_divideComplex, real part

Percentage Accurate: 60.7% → 81.3%

Time: 3.8s

Alternatives: 12

Speedup: 1.6×

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: 60.7% 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: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(\frac{x.im}{y.re}, \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{x.re}, y.re\right)}{t\_0} \cdot x.re\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y.re, x.re, \frac{\left(y.re \cdot y.re\right) \cdot x.im}{-y.im}\right)}{y.im}, -1, -x.im\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ x.im y.re) (/ y.im y.re) (/ x.re y.re))))
   (if (<= y.re -1.25e+150)
     t_1
     (if (<= y.re -2.3e-83)
       (* (/ (fma x.im (/ y.im x.re) y.re) t_0) x.re)
       (if (<= y.re 7.5e-118)
         (/
          (fma
           (/ (fma y.re x.re (/ (* (* y.re y.re) x.im) (- y.im))) y.im)
           -1.0
           (- x.im))
          (- y.im))
         (if (<= y.re 5.1e+75) (/ (fma y.im x.im (* y.re x.re)) t_0) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_re), (y_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.25e+150) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-83) {
		tmp = (fma(x_46_im, (y_46_im / x_46_re), y_46_re) / t_0) * x_46_re;
	} else if (y_46_re <= 7.5e-118) {
		tmp = fma((fma(y_46_re, x_46_re, (((y_46_re * y_46_re) * x_46_im) / -y_46_im)) / y_46_im), -1.0, -x_46_im) / -y_46_im;
	} else if (y_46_re <= 5.1e+75) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(x_46_im / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.25e+150)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-83)
		tmp = Float64(Float64(fma(x_46_im, Float64(y_46_im / x_46_re), y_46_re) / t_0) * x_46_re);
	elseif (y_46_re <= 7.5e-118)
		tmp = Float64(fma(Float64(fma(y_46_re, x_46_re, Float64(Float64(Float64(y_46_re * y_46_re) * x_46_im) / Float64(-y_46_im))) / y_46_im), -1.0, Float64(-x_46_im)) / Float64(-y_46_im));
	elseif (y_46_re <= 5.1e+75)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e+150], t$95$1, If[LessEqual[y$46$re, -2.3e-83], N[(N[(N[(x$46$im * N[(y$46$im / x$46$re), $MachinePrecision] + y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-118], N[(N[(N[(N[(y$46$re * x$46$re + N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] / (-y$46$im)), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] * -1.0 + (-x$46$im)), $MachinePrecision] / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 5.1e+75], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.25000000000000002e150 or 5.10000000000000037e75 < y.re

   1. Initial program 34.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.25000000000000002e150 < y.re < -2.2999999999999999e-83

   1. Initial program 74.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.2999999999999999e-83 < y.re < 7.49999999999999978e-118

   1. Initial program 67.2%

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

   3. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 89.4%

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

   if 7.49999999999999978e-118 < y.re < 5.10000000000000037e75

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 84.1

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

   4. Applied rewrites 84.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.re}, \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{x.re}, y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y.re, x.re, \frac{\left(y.re \cdot y.re\right) \cdot x.im}{-y.im}\right)}{y.im}, -1, -x.im\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.re}, \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(\frac{x.im}{y.re}, \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{x.re}, y.re\right)}{t\_0} \cdot x.re\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ x.im y.re) (/ y.im y.re) (/ x.re y.re))))
   (if (<= y.re -1.25e+150)
     t_1
     (if (<= y.re -2.3e-83)
       (* (/ (fma x.im (/ y.im x.re) y.re) t_0) x.re)
       (if (<= y.re 6.6e-117)
         (/ (fma x.re (/ y.re y.im) x.im) y.im)
         (if (<= y.re 5.1e+75) (/ (fma y.im x.im (* y.re x.re)) t_0) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_re), (y_46_im / y_46_re), (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.25e+150) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-83) {
		tmp = (fma(x_46_im, (y_46_im / x_46_re), y_46_re) / t_0) * x_46_re;
	} else if (y_46_re <= 6.6e-117) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 5.1e+75) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(x_46_im / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.25e+150)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-83)
		tmp = Float64(Float64(fma(x_46_im, Float64(y_46_im / x_46_re), y_46_re) / t_0) * x_46_re);
	elseif (y_46_re <= 6.6e-117)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_re <= 5.1e+75)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e+150], t$95$1, If[LessEqual[y$46$re, -2.3e-83], N[(N[(N[(x$46$im * N[(y$46$im / x$46$re), $MachinePrecision] + y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.6e-117], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.1e+75], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.25000000000000002e150 or 5.10000000000000037e75 < y.re

