_divideComplex, real part

Percentage Accurate: 63.0% → 84.1%

Time: 6.2s

Alternatives: 9

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: 63.0% 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: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(\frac{y.im}{t\_0}, x.im, \frac{x.re}{t\_0} \cdot y.re\right)\\ t_2 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -1.16 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma (/ y.im t_0) x.im (* (/ x.re t_0) y.re)))
        (t_2 (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))))
   (if (<= y.re -2.25e+122)
     t_2
     (if (<= y.re -1.16e-122)
       t_1
       (if (<= y.re 4.5e-40)
         (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
         (if (<= y.re 5.4e+107) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((y_46_im / t_0), x_46_im, ((x_46_re / t_0) * y_46_re));
	double t_2 = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -2.25e+122) {
		tmp = t_2;
	} else if (y_46_re <= -1.16e-122) {
		tmp = t_1;
	} else if (y_46_re <= 4.5e-40) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 5.4e+107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(y_46_im / t_0), x_46_im, Float64(Float64(x_46_re / t_0) * y_46_re))
	t_2 = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.25e+122)
		tmp = t_2;
	elseif (y_46_re <= -1.16e-122)
		tmp = t_1;
	elseif (y_46_re <= 4.5e-40)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 5.4e+107)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$im / t$95$0), $MachinePrecision] * x$46$im + N[(N[(x$46$re / t$95$0), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.25e+122], t$95$2, If[LessEqual[y$46$re, -1.16e-122], t$95$1, If[LessEqual[y$46$re, 4.5e-40], 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$46$re, 5.4e+107], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.24999999999999999e122 or 5.4000000000000003e107 < y.re

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   4. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   5. 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 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. add-to-fraction N/A

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

      7. +-commutative N/A

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

      8. div-add N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-rgt-in N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l/ N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 53.0%

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

   if -2.24999999999999999e122 < y.re < -1.16000000000000001e-122 or 4.5000000000000001e-40 < y.re < 5.4000000000000003e107

   1. Initial program 63.0%

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

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

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

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

      10. div-add N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity N/A

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

      17. associate-*r/ N/A

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

      18. mult-flip N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 62.5%

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

   if -1.16000000000000001e-122 < y.re < 4.5000000000000001e-40

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

Alternative 2: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{y.im}{x.re}}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -8e+80)
   (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))
   (if (<= y.im -1.8e-120)
     (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
     (if (<= y.im 1.3e+22)
       (/ (fma (/ y.im y.re) x.im x.re) y.re)
       (/ (fma (/ 1.0 (/ y.im 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 <= -8e+80) {
		tmp = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	} else if (y_46_im <= -1.8e-120) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 1.3e+22) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((1.0 / (y_46_im / x_46_re)), 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 <= -8e+80)
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im));
	elseif (y_46_im <= -1.8e-120)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 1.3e+22)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(y_46_im / x_46_re)), y_46_re, 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, -8e+80], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.8e-120], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+22], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8e80

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if -8e80 < y.im < -1.8000000000000001e-120

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   if -1.8000000000000001e-120 < y.im < 1.3e22

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   4. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   5. 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 52.6

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

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   7. 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 54.4

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

   8. Applied rewrites 54.4%

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

   if 1.3e22 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 3: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -32000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{y.im}{x.re}}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -32000.0)
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (if (<= y.im 1.3e+22)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (/ (fma (/ 1.0 (/ y.im 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 <= -32000.0) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_im <= 1.3e+22) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((1.0 / (y_46_im / x_46_re)), 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 <= -32000.0)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_im <= 1.3e+22)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(y_46_im / x_46_re)), y_46_re, 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, -32000.0], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+22], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -32000

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

   if -32000 < y.im < 1.3e22

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   4. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   5. 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 52.6

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

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   7. 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 54.4

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

   8. Applied rewrites 54.4%

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

   if 1.3e22 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-special-/.f64 N/A

