_divideComplex, real part

Percentage Accurate: 61.2% → 93.7%

Time: 4.0s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}\\ t_1 := \frac{1}{\mathsf{fma}\left(y.re, \frac{1}{x.re}, t\_0\right)}\\ \mathbf{if}\;x.re \leq -1.4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.re \leq 2.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.re, \frac{y.re}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (fma (/ y.re y.im) x.re x.im)))
        (t_1 (/ 1.0 (fma y.re (/ 1.0 x.re) t_0))))
   (if (<= x.re -1.4e+20)
     t_1
     (if (<= x.re 2.1e+74)
       (/ 1.0 (fma y.re (/ y.re (fma x.re y.re (* x.im y.im))) t_0))
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im / fma((y_46_re / y_46_im), x_46_re, x_46_im);
	double t_1 = 1.0 / fma(y_46_re, (1.0 / x_46_re), t_0);
	double tmp;
	if (x_46_re <= -1.4e+20) {
		tmp = t_1;
	} else if (x_46_re <= 2.1e+74) {
		tmp = 1.0 / fma(y_46_re, (y_46_re / fma(x_46_re, y_46_re, (x_46_im * y_46_im))), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im / fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im))
	t_1 = Float64(1.0 / fma(y_46_re, Float64(1.0 / x_46_re), t_0))
	tmp = 0.0
	if (x_46_re <= -1.4e+20)
		tmp = t_1;
	elseif (x_46_re <= 2.1e+74)
		tmp = Float64(1.0 / fma(y_46_re, Float64(y_46_re / fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im))), t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(y$46$re * N[(1.0 / x$46$re), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.4e+20], t$95$1, If[LessEqual[x$46$re, 2.1e+74], N[(1.0 / N[(y$46$re * N[(y$46$re / N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.4e20 or 2.0999999999999999e74 < x.re

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-/.f64 76.1

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

   8. Applied rewrites 76.1%

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

   if -1.4e20 < x.re < 2.0999999999999999e74

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y.re, \frac{y.re}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}, \frac{y.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(y.re, \frac{1}{x.re}, \frac{y.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}\right)}\\ \mathbf{if}\;x.re \leq -1.05 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.re \leq 1.62 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.re, \frac{y.re}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}, \frac{y.im}{x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (fma y.re (/ 1.0 x.re) (/ y.im (fma (/ y.re y.im) x.re x.im))))))
   (if (<= x.re -1.05e-54)
     t_0
     (if (<= x.re 1.62e-65)
       (/ 1.0 (fma y.re (/ y.re (fma x.re y.re (* x.im y.im))) (/ y.im x.im)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / fma(y_46_re, (1.0 / x_46_re), (y_46_im / fma((y_46_re / y_46_im), x_46_re, x_46_im)));
	double tmp;
	if (x_46_re <= -1.05e-54) {
		tmp = t_0;
	} else if (x_46_re <= 1.62e-65) {
		tmp = 1.0 / fma(y_46_re, (y_46_re / fma(x_46_re, y_46_re, (x_46_im * y_46_im))), (y_46_im / x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / fma(y_46_re, Float64(1.0 / x_46_re), Float64(y_46_im / fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im))))
	tmp = 0.0
	if (x_46_re <= -1.05e-54)
		tmp = t_0;
	elseif (x_46_re <= 1.62e-65)
		tmp = Float64(1.0 / fma(y_46_re, Float64(y_46_re / fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im))), Float64(y_46_im / x_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[(y$46$re * N[(1.0 / x$46$re), $MachinePrecision] + N[(y$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.05e-54], t$95$0, If[LessEqual[x$46$re, 1.62e-65], N[(1.0 / N[(y$46$re * N[(y$46$re / N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.05e-54 or 1.6200000000000001e-65 < x.re

