_divideComplex, real part

Percentage Accurate: 62.3% → 95.1%

Time: 3.9s

Alternatives: 14

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: 62.3% 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: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-x.re, \frac{1}{-\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)}, \frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma
  (- x.re)
  (/ 1.0 (- (fma (/ y.im y.re) y.im y.re)))
  (/ x.im (fma (/ y.re y.im) y.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(-x_46_re, (1.0 / -fma((y_46_im / y_46_re), y_46_im, y_46_re)), (x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im)));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(Float64(-x_46_re), Float64(1.0 / Float64(-fma(Float64(y_46_im / y_46_re), y_46_im, y_46_re))), Float64(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im)))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[((-x$46$re) * N[(1.0 / (-N[(N[(y$46$im / y$46$re), $MachinePrecision] * y$46$im + y$46$re), $MachinePrecision])), $MachinePrecision] + N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 62.4

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lift-*.f64 N/A

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

   21. lower-fma.f64 62.4

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

3. Applied rewrites 62.4%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. lower-+.f64 N/A

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

5. Applied rewrites 88.3%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. mult-flip N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 88.3

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lift-/.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 91.8

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

7. Applied rewrites 95.1%

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

Alternative 2: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.2e+187)
   (/ (fma y.im (/ x.im y.re) x.re) y.re)
   (if (<= y.re 1.9e+147)
     (+
      (/ x.re (+ y.re (/ (* y.im y.im) y.re)))
      (/ x.im (+ y.im (/ (* y.re y.re) y.im))))
     (/ (fma (/ y.im y.re) x.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 <= -6.2e+187) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_re <= 1.9e+147) {
		tmp = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / (y_46_im + ((y_46_re * y_46_re) / y_46_im)));
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, 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 <= -6.2e+187)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_re <= 1.9e+147)
		tmp = Float64(Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / Float64(y_46_im + Float64(Float64(y_46_re * y_46_re) / y_46_im))));
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, 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, -6.2e+187], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+147], N[(N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / N[(y$46$im + N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.20000000000000024e187

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.4

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

   6. Applied rewrites 54.4%

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

   if -6.20000000000000024e187 < y.re < 1.89999999999999985e147

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   if 1.89999999999999985e147 < y.re

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

Alternative 3: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.re}{x.im \cdot y.im}, y.re, 1\right) \cdot \frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (fma (/ x.re (* x.im y.im)) y.re 1.0)
          (/ x.im (fma (/ y.re y.im) y.re y.im)))))
   (if (<= y.im -2e-15)
     t_0
     (if (<= y.im 8.6e-159)
       (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
       (if (<= y.im 3.4e+49)
         (/
          1.0
          (/ (fma y.im y.im (* y.re y.re)) (fma y.im x.im (* y.re x.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_re / (x_46_im * y_46_im)), y_46_re, 1.0) * (x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -2e-15) {
		tmp = t_0;
	} else if (y_46_im <= 8.6e-159) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.4e+49) {
		tmp = 1.0 / (fma(y_46_im, y_46_im, (y_46_re * y_46_re)) / fma(y_46_im, x_46_im, (y_46_re * x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_re / Float64(x_46_im * y_46_im)), y_46_re, 1.0) * Float64(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2e-15)
		tmp = t_0;
	elseif (y_46_im <= 8.6e-159)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.4e+49)
		tmp = Float64(1.0 / Float64(fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)) / fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re / N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] * y$46$re + 1.0), $MachinePrecision] * N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2e-15], t$95$0, If[LessEqual[y$46$im, 8.6e-159], 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$46$im, 3.4e+49], N[(1.0 / N[(N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.0000000000000002e-15 or 3.4000000000000001e49 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 65.7%

