y.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 <= -3e+44) {
		tmp = t_0;
	} else if (y_46_im <= 2.75e+52) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_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 <= (-3d+44)) then
        tmp = t_0
    else if (y_46im <= 2.75d+52) then
        tmp = (x_46im - ((x_46re * y_46im) / y_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 <= -3e+44) {
		tmp = t_0;
	} else if (y_46_im <= 2.75e+52) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_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 <= -3e+44:
		tmp = t_0
	elif y_46_im <= 2.75e+52:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_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 <= -3e+44)
		tmp = t_0;
	elseif (y_46_im <= 2.75e+52)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_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 <= -3e+44)
		tmp = t_0;
	elseif (y_46_im <= 2.75e+52)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_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, -3e+44], t$95$0, If[LessEqual[y$46$im, 2.75e+52], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.99999999999999987e44 or 2.74999999999999998e52 < y.im

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 42.6

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

   7. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 42.6

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

   9. Applied rewrites 42.6%

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

   if -2.99999999999999987e44 < y.im < 2.74999999999999998e52

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

Alternative 11: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -3e+44)
     t_0
     (if (<= y.im 8e+89) (/ (- 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 <= -3e+44) {
		tmp = t_0;
	} else if (y_46_im <= 8e+89) {
		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 <= (-3d+44)) then
        tmp = t_0
    else if (y_46im <= 8d+89) 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 <= -3e+44) {
		tmp = t_0;
	} else if (y_46_im <= 8e+89) {
		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 <= -3e+44:
		tmp = t_0
	elif y_46_im <= 8e+89:
		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 <= -3e+44)
		tmp = t_0;
	elseif (y_46_im <= 8e+89)
		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 <= -3e+44)
		tmp = t_0;
	elseif (y_46_im <= 8e+89)
		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, -3e+44], t$95$0, If[LessEqual[y$46$im, 8e+89], 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 < -2.99999999999999987e44 or 7.99999999999999996e89 < y.im

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 42.6

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

   7. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 42.6

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

   9. Applied rewrites 42.6%

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

   if -2.99999999999999987e44 < y.im < 7.99999999999999996e89

   1. Initial program 61.2%

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

   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 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 12: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.8e+60)
   (/ x.im y.re)
   (if (<= y.re 1.25e+19) (/ (- x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.8e+60) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.25e+19) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.8d+60)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 1.25d+19) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.8e+60) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.25e+19) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.8e+60:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 1.25e+19:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.8e+60)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.25e+19)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.8e+60)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 1.25e+19)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.8e+60], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+19], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.8e60 or 1.25e19 < y.re

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0

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

   4. Applied rewrites 42.0%

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

   if -4.8e60 < y.re < 1.25e19

   1. Initial program 61.2%

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

   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 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 42.6

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

   7. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 42.6

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

   9. Applied rewrites 42.6%

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

Alternative 13: 42.0% 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 61.2%

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

2. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.0

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

4. Applied rewrites 42.0%

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

Reproduce

herbie shell --seed 2025155 
(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))))