   1. Initial program 34.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.25000000000000002e150 < y.re < -2.2999999999999999e-83

   1. Initial program 74.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.2999999999999999e-83 < y.re < 6.6000000000000003e-117

   1. Initial program 67.2%

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

   3. Taylor expanded in y.im around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 6.6000000000000003e-117 < y.re < 5.10000000000000037e75

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 84.1

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

   4. Applied rewrites 84.1%

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

Alternative 3: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+38)
   (/ (+ x.im (* (/ x.re y.im) y.re)) y.im)
   (if (<= y.im 9.8e-172)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 4.2e+68)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (fma (/ x.re y.im) y.re x.im) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9e+38) {
		tmp = (x_46_im + ((x_46_re / y_46_im) * y_46_re)) / y_46_im;
	} else if (y_46_im <= 9.8e-172) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 4.2e+68) {
		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));
	} else {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e+38)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re / y_46_im) * y_46_re)) / y_46_im);
	elseif (y_46_im <= 9.8e-172)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 4.2e+68)
		tmp = 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)));
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), 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, -9e+38], N[(N[(x$46$im + N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 9.8e-172], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+68], 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], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.99999999999999961e38

   1. Initial program 48.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -8.99999999999999961e38 < y.im < 9.8000000000000001e-172

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 9.8000000000000001e-172 < y.im < 4.20000000000000002e68

   1. Initial program 84.6%

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

   if 4.20000000000000002e68 < y.im

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 4: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+79}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e-15)
   (/ x.re y.re)
   (if (<= y.re 7.5e-117)
     (/ x.im y.im)
     (if (<= y.re 8e+79)
       (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))
       (if (<= y.re 1.2e+131)
         (/ (fma x.re y.re (* x.im y.im)) (* y.re y.re))
         (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.6e-15) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 7.5e-117) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 8e+79) {
		tmp = x_46_re * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_re <= 1.2e+131) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.6e-15)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 7.5e-117)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 8e+79)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 1.2e+131)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.6e-15], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-117], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8e+79], N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.2e+131], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.60000000000000028e-15 or 1.2e131 < y.re

   1. Initial program 41.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -5.60000000000000028e-15 < y.re < 7.50000000000000066e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if 7.50000000000000066e-117 < y.re < 7.99999999999999974e79

   1. Initial program 82.0%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 7.99999999999999974e79 < y.re < 1.2e131

   1. Initial program 84.9%

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

   3. Taylor expanded in y.re around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

Alternative 5: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+38)
   (/ (+ x.im (* (/ x.re y.im) y.re)) y.im)
   (if (<= y.im 9.8e-172)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 4.2e+68)
       (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))
       (/ (fma (/ x.re y.im) y.re x.im) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9e+38) {
		tmp = (x_46_im + ((x_46_re / y_46_im) * y_46_re)) / y_46_im;
	} else if (y_46_im <= 9.8e-172) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 4.2e+68) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e+38)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re / y_46_im) * y_46_re)) / y_46_im);
	elseif (y_46_im <= 9.8e-172)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 4.2e+68)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), 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, -9e+38], N[(N[(x$46$im + N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 9.8e-172], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+68], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.99999999999999961e38

   1. Initial program 48.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -8.99999999999999961e38 < y.im < 9.8000000000000001e-172

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 9.8000000000000001e-172 < y.im < 4.20000000000000002e68

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.6

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

   if 4.20000000000000002e68 < y.im

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

Alternative 6: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+79}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -3.6e-15)
     t_0
     (if (<= y.re 7.5e-117)
       (/ x.im y.im)
       (if (<= y.re 7.2e+79)
         (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -3.6e-15) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e-117) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 7.2e+79) {
		tmp = x_46_re * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.6e-15)
		tmp = t_0;
	elseif (y_46_re <= 7.5e-117)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 7.2e+79)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e-15], t$95$0, If[LessEqual[y$46$re, 7.5e-117], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+79], N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.6000000000000001e-15 or 7.1999999999999999e79 < y.re