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-/.f64 76.1

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

   8. Applied rewrites 76.1%

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

   if -1.05e-54 < x.re < 1.6200000000000001e-65

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 68.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(y.re, \frac{y.re}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}, \frac{y.im}{\color{blue}{x.im}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im + t\_0}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.re, \frac{y.re}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}, \frac{y.im}{x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.re, \frac{1}{x.re}, \frac{y.im}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.re y.re) y.im)))
   (if (<= y.re -4.3e+42)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (if (<= y.re -1.75e-134)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (if (<= y.re 3.2e-131)
         (/ (+ x.im t_0) y.im)
         (if (<= y.re 6.6e+71)
           (/
            1.0
            (fma y.re (/ y.re (fma x.re y.re (* x.im y.im))) (/ y.im x.im)))
           (/ 1.0 (fma y.re (/ 1.0 x.re) (/ y.im t_0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -4.3e+42) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= -1.75e-134) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_re <= 3.2e-131) {
		tmp = (x_46_im + t_0) / y_46_im;
	} else if (y_46_re <= 6.6e+71) {
		tmp = 1.0 / fma(y_46_re, (y_46_re / fma(x_46_re, y_46_re, (x_46_im * y_46_im))), (y_46_im / x_46_im));
	} else {
		tmp = 1.0 / fma(y_46_re, (1.0 / x_46_re), (y_46_im / t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.3e+42)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= -1.75e-134)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 3.2e-131)
		tmp = Float64(Float64(x_46_im + t_0) / y_46_im);
	elseif (y_46_re <= 6.6e+71)
		tmp = Float64(1.0 / fma(y_46_re, Float64(y_46_re / fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im))), Float64(y_46_im / x_46_im)));
	else
		tmp = Float64(1.0 / fma(y_46_re, Float64(1.0 / x_46_re), Float64(y_46_im / t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -4.3e+42], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.75e-134], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.2e-131], N[(N[(x$46$im + t$95$0), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+71], N[(1.0 / N[(y$46$re * N[(y$46$re / N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$46$re * N[(1.0 / x$46$re), $MachinePrecision] + N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.2999999999999998e42

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.4

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

   4. Applied rewrites 53.4%

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

   if -4.2999999999999998e42 < y.re < -1.7499999999999999e-134

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -1.7499999999999999e-134 < y.re < 3.2e-131

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

   if 3.2e-131 < y.re < 6.5999999999999996e71

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 68.2%

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

   if 6.5999999999999996e71 < y.re

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-/.f64 76.1

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x.re around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 56.1

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

   11. Applied rewrites 56.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(y.re, \frac{1}{x.re}, \frac{y.im}{\color{blue}{\frac{x.re \cdot y.re}{y.im}}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im + t\_0}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.re, \frac{1}{x.re}, \frac{y.im}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.re y.re) y.im)))
   (if (<= y.re -4.3e+42)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (if (<= y.re -1.75e-134)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (if (<= y.re 3.5e+71)
         (/ (+ x.im t_0) y.im)
         (/ 1.0 (fma y.re (/ 1.0 x.re) (/ y.im t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -4.3e+42) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= -1.75e-134) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_re <= 3.5e+71) {
		tmp = (x_46_im + t_0) / y_46_im;
	} else {
		tmp = 1.0 / fma(y_46_re, (1.0 / x_46_re), (y_46_im / t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.3e+42)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= -1.75e-134)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 3.5e+71)
		tmp = Float64(Float64(x_46_im + t_0) / y_46_im);
	else
		tmp = Float64(1.0 / fma(y_46_re, Float64(1.0 / x_46_re), Float64(y_46_im / t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -4.3e+42], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.75e-134], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+71], N[(N[(x$46$im + t$95$0), $MachinePrecision] / y$46$im), $MachinePrecision], N[(1.0 / N[(y$46$re * N[(1.0 / x$46$re), $MachinePrecision] + N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.2999999999999998e42

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.4

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

   4. Applied rewrites 53.4%

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

   if -4.2999999999999998e42 < y.re < -1.7499999999999999e-134

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -1.7499999999999999e-134 < y.re < 3.4999999999999999e71

