_divideComplex, imaginary part

Percentage Accurate: 61.0% → 81.6%

Time: 4.4s

Alternatives: 11

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: 61.0% 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: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{x.re}, -y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9.5e+136)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im -3.5e-18)
     (*
      (/ (fma x.im (/ y.re x.re) (- y.im)) (fma y.im y.im (* y.re y.re)))
      x.re)
     (if (<= y.im 5.6e-123)
       (/ (fma x.re (/ y.im y.re) (- x.im)) (- y.re))
       (if (<= y.im 2e+146)
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (/ (fma 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_im <= -9.5e+136) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -3.5e-18) {
		tmp = (fma(x_46_im, (y_46_re / x_46_re), -y_46_im) / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_re;
	} else if (y_46_im <= 5.6e-123) {
		tmp = fma(x_46_re, (y_46_im / y_46_re), -x_46_im) / -y_46_re;
	} else if (y_46_im <= 2e+146) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = fma(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_im <= -9.5e+136)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -3.5e-18)
		tmp = Float64(Float64(fma(x_46_im, Float64(y_46_re / x_46_re), Float64(-y_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_re);
	elseif (y_46_im <= 5.6e-123)
		tmp = Float64(fma(x_46_re, Float64(y_46_im / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	elseif (y_46_im <= 2e+146)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-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[LessEqual[y$46$im, -9.5e+136], 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, -3.5e-18], N[(N[(N[(x$46$im * N[(y$46$re / x$46$re), $MachinePrecision] + (-y$46$im)), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.6e-123], N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], If[LessEqual[y$46$im, 2e+146], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.49999999999999907e136

   1. Initial program 36.8%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 85.3

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

   8. Applied rewrites 85.3%

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

   if -9.49999999999999907e136 < y.im < -3.4999999999999999e-18

   1. Initial program 59.4%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.1%

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

   if -3.4999999999999999e-18 < y.im < 5.5999999999999998e-123

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if 5.5999999999999998e-123 < y.im < 1.99999999999999987e146

   1. Initial program 79.4%

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

   if 1.99999999999999987e146 < y.im

   1. Initial program 23.3%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{x.re}, -y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -180:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -180.0)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 5.6e-123)
     (/ (fma x.re (/ y.im y.re) (- x.im)) (- y.re))
     (if (<= y.im 2e+146)
       (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (fma 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_im <= -180.0) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 5.6e-123) {
		tmp = fma(x_46_re, (y_46_im / y_46_re), -x_46_im) / -y_46_re;
	} else if (y_46_im <= 2e+146) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = fma(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_im <= -180.0)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 5.6e-123)
		tmp = Float64(fma(x_46_re, Float64(y_46_im / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	elseif (y_46_im <= 2e+146)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-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[LessEqual[y$46$im, -180.0], 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, 5.6e-123], N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], If[LessEqual[y$46$im, 2e+146], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -180

   1. Initial program 44.8%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -180 < y.im < 5.5999999999999998e-123

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 5.5999999999999998e-123 < y.im < 1.99999999999999987e146

   1. Initial program 79.4%

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

   if 1.99999999999999987e146 < y.im

   1. Initial program 23.3%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -180:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.15 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (- x.re) y.im (* y.re x.im)) (* y.re y.re))))
   (if (<= y.re -1e+116)
     (/ x.im y.re)
     (if (<= y.re -2.15e-93)
       t_0
       (if (<= y.re 1.9e-44)
         (/ (- x.re) y.im)
         (if (<= y.re 3.5e+146) t_0 (/ 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(-x_46_re, y_46_im, (y_46_re * x_46_im)) / (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -1e+116) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.15e-93) {
		tmp = t_0;
	} else if (y_46_re <= 1.9e-44) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 3.5e+146) {
		tmp = t_0;
	} 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)
	t_0 = Float64(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1e+116)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -2.15e-93)
		tmp = t_0;
	elseif (y_46_re <= 1.9e-44)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 3.5e+146)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1e+116], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.15e-93], t$95$0, If[LessEqual[y$46$re, 1.9e-44], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+146], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.00000000000000002e116 or 3.5000000000000001e146 < y.re

   1. Initial program 33.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -1.00000000000000002e116 < y.re < -2.14999999999999981e-93 or 1.9e-44 < y.re < 3.5000000000000001e146

   1. Initial program 77.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. cancel-sign-sub-inv N/A

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

      5. lower-/.f64 N/A

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

      6. cancel-sign-sub-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -2.14999999999999981e-93 < y.re < 1.9e-44

   1. Initial program 65.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 72.1

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

   5. Applied rewrites 72.1%

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

Alternative 4: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))))
   (if (<= y.re -5.4e+153)
     (/ x.im y.re)
     (if (<= y.re -3.6e-109)
       t_0
       (if (<= y.re 3.2e-97)
         (/ (- x.re) y.im)
         (if (<= y.re 1.55e+146) t_0 (/ 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 = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_re <= -5.4e+153) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.6e-109) {
		tmp = t_0;
	} else if (y_46_re <= 3.2e-97) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.55e+146) {
		tmp = t_0;
	} 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)
	t_0 = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_re <= -5.4e+153)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.6e-109)
		tmp = t_0;
	elseif (y_46_re <= 3.2e-97)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.55e+146)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.4e+153], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.6e-109], t$95$0, If[LessEqual[y$46$re, 3.2e-97], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.55e+146], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.4000000000000001e153 or 1.5500000000000001e146 < y.re

