_divideComplex, imaginary part

Percentage Accurate: 61.7% → 81.9%

Time: 4.8s

Alternatives: 8

Speedup: 1.5×

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: 61.7% 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: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1850:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (fma (- x.re) y.im (* y.re x.im)) (fma y.im y.im (* y.re y.re))))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -2.6e+111)
     t_1
     (if (<= y.im -1850.0)
       t_0
       (if (<= y.im 5.6e-113)
         (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)
         (if (<= y.im 2.6e+75) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(-x_46_re, y_46_im, (y_46_re * x_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.6e+111) {
		tmp = t_1;
	} else if (y_46_im <= -1850.0) {
		tmp = t_0;
	} else if (y_46_im <= 5.6e-113) {
		tmp = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	} else if (y_46_im <= 2.6e+75) {
		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(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.6e+111)
		tmp = t_1;
	elseif (y_46_im <= -1850.0)
		tmp = t_0;
	elseif (y_46_im <= 5.6e-113)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 2.6e+75)
		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[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.6e+111], t$95$1, If[LessEqual[y$46$im, -1850.0], t$95$0, If[LessEqual[y$46$im, 5.6e-113], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.6e+75], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.5999999999999999e111 or 2.59999999999999985e75 < y.im

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

      7. div-add N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. mul-1-neg N/A

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

      22. lower-neg.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 83.9%

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

   if -2.5999999999999999e111 < y.im < -1850 or 5.6e-113 < y.im < 2.59999999999999985e75

   1. Initial program 91.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 91.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 91.0

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

   4. Applied rewrites 91.0%

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

   if -1850 < y.im < 5.6e-113

   1. Initial program 70.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 70.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 87.3

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

   10. Applied rewrites 87.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1850:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\left(-x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -3.85e+158)
     t_0
     (if (<= y.im -9600.0)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 1e-9)
         (/ x.im y.re)
         (if (<= y.im 1.9e+139)
           (* (- x.re) (/ y.im (fma y.im 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 <= -3.85e+158) {
		tmp = t_0;
	} else if (y_46_im <= -9600.0) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 1e-9) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.9e+139) {
		tmp = -x_46_re * (y_46_im / fma(y_46_im, 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 <= -3.85e+158)
		tmp = t_0;
	elseif (y_46_im <= -9600.0)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 1e-9)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.9e+139)
		tmp = Float64(Float64(-x_46_re) * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.85e+158], t$95$0, If[LessEqual[y$46$im, -9600.0], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1e-9], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.9e+139], N[((-x$46$re) * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.84999999999999996e158 or 1.9e139 < y.im

   1. Initial program 32.2%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -3.84999999999999996e158 < y.im < -9600

   1. Initial program 83.8%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -9600 < y.im < 1.00000000000000006e-9

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.00000000000000006e-9 < y.im < 1.9e139

   1. Initial program 76.2%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\left(-x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \left(-x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1 (* (- x.re) (/ y.im (fma y.im y.im (* y.re y.re))))))
   (if (<= y.im -3.85e+158)
     t_0
     (if (<= y.im -1.4e-115)
       t_1
       (if (<= y.im 1e-9) (/ x.im y.re) (if (<= y.im 1.9e+139) t_1 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 t_1 = -x_46_re * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_im <= -3.85e+158) {
		tmp = t_0;
	} else if (y_46_im <= -1.4e-115) {
		tmp = t_1;
	} else if (y_46_im <= 1e-9) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.9e+139) {
		tmp = t_1;
	} 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)
	t_1 = Float64(Float64(-x_46_re) * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_im <= -3.85e+158)
		tmp = t_0;
	elseif (y_46_im <= -1.4e-115)
		tmp = t_1;
	elseif (y_46_im <= 1e-9)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.9e+139)
		tmp = t_1;
	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[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[((-x$46$re) * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.85e+158], t$95$0, If[LessEqual[y$46$im, -1.4e-115], t$95$1, If[LessEqual[y$46$im, 1e-9], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.9e+139], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.84999999999999996e158 or 1.9e139 < y.im

