_divideComplex, imaginary part

Percentage Accurate: 62.1% → 80.9%

Time: 5.6s

Alternatives: 9

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% 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: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{-y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-140}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t\_0} - x.re \cdot \frac{y.im}{t\_0}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -1.45e+90)
     (/ (fma (/ y.im (- y.re)) x.re x.im) y.re)
     (if (<= y.re -1.3e-140)
       (- (* x.im (/ y.re t_0)) (* x.re (/ y.im t_0)))
       (if (<= y.re 2.7e+88)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -1.45e+90) {
		tmp = fma((y_46_im / -y_46_re), x_46_re, x_46_im) / y_46_re;
	} else if (y_46_re <= -1.3e-140) {
		tmp = (x_46_im * (y_46_re / t_0)) - (x_46_re * (y_46_im / t_0));
	} else if (y_46_re <= 2.7e+88) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.45e+90)
		tmp = Float64(fma(Float64(y_46_im / Float64(-y_46_re)), x_46_re, x_46_im) / y_46_re);
	elseif (y_46_re <= -1.3e-140)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / t_0)) - Float64(x_46_re * Float64(y_46_im / t_0)));
	elseif (y_46_re <= 2.7e+88)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+90], N[(N[(N[(y$46$im / (-y$46$re)), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.3e-140], N[(N[(x$46$im * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+88], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.4500000000000001e90

   1. Initial program 37.8%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 41.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 41.1

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

   4. Applied rewrites 41.1%

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

   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

   7. Applied rewrites 90.6%

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

   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

   10. Applied rewrites 90.7%

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

   if -1.4500000000000001e90 < y.re < -1.2999999999999999e-140

   1. Initial program 82.0%

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

      1. lift-/.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 83.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -1.2999999999999999e-140 < y.re < 2.70000000000000016e88

   1. Initial program 66.1%

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

   3. Taylor expanded in y.re around 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

   5. Applied rewrites 80.3%

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

   if 2.70000000000000016e88 < y.re

   1. Initial program 35.3%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 43.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 43.0

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

   4. Applied rewrites 43.0%

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

   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

   7. Applied rewrites 89.4%

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

Alternative 2: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.56 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{-y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.56e+90)
   (/ (fma (/ y.im (- y.re)) x.re x.im) y.re)
   (if (<= y.re -1.2e-129)
     (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 2.7e+88)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.56e+90) {
		tmp = fma((y_46_im / -y_46_re), x_46_re, x_46_im) / y_46_re;
	} else if (y_46_re <= -1.2e-129) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.7e+88) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.56e+90)
		tmp = Float64(fma(Float64(y_46_im / Float64(-y_46_re)), x_46_re, x_46_im) / y_46_re);
	elseif (y_46_re <= -1.2e-129)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.7e+88)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-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, -1.56e+90], N[(N[(N[(y$46$im / (-y$46$re)), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.2e-129], 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], If[LessEqual[y$46$re, 2.7e+88], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.56000000000000004e90

   1. Initial program 36.6%

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

      1. lift-/.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 39.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 39.9

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

   4. Applied rewrites 39.9%

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

   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

   7. Applied rewrites 90.4%

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

   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

   10. Applied rewrites 90.5%

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

   if -1.56000000000000004e90 < y.re < -1.19999999999999994e-129

   1. Initial program 82.0%

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

   if -1.19999999999999994e-129 < y.re < 2.70000000000000016e88

   1. Initial program 66.4%

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

   3. 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

   5. Applied rewrites 80.5%

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

   if 2.70000000000000016e88 < y.re

   1. Initial program 35.3%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 43.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 43.0

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

   4. Applied rewrites 43.0%

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

   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

   7. Applied rewrites 89.4%

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

Alternative 3: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{-y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.2e+62)
   (/ (fma (/ y.im (- y.re)) x.re x.im) y.re)
   (if (<= y.re 2.7e+88)
     (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
     (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.2e+62) {
		tmp = fma((y_46_im / -y_46_re), x_46_re, x_46_im) / y_46_re;
	} else if (y_46_re <= 2.7e+88) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.2e+62)
		tmp = Float64(fma(Float64(y_46_im / Float64(-y_46_re)), x_46_re, x_46_im) / y_46_re);
	elseif (y_46_re <= 2.7e+88)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-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, -7.2e+62], N[(N[(N[(y$46$im / (-y$46$re)), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+88], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.2e62

   1. Initial program 43.3%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 46.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 46.0

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

   4. Applied rewrites 46.0%

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

   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

   7. Applied rewrites 87.0%

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

   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

   10. Applied rewrites 87.1%

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

   if -7.2e62 < y.re < 2.70000000000000016e88

   1. Initial program 70.8%

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

   3. Taylor expanded in y.re around 0

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

   5. Applied rewrites 76.6%

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

   if 2.70000000000000016e88 < y.re

   1. Initial program 35.3%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 43.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 43.0

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

   4. Applied rewrites 43.0%

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

   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

   7. Applied rewrites 89.4%

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

Alternative 4: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+62} \lor \neg \left(y.re \leq 2.7 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{-y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -7.2e+62) (not (<= y.re 2.7e+88)))
   (/ (fma (/ y.im (- y.re)) x.re x.im) y.re)
   (/ (- (/ (* x.im y.re) y.im) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -7.2e+62) || !(y_46_re <= 2.7e+88)) {
		tmp = fma((y_46_im / -y_46_re), x_46_re, x_46_im) / y_46_re;
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -7.2e+62) || !(y_46_re <= 2.7e+88))
		tmp = Float64(fma(Float64(y_46_im / Float64(-y_46_re)), x_46_re, x_46_im) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -7.2e+62], N[Not[LessEqual[y$46$re, 2.7e+88]], $MachinePrecision]], N[(N[(N[(y$46$im / (-y$46$re)), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.2e62 or 2.70000000000000016e88 < y.re