   1. Initial program 31.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -5.4000000000000001e153 < y.re < -3.6000000000000001e-109 or 3.1999999999999998e-97 < y.re < 1.5500000000000001e146

   1. Initial program 77.6%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -3.6000000000000001e-109 < y.re < 3.1999999999999998e-97

   1. Initial program 63.5%

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 5: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1080000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1080000000000.0)
   (/ x.im y.re)
   (if (<= y.re 1.85e+20)
     (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
     (if (<= y.re 7e+145)
       (/ (fma (- x.re) y.im (* y.re x.im)) (* y.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 <= -1080000000000.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.85e+20) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7e+145) {
		tmp = fma(-x_46_re, y_46_im, (y_46_re * x_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1080000000000.0)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.85e+20)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 7e+145)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1080000000000.0], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+20], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7e+145], N[(N[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.08e12 or 7.0000000000000002e145 < y.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -1.08e12 < y.re < 1.85e20

   1. Initial program 69.0%

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

   3. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 76.7

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

   8. Applied rewrites 76.7%

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

   if 1.85e20 < y.re < 7.0000000000000002e145

   1. Initial program 81.7%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. cancel-sign-sub-inv N/A

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

      5. lower-/.f64 N/A

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

      6. cancel-sign-sub-inv N/A

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

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

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -180:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -180.0)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 1.6e+92)
     (/ (fma x.re (/ y.im y.re) (- x.im)) (- y.re))
     (/ (fma 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_im <= -180.0) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 1.6e+92) {
		tmp = fma(x_46_re, (y_46_im / y_46_re), -x_46_im) / -y_46_re;
	} else {
		tmp = fma(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_im <= -180.0)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 1.6e+92)
		tmp = Float64(fma(x_46_re, Float64(y_46_im / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-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[LessEqual[y$46$im, -180.0], 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, 1.6e+92], N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -180

   1. Initial program 44.8%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -180 < y.im < 1.60000000000000013e92

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 1.60000000000000013e92 < y.im

   1. Initial program 35.5%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -180:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.im}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -180 \lor \neg \left(y.im \leq 1.6 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -180.0) (not (<= y.im 1.6e+92)))
   (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
   (/ (- x.im (/ (* y.im x.re) 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 <= -180.0) || !(y_46_im <= 1.6e+92)) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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 <= -180.0) || !(y_46_im <= 1.6e+92))
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / 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, -180.0], N[Not[LessEqual[y$46$im, 1.6e+92]], $MachinePrecision]], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -180 or 1.60000000000000013e92 < y.im

   1. Initial program 40.7%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -180 < y.im < 1.60000000000000013e92

   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

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -180 \lor \neg \left(y.im \leq 1.6 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-92} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8e-92) (not (<= y.re 1.7e+20)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ (- (/ (* y.re x.im) 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 <= -8e-92) || !(y_46_re <= 1.7e+20)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-8d-92)) .or. (.not. (y_46re <= 1.7d+20))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8e-92) || !(y_46_re <= 1.7e+20)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -8e-92) or not (y_46_re <= 1.7e+20):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = (((y_46_re * x_46_im) / 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 <= -8e-92) || !(y_46_re <= 1.7e+20))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -8e-92) || ~((y_46_re <= 1.7e+20)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -8e-92], N[Not[LessEqual[y$46$re, 1.7e+20]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.9999999999999999e-92 or 1.7e20 < y.re

   1. Initial program 55.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if -7.9999999999999999e-92 < y.re < 1.7e20

   1. Initial program 66.0%

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

   3. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 82.4

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

   8. Applied rewrites 82.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-92} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -180:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -180.0)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 1.6e+92)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (/ (fma 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_im <= -180.0) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 1.6e+92) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(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_im <= -180.0)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 1.6e+92)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-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[LessEqual[y$46$im, -180.0], 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, 1.6e+92], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -180

   1. Initial program 44.8%

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

   3. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -180 < y.im < 1.60000000000000013e92

   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

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if 1.60000000000000013e92 < y.im

   1. Initial program 35.5%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 80.8

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

   5. Applied rewrites 80.8%

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

Alternative 10: 63.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -800000000000 \lor \neg \left(y.re \leq 2.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -800000000000.0) (not (<= y.re 2.5e-44)))
   (/ x.im y.re)
   (/ (- 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 <= -800000000000.0) || !(y_46_re <= 2.5e-44)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-800000000000.0d0)) .or. (.not. (y_46re <= 2.5d-44))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -800000000000.0) || !(y_46_re <= 2.5e-44)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -800000000000.0) or not (y_46_re <= 2.5e-44):
		tmp = x_46_im / y_46_re
	else:
		tmp = -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 <= -800000000000.0) || !(y_46_re <= 2.5e-44))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -800000000000.0) || ~((y_46_re <= 2.5e-44)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -800000000000.0], N[Not[LessEqual[y$46$re, 2.5e-44]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8e11 or 2.50000000000000019e-44 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -8e11 < y.re < 2.50000000000000019e-44

   1. Initial program 68.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -800000000000 \lor \neg \left(y.re \leq 2.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.9% 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 59.6%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 41.2

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

5. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2025064 
(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))))