   1. Initial program 32.2%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -3.84999999999999996e158 < y.im < -1.39999999999999994e-115 or 1.00000000000000006e-9 < y.im < 1.9e139

   1. Initial program 77.5%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if -1.39999999999999994e-115 < y.im < 1.00000000000000006e-9

   1. Initial program 76.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;\left(-x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\left(-x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -3.85e+158)
     t_0
     (if (<= y.im -9600.0)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 2.65e+42)
         (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -3.85e+158) {
		tmp = t_0;
	} else if (y_46_im <= -9600.0) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.65e+42) {
		tmp = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.85e+158)
		tmp = t_0;
	elseif (y_46_im <= -9600.0)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.65e+42)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.85e+158], t$95$0, If[LessEqual[y$46$im, -9600.0], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+42], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.84999999999999996e158 or 2.65000000000000014e42 < y.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -3.84999999999999996e158 < y.im < -9600

   1. Initial program 83.8%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -9600 < y.im < 2.65000000000000014e42

   1. Initial program 76.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 76.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 80.0

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 81.6

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

   10. Applied rewrites 81.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -3.85e+158)
     t_0
     (if (<= y.im -9600.0)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 2.65e+42)
         (/ (fma (- y.im) (/ x.re y.re) x.im) y.re)
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -3.85e+158) {
		tmp = t_0;
	} else if (y_46_im <= -9600.0) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.65e+42) {
		tmp = fma(-y_46_im, (x_46_re / y_46_re), x_46_im) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.85e+158)
		tmp = t_0;
	elseif (y_46_im <= -9600.0)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.65e+42)
		tmp = Float64(fma(Float64(-y_46_im), Float64(x_46_re / y_46_re), x_46_im) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.85e+158], t$95$0, If[LessEqual[y$46$im, -9600.0], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+42], N[(N[((-y$46$im) * N[(x$46$re / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.84999999999999996e158 or 2.65000000000000014e42 < y.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -3.84999999999999996e158 < y.im < -9600

   1. Initial program 83.8%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -9600 < y.im < 2.65000000000000014e42

   1. Initial program 76.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 76.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 80.0

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 81.6

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

   10. Applied rewrites 81.6%

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

   12. Applied rewrites 80.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9600:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9600 \lor \neg \left(y.im \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9600.0) (not (<= y.im 6e-5)))
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9600.0) || !(y_46_im <= 6e-5)) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else {
		tmp = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9600.0) || !(y_46_im <= 6e-5))
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9600.0], N[Not[LessEqual[y$46$im, 6e-5]], $MachinePrecision]], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9600 or 6.00000000000000015e-5 < y.im

   1. Initial program 56.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

      7. div-add N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. mul-1-neg N/A

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

      22. lower-neg.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 79.2%

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

   if -9600 < y.im < 6.00000000000000015e-5

   1. Initial program 74.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 74.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 84.3

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

   10. Applied rewrites 84.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9600 \lor \neg \left(y.im \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+16} \lor \neg \left(y.im \leq 2.1 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1e+16) (not (<= y.im 2.1e-9)))
   (/ (- 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_im <= -1e+16) || !(y_46_im <= 2.1e-9)) {
		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_46im <= (-1d+16)) .or. (.not. (y_46im <= 2.1d-9))) 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_im <= -1e+16) || !(y_46_im <= 2.1e-9)) {
		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_im <= -1e+16) or not (y_46_im <= 2.1e-9):
		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_im <= -1e+16) || !(y_46_im <= 2.1e-9))
		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_im <= -1e+16) || ~((y_46_im <= 2.1e-9)))
		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[Or[LessEqual[y$46$im, -1e+16], N[Not[LessEqual[y$46$im, 2.1e-9]], $MachinePrecision]], 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.im < -1e16 or 2.10000000000000019e-9 < y.im

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -1e16 < y.im < 2.10000000000000019e-9

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+16} \lor \neg \left(y.im \leq 2.1 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.3% accurate, 3.2× 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 65.3%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 45.7

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

5. Applied rewrites 45.7%

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

Reproduce

herbie shell --seed 2025017 
(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))))