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

   if -2.0000000000000002e-15 < y.im < 8.6e-159

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

   if 8.6e-159 < y.im < 3.4000000000000001e49

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.0

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

      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 62.0

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

      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 62.0

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

      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 62.0

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

   3. Applied rewrites 62.0%

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

Alternative 4: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{-\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)}, \frac{x.im}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e-43)
   (+ (/ x.re (+ y.re (/ (* y.im y.im) y.re))) (/ x.im y.im))
   (if (<= y.im 8.6e-159)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (if (<= y.im 5e+77)
       (/ 1.0 (/ (fma y.im y.im (* y.re y.re)) (fma y.im x.im (* y.re x.re))))
       (fma
        (- x.re)
        (/ 1.0 (- (fma (/ y.im y.re) y.im 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 <= -9e-43) {
		tmp = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	} else if (y_46_im <= 8.6e-159) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 5e+77) {
		tmp = 1.0 / (fma(y_46_im, y_46_im, (y_46_re * y_46_re)) / fma(y_46_im, x_46_im, (y_46_re * x_46_re)));
	} else {
		tmp = fma(-x_46_re, (1.0 / -fma((y_46_im / y_46_re), y_46_im, y_46_re)), (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 <= -9e-43)
		tmp = Float64(Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / y_46_im));
	elseif (y_46_im <= 8.6e-159)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 5e+77)
		tmp = Float64(1.0 / Float64(fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)) / fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re))));
	else
		tmp = fma(Float64(-x_46_re), Float64(1.0 / Float64(-fma(Float64(y_46_im / y_46_re), y_46_im, y_46_re))), Float64(x_46_im / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9e-43], N[(N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.6e-159], 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$46$im, 5e+77], N[(1.0 / N[(N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x$46$re) * N[(1.0 / (-N[(N[(y$46$im / y$46$re), $MachinePrecision] * y$46$im + y$46$re), $MachinePrecision])), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.0000000000000005e-43

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.2%

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

   if -9.0000000000000005e-43 < y.im < 8.6e-159

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

   if 8.6e-159 < y.im < 5.00000000000000004e77

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.0

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

      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 62.0

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

      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 62.0

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

      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 62.0

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

   3. Applied rewrites 62.0%

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

   if 5.00000000000000004e77 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 88.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 91.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 61.7%

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

Alternative 5: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right) \cdot \frac{1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{-\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)}, \frac{x.im}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e-43)
   (+ (/ x.re (+ y.re (/ (* y.im y.im) y.re))) (/ x.im y.im))
   (if (<= y.im 8.6e-159)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (if (<= y.im 5e+77)
       (* (fma y.im x.im (* y.re x.re)) (/ 1.0 (fma y.im y.im (* y.re y.re))))
       (fma
        (- x.re)
        (/ 1.0 (- (fma (/ y.im y.re) y.im 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 <= -9e-43) {
		tmp = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	} else if (y_46_im <= 8.6e-159) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 5e+77) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) * (1.0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = fma(-x_46_re, (1.0 / -fma((y_46_im / y_46_re), y_46_im, y_46_re)), (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 <= -9e-43)
		tmp = Float64(Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / y_46_im));
	elseif (y_46_im <= 8.6e-159)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 5e+77)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) * Float64(1.0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = fma(Float64(-x_46_re), Float64(1.0 / Float64(-fma(Float64(y_46_im / y_46_re), y_46_im, y_46_re))), Float64(x_46_im / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9e-43], N[(N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.6e-159], 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$46$im, 5e+77], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x$46$re) * N[(1.0 / (-N[(N[(y$46$im / y$46$re), $MachinePrecision] * y$46$im + y$46$re), $MachinePrecision])), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.0000000000000005e-43

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.2%

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

   if -9.0000000000000005e-43 < y.im < 8.6e-159

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

   if 8.6e-159 < y.im < 5.00000000000000004e77

   1. Initial program 62.3%

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

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 62.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 62.1

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

   3. Applied rewrites 62.1%

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

   if 5.00000000000000004e77 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 88.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 91.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 61.7%