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

      4. lower-special-/.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 4: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -32000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -32000.0)
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (if (<= y.im 1.3e+22)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (/ (fma (/ y.re y.im) x.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 <= -32000.0) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_im <= 1.3e+22) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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 <= -32000.0)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_im <= 1.3e+22)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, 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, -32000.0], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+22], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -32000

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

   if -32000 < y.im < 1.3e22

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   4. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   5. 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 52.6

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

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   7. 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 54.4

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

   8. Applied rewrites 54.4%

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

   if 1.3e22 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. associate-*l* N/A

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

      16. inv-pow N/A

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

      17. pow-plus N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.re + x.im \cdot {y.im}^{0}}{y.im} \]

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 5: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -32000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -32000.0)
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (if (<= y.im 1.3e+22)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (/ (fma (/ y.re y.im) x.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 <= -32000.0) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_im <= 1.3e+22) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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 <= -32000.0)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_im <= 1.3e+22)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, 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, -32000.0], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+22], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -32000

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

   if -32000 < y.im < 1.3e22

   1. Initial program 63.0%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-rgt-identity N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. sqr-abs-rev N/A

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

      19. sqr-neg-rev N/A

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

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

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

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

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

      22. lift-*.f64 N/A

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

      23. distribute-lft-neg-in N/A

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

      24. distribute-rgt-neg-out N/A

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

      25. sqr-neg-rev N/A

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

      26. sqr-neg-rev N/A

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

      27. sqr-abs-rev N/A

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

   3. Applied rewrites 63.0%

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

   4. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   5. 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 52.6

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

   6. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   7. 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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if 1.3e22 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. associate-*l* N/A

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

      16. inv-pow N/A

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

      17. pow-plus N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.re + x.im \cdot {y.im}^{0}}{y.im} \]

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 6: 70.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 26:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e-84)
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (if (<= y.im 26.0) (/ x.re y.re) (/ (fma (/ y.re y.im) x.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 <= -2.1e-84) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_im <= 26.0) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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 <= -2.1e-84)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_im <= 26.0)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, 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, -2.1e-84], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 26.0], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.09999999999999998e-84

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

   if -2.09999999999999998e-84 < y.im < 26

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   if 26 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. associate-*l* N/A

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

      16. inv-pow N/A

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

      17. pow-plus N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{\frac{y.re}{y.im} \cdot x.re + x.im \cdot {y.im}^{0}}{y.im} \]

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 7: 70.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 26:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -2.1e-84) t_0 (if (<= y.im 26.0) (/ 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 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -2.1e-84) {
		tmp = t_0;
	} else if (y_46_im <= 26.0) {
		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(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.1e-84)
		tmp = t_0;
	elseif (y_46_im <= 26.0)
		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[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.1e-84], t$95$0, If[LessEqual[y$46$im, 26.0], N[(x$46$re / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.09999999999999998e-84 or 26 < y.im

   1. Initial program 63.0%

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

   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 52.5

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

   4. Applied rewrites 52.5%

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

   6. Applied rewrites 53.7%

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

   if -2.09999999999999998e-84 < y.im < 26

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

Alternative 8: 63.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.7e-57)
   (/ x.im y.im)
   (if (<= y.im 1.35e+22) (/ 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 <= -2.7e-57) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.35e+22) {
		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 <= (-2.7d-57)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.35d+22) 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 <= -2.7e-57) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.35e+22) {
		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 <= -2.7e-57:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.35e+22:
		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 <= -2.7e-57)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.35e+22)
		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 <= -2.7e-57)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.35e+22)
		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, -2.7e-57], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+22], 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 < -2.7000000000000002e-57 or 1.3500000000000001e22 < y.im

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 42.6

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

   4. Applied rewrites 42.6%

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

   if -2.7000000000000002e-57 < y.im < 1.3500000000000001e22

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

Alternative 9: 42.6% 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 63.0%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 42.6

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

4. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2025151 
(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))))