   1. Initial program 46.6%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -3.6000000000000001e-15 < y.re < 7.50000000000000066e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if 7.50000000000000066e-117 < y.re < 7.1999999999999999e79

   1. Initial program 82.0%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 7: 64.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+79}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e-15)
   (/ x.re y.re)
   (if (<= y.re 7.5e-117)
     (/ x.im y.im)
     (if (<= y.re 8e+79)
       (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))
       (/ x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.6e-15) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 7.5e-117) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 8e+79) {
		tmp = x_46_re * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.6e-15)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 7.5e-117)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 8e+79)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.6e-15], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-117], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8e+79], N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.60000000000000028e-15 or 7.99999999999999974e79 < y.re

   1. Initial program 46.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   if -5.60000000000000028e-15 < y.re < 7.50000000000000066e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if 7.50000000000000066e-117 < y.re < 7.99999999999999974e79

   1. Initial program 82.0%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 8: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+38} \lor \neg \left(y.im \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9e+38) (not (<= y.im 2.7e-8)))
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (/ (fma x.im (/ y.im y.re) x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9e+38) || !(y_46_im <= 2.7e-8)) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9e+38) || !(y_46_im <= 2.7e-8))
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9e+38], N[Not[LessEqual[y$46$im, 2.7e-8]], $MachinePrecision]], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.99999999999999961e38 or 2.70000000000000002e-8 < y.im

   1. Initial program 48.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -8.99999999999999961e38 < y.im < 2.70000000000000002e-8

   1. Initial program 73.6%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+38} \lor \neg \left(y.im \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+39} \lor \neg \left(y.im \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1e+39) (not (<= y.im 2.7e-8)))
   (/ (fma x.re (/ y.re y.im) x.im) y.im)
   (/ (fma x.im (/ y.im y.re) x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1e+39) || !(y_46_im <= 2.7e-8)) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1e+39) || !(y_46_im <= 2.7e-8))
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1e+39], N[Not[LessEqual[y$46$im, 2.7e-8]], $MachinePrecision]], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.9999999999999994e38 or 2.70000000000000002e-8 < y.im

   1. Initial program 48.6%

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

   3. Taylor expanded in y.im around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -9.9999999999999994e38 < y.im < 2.70000000000000002e-8

   1. Initial program 73.6%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+39} \lor \neg \left(y.im \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+38)
   (/ (+ x.im (* (/ x.re y.im) y.re)) y.im)
   (if (<= y.im 2.7e-8)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (/ (fma (/ x.re y.im) y.re x.im) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9e+38) {
		tmp = (x_46_im + ((x_46_re / y_46_im) * y_46_re)) / y_46_im;
	} else if (y_46_im <= 2.7e-8) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e+38)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re / y_46_im) * y_46_re)) / y_46_im);
	elseif (y_46_im <= 2.7e-8)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), 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, -9e+38], N[(N[(x$46$im + N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.7e-8], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.99999999999999961e38

   1. Initial program 48.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. associate-*r/ N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -8.99999999999999961e38 < y.im < 2.70000000000000002e-8

   1. Initial program 73.6%

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

   3. Taylor expanded in y.re around inf

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

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 2.70000000000000002e-8 < y.im

   1. Initial program 48.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

Alternative 11: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-15} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.6e-15) (not (<= y.re 8.5e-117)))
   (/ 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_re <= -5.6e-15) || !(y_46_re <= 8.5e-117)) {
		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_46re <= (-5.6d-15)) .or. (.not. (y_46re <= 8.5d-117))) 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_re <= -5.6e-15) || !(y_46_re <= 8.5e-117)) {
		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_re <= -5.6e-15) or not (y_46_re <= 8.5e-117):
		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_re <= -5.6e-15) || !(y_46_re <= 8.5e-117))
		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_re <= -5.6e-15) || ~((y_46_re <= 8.5e-117)))
		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[Or[LessEqual[y$46$re, -5.6e-15], N[Not[LessEqual[y$46$re, 8.5e-117]], $MachinePrecision]], 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.re < -5.60000000000000028e-15 or 8.49999999999999981e-117 < y.re

   1. Initial program 55.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -5.60000000000000028e-15 < y.re < 8.49999999999999981e-117

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-15} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.6% accurate, 3.2× 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 60.9%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 40.8

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

5. Applied rewrites 40.8%

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

Reproduce

herbie shell --seed 2025056 
(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))))