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

Alternative 6: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{-\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)}, \frac{x.im}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e-43)
   (+ (/ x.re (+ y.re (/ (* y.im y.im) y.re))) (/ x.im y.im))
   (if (<= y.im 8.6e-159)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (if (<= y.im 5e+77)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (fma
        (- x.re)
        (/ 1.0 (- (fma (/ y.im y.re) y.im 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 <= -9e-43) {
		tmp = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	} else if (y_46_im <= 8.6e-159) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 5e+77) {
		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 {
		tmp = fma(-x_46_re, (1.0 / -fma((y_46_im / y_46_re), y_46_im, y_46_re)), (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 <= -9e-43)
		tmp = Float64(Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / y_46_im));
	elseif (y_46_im <= 8.6e-159)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 5e+77)
		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)));
	else
		tmp = fma(Float64(-x_46_re), Float64(1.0 / Float64(-fma(Float64(y_46_im / y_46_re), y_46_im, y_46_re))), Float64(x_46_im / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9e-43], N[(N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.6e-159], 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$46$im, 5e+77], 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], N[((-x$46$re) * N[(1.0 / (-N[(N[(y$46$im / y$46$re), $MachinePrecision] * y$46$im + y$46$re), $MachinePrecision])), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.0000000000000005e-43

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.2%

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

   if -9.0000000000000005e-43 < y.im < 8.6e-159

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

   if 8.6e-159 < y.im < 5.00000000000000004e77

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.3

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

      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 62.3

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

      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 62.3

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

   3. Applied rewrites 62.3%

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

   if 5.00000000000000004e77 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 88.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 91.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 61.7%

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

Alternative 7: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re (+ y.re (/ (* y.im y.im) y.re))) (/ x.im y.im))))
   (if (<= y.im -9e-43)
     t_0
     (if (<= y.im 8.6e-159)
       (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
       (if (<= y.im 5e+77)
         (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -9e-43) {
		tmp = t_0;
	} else if (y_46_im <= 8.6e-159) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 5e+77) {
		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 {
		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(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -9e-43)
		tmp = t_0;
	elseif (y_46_im <= 8.6e-159)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 5e+77)
		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)));
	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[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9e-43], t$95$0, If[LessEqual[y$46$im, 8.6e-159], 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$46$im, 5e+77], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.0000000000000005e-43 or 5.00000000000000004e77 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.2%

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

   if -9.0000000000000005e-43 < y.im < 8.6e-159

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

   if 8.6e-159 < y.im < 5.00000000000000004e77

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.3

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

      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 62.3

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

      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 62.3

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

   3. Applied rewrites 62.3%

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

Alternative 8: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}} + \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re (+ y.re (/ (* y.im y.im) y.re))) (/ x.im y.im))))
   (if (<= y.im -9e-43)
     t_0
     (if (<= y.im 4.2e+44) (/ (+ 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 = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -9e-43) {
		tmp = t_0;
	} else if (y_46_im <= 4.2e+44) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / (y_46re + ((y_46im * y_46im) / y_46re))) + (x_46im / y_46im)
    if (y_46im <= (-9d-43)) then
        tmp = t_0
    else if (y_46im <= 4.2d+44) then
        tmp = (x_46re + ((x_46im * y_46im) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -9e-43) {
		tmp = t_0;
	} else if (y_46_im <= 4.2e+44) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im)
	tmp = 0
	if y_46_im <= -9e-43:
		tmp = t_0
	elif y_46_im <= 4.2e+44:
		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(Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re))) + Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -9e-43)
		tmp = t_0;
	elseif (y_46_im <= 4.2e+44)
		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

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))) + (x_46_im / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -9e-43)
		tmp = t_0;
	elseif (y_46_im <= 4.2e+44)
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9e-43], t$95$0, If[LessEqual[y$46$im, 4.2e+44], 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 2 regimes

2. if y.im < -9.0000000000000005e-43 or 4.19999999999999974e44 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.2%