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

   if 3.4999999999999999e71 < y.re

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 61.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 61.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.0

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

   3. Applied rewrites 61.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. div-flip N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l/ N/A

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-/.f64 76.1

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x.re around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 56.1

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

   11. Applied rewrites 56.1%

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

Alternative 5: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
   (if (<= y.im -2.55e+148)
     t_0
     (if (<= y.im -6.4e-34)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (if (<= y.im 2.65e+72) (/ (+ x.re (/ (* x.im y.im) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -2.55e+148) {
		tmp = t_0;
	} else if (y_46_im <= -6.4e-34) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 2.65e+72) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.55e+148)
		tmp = t_0;
	elseif (y_46_im <= -6.4e-34)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 2.65e+72)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.55e+148], t$95$0, If[LessEqual[y$46$im, -6.4e-34], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+72], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.54999999999999993e148 or 2.6500000000000001e72 < y.im

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 52.6

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

   6. Applied rewrites 52.6%

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

   if -2.54999999999999993e148 < y.im < -6.40000000000000005e-34

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -6.40000000000000005e-34 < y.im < 2.6500000000000001e72

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.4

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

   4. Applied rewrites 53.4%

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

Alternative 6: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)))
   (if (<= y.re -7.2e+33)
     t_0
     (if (<= y.re 1.15e+86) (/ (fma (/ y.re y.im) x.re x.im) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.2e+33) {
		tmp = t_0;
	} else if (y_46_re <= 1.15e+86) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -7.2e+33)
		tmp = t_0;
	elseif (y_46_re <= 1.15e+86)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e+33], t$95$0, If[LessEqual[y$46$re, 1.15e+86], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.2000000000000005e33 or 1.14999999999999995e86 < y.re

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.4

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

   4. Applied rewrites 53.4%

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

   if -7.2000000000000005e33 < y.re < 1.14999999999999995e86

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.0

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

   6. Applied rewrites 53.0%

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

Alternative 7: 73.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.2e+33)
   (/ x.re y.re)
   (if (<= y.re 1.15e+86)
     (/ (fma (/ y.re y.im) x.re x.im) y.im)
     (/ x.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_re <= -7.2e+33) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e+86) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = x_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_re <= -7.2e+33)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.15e+86)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.2e+33], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+86], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.2000000000000005e33 or 1.14999999999999995e86 < y.re

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   if -7.2000000000000005e33 < y.re < 1.14999999999999995e86

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.0

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

   6. Applied rewrites 53.0%

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

Alternative 8: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.2e+33)
   (/ x.re y.re)
   (if (<= y.re 1.15e+86)
     (/ (fma y.re (/ x.re y.im) x.im) y.im)
     (/ x.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_re <= -7.2e+33) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e+86) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = x_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_re <= -7.2e+33)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.15e+86)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.2e+33], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+86], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.2000000000000005e33 or 1.14999999999999995e86 < y.re

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   if -7.2000000000000005e33 < y.re < 1.14999999999999995e86

   1. Initial program 61.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 9: 62.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.4e+90)
   (/ x.im y.im)
   (if (<= y.im 2.65e+72) (/ x.re y.re) (/ x.im 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 <= -1.4e+90) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.65e+72) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / 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_46im <= (-1.4d+90)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2.65d+72) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / 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_im <= -1.4e+90) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.65e+72) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / 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_im <= -1.4e+90:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2.65e+72:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / 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 <= -1.4e+90)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.65e+72)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / 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_im <= -1.4e+90)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2.65e+72)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.4e+90], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+72], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.4e90 or 2.6500000000000001e72 < y.im

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 41.5

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

   4. Applied rewrites 41.5%

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

   if -1.4e90 < y.im < 2.6500000000000001e72

   1. Initial program 61.2%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

Alternative 10: 41.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_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_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 61.2%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 41.5

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

4. Applied rewrites 41.5%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))