   1. Initial program 39.8%

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

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 44.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 44.7

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

   4. Applied rewrites 44.7%

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

   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

   7. Applied rewrites 88.1%

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

   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

   10. Applied rewrites 88.1%

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

   if -7.2e62 < y.re < 2.70000000000000016e88

   1. Initial program 70.8%

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

   3. Taylor expanded in y.re around 0

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

   5. Applied rewrites 76.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+62} \lor \neg \left(y.re \leq 2.7 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{-y.re}, x.re, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-29} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.12e-29) (not (<= y.im 3.5e+44)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ (* x.re y.im) y.re)) 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 <= -1.12e-29) || !(y_46_im <= 3.5e+44)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= (-1.12d-29)) .or. (.not. (y_46im <= 3.5d+44))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / 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 <= -1.12e-29) || !(y_46_im <= 3.5e+44)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= -1.12e-29) or not (y_46_im <= 3.5e+44):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= -1.12e-29) || !(y_46_im <= 3.5e+44))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / 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 <= -1.12e-29) || ~((y_46_im <= 3.5e+44)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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, -1.12e-29], N[Not[LessEqual[y$46$im, 3.5e+44]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.11999999999999995e-29 or 3.4999999999999999e44 < y.im

   1. Initial program 42.0%

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 77.4%

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

   if -1.11999999999999995e-29 < y.im < 3.4999999999999999e44

   1. Initial program 71.5%

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

   3. Taylor expanded in y.re around inf

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

   5. Applied rewrites 83.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-29} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+131} \lor \neg \left(y.im \leq 6 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.8e+131) (not (<= y.im 6e+44)))
   (/ (- x.re) y.im)
   (/ (- x.im (/ (* x.re y.im) y.re)) 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 <= -4.8e+131) || !(y_46_im <= 6e+44)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= (-4.8d+131)) .or. (.not. (y_46im <= 6d+44))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / 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 <= -4.8e+131) || !(y_46_im <= 6e+44)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= -4.8e+131) or not (y_46_im <= 6e+44):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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 <= -4.8e+131) || !(y_46_im <= 6e+44))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / 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 <= -4.8e+131) || ~((y_46_im <= 6e+44)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / 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, -4.8e+131], N[Not[LessEqual[y$46$im, 6e+44]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.7999999999999999e131 or 5.99999999999999974e44 < y.im

   1. Initial program 36.5%

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

   3. Taylor expanded in y.re around 0

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

   5. Applied rewrites 75.6%

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

   if -4.7999999999999999e131 < y.im < 5.99999999999999974e44

   1. Initial program 69.6%

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

   3. Taylor expanded in y.re around inf

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

   5. Applied rewrites 77.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+131} \lor \neg \left(y.im \leq 6 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -7.5e+150)
     t_0
     (if (<= y.im -5.5e-30)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 5.2e+44) (/ 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 <= -7.5e+150) {
		tmp = t_0;
	} else if (y_46_im <= -5.5e-30) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 5.2e+44) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-7.5d+150)) then
        tmp = t_0
    else if (y_46im <= (-5.5d-30)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    else if (y_46im <= 5.2d+44) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -7.5e+150) {
		tmp = t_0;
	} else if (y_46_im <= -5.5e-30) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 5.2e+44) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -7.5e+150:
		tmp = t_0
	elif y_46_im <= -5.5e-30:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	elif y_46_im <= 5.2e+44:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.5e+150)
		tmp = t_0;
	elseif (y_46_im <= -5.5e-30)
		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 <= 5.2e+44)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.5e+150)
		tmp = t_0;
	elseif (y_46_im <= -5.5e-30)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	elseif (y_46_im <= 5.2e+44)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+150], t$95$0, If[LessEqual[y$46$im, -5.5e-30], 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, 5.2e+44], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.4999999999999998e150 or 5.1999999999999998e44 < y.im

   1. Initial program 34.3%

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

   5. Applied rewrites 77.0%

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

   if -7.4999999999999998e150 < y.im < -5.49999999999999976e-30

   1. Initial program 63.7%

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

   5. Applied rewrites 57.8%

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

   if -5.49999999999999976e-30 < y.im < 5.1999999999999998e44

   1. Initial program 71.5%

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

   3. Taylor expanded in y.re around inf

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

   5. Applied rewrites 68.4%

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

Alternative 8: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+131} \lor \neg \left(y.im \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2e+131) (not (<= y.im 5.2e+44)))
   (/ (- 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 <= -2e+131) || !(y_46_im <= 5.2e+44)) {
		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 <= (-2d+131)) .or. (.not. (y_46im <= 5.2d+44))) 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 <= -2e+131) || !(y_46_im <= 5.2e+44)) {
		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 <= -2e+131) or not (y_46_im <= 5.2e+44):
		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 <= -2e+131) || !(y_46_im <= 5.2e+44))
		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 <= -2e+131) || ~((y_46_im <= 5.2e+44)))
		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, -2e+131], N[Not[LessEqual[y$46$im, 5.2e+44]], $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 < -1.9999999999999998e131 or 5.1999999999999998e44 < y.im

   1. Initial program 36.5%

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

   3. Taylor expanded in y.re around 0

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

   5. Applied rewrites 75.6%

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

   if -1.9999999999999998e131 < y.im < 5.1999999999999998e44

   1. Initial program 69.6%

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

   3. Taylor expanded in y.re around inf

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

   5. Applied rewrites 63.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+131} \lor \neg \left(y.im \leq 5.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.3% 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 58.2%

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

3. Taylor expanded in y.re around inf

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

5. Applied rewrites 46.7%

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

Reproduce

herbie shell --seed 2025019 
(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))))