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

   if -9.0000000000000005e-43 < y.im < 4.19999999999999974e44

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

Alternative 9: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.6e+42)
   (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
   (if (<= y.im 1.7e+82)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (/ x.im (fma (/ y.re y.im) y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.6e+42) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_im <= 1.7e+82) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.6e+42)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.7e+82)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.6e+42], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+82], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.5999999999999999e42

   1. Initial program 62.3%

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

   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 51.7

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

   4. Applied rewrites 51.7%

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

   if -5.5999999999999999e42 < y.im < 1.69999999999999997e82

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

   if 1.69999999999999997e82 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 53.6

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 56.2%

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

Alternative 10: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (fma (/ y.re y.im) y.re y.im))))
   (if (<= y.im -1.05e+26)
     t_0
     (if (<= y.im 1.7e+82) (/ (fma (/ y.im y.re) x.im x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.05e+26) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e+82) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_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(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.05e+26)
		tmp = t_0;
	elseif (y_46_im <= 1.7e+82)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+26], t$95$0, If[LessEqual[y$46$im, 1.7e+82], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.05e26 or 1.69999999999999997e82 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 53.6

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 56.2%

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

   if -1.05e26 < y.im < 1.69999999999999997e82

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

Alternative 11: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (fma (/ y.re y.im) y.re y.im))))
   (if (<= y.im -1.05e+26)
     t_0
     (if (<= y.im 1.7e+82) (/ (fma y.im (/ x.im y.re) x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.05e+26) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e+82) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_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(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.05e+26)
		tmp = t_0;
	elseif (y_46_im <= 1.7e+82)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+26], t$95$0, If[LessEqual[y$46$im, 1.7e+82], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.05e26 or 1.69999999999999997e82 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 53.6

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 56.2%

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

   if -1.05e26 < y.im < 1.69999999999999997e82

   1. Initial program 62.3%

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

   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 53.3

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

   4. Applied rewrites 53.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.4

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

   6. Applied rewrites 54.4%

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

Alternative 12: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (fma (/ y.re y.im) y.re y.im))))
   (if (<= y.im -9e+24) t_0 (if (<= y.im 1.75e-141) (/ x.re y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im / fma((y_46_re / y_46_im), y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -9e+24) {
		tmp = t_0;
	} else if (y_46_im <= 1.75e-141) {
		tmp = x_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(x_46_im / fma(Float64(y_46_re / y_46_im), y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -9e+24)
		tmp = t_0;
	elseif (y_46_im <= 1.75e-141)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * y$46$re + y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9e+24], t$95$0, If[LessEqual[y$46$im, 1.75e-141], N[(x$46$re / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.00000000000000039e24 or 1.7500000000000001e-141 < y.im

   1. Initial program 62.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 62.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 62.4

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

   3. Applied rewrites 62.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 53.6

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 56.2%

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

   if -9.00000000000000039e24 < y.im < 1.7500000000000001e-141

   1. Initial program 62.3%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.7

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

   4. Applied rewrites 43.7%

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

Alternative 13: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.45e+41)
   (/ x.im y.im)
   (if (<= y.im 1.7e+82) (/ 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 <= -1.45e+41) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.7e+82) {
		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 <= (-1.45d+41)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.7d+82) 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 <= -1.45e+41) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.7e+82) {
		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 <= -1.45e+41:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.7e+82:
		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 <= -1.45e+41)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.7e+82)
		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 <= -1.45e+41)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.7e+82)
		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, -1.45e+41], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+82], 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 < -1.44999999999999994e41 or 1.69999999999999997e82 < y.im

   1. Initial program 62.3%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 42.0

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

   4. Applied rewrites 42.0%

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

   if -1.44999999999999994e41 < y.im < 1.69999999999999997e82

   1. Initial program 62.3%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.7

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

   4. Applied rewrites 43.7%

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

Alternative 14: 42.0% 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 62.3%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 42.0

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

4. Applied rewrites 42.0%

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

Reproduce

herbie shell --seed 2025156 
(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))))