_divideComplex, imaginary part

Percentage Accurate: 60.9% → 83.5%

Time: 4.2s

Alternatives: 12

Speedup: 1.5×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

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

Initial Program: 60.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

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

Alternative 1: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \left(-x.re\right) \cdot \mathsf{fma}\left(\frac{-1}{x.re}, x.im \cdot \frac{y.re}{t\_0}, \frac{y.im}{t\_0}\right)\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 6.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.im}, \frac{y.re}{y.im}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

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
         (* (- x.re) (fma (/ -1.0 x.re) (* x.im (/ y.re t_0)) (/ y.im t_0)))))
   (if (<= y.im -1.55e+126)
     (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
     (if (<= y.im -4.6e-41)
       t_1
       (if (<= y.im 6.7e-59)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 1e+132)
           t_1
           (fma (/ x.im y.im) (/ y.re y.im) (/ (- x.re) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = -x_46_re * fma((-1.0 / x_46_re), (x_46_im * (y_46_re / t_0)), (y_46_im / t_0));
	double tmp;
	if (y_46_im <= -1.55e+126) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -4.6e-41) {
		tmp = t_1;
	} else if (y_46_im <= 6.7e-59) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1e+132) {
		tmp = t_1;
	} else {
		tmp = fma((x_46_im / y_46_im), (y_46_re / y_46_im), (-x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(Float64(-x_46_re) * fma(Float64(-1.0 / x_46_re), Float64(x_46_im * Float64(y_46_re / t_0)), Float64(y_46_im / t_0)))
	tmp = 0.0
	if (y_46_im <= -1.55e+126)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -4.6e-41)
		tmp = t_1;
	elseif (y_46_im <= 6.7e-59)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1e+132)
		tmp = t_1;
	else
		tmp = fma(Float64(x_46_im / y_46_im), Float64(y_46_re / y_46_im), Float64(Float64(-x_46_re) / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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[((-x$46$re) * N[(N[(-1.0 / x$46$re), $MachinePrecision] * N[(x$46$im * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e+126], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.6e-41], t$95$1, If[LessEqual[y$46$im, 6.7e-59], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1e+132], t$95$1, N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.55e126

   1. Initial program 29.6%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -1.55e126 < y.im < -4.6000000000000002e-41 or 6.7e-59 < y.im < 9.99999999999999991e131

   1. Initial program 74.5%

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

   3. Taylor expanded in x.re around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   if -4.6000000000000002e-41 < y.im < 6.7e-59

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if 9.99999999999999991e131 < y.im

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 2: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.im}, \frac{y.re}{y.im}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.6e+35)
   (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
   (if (<= y.im 2e-88)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (if (<= y.im 1.6e+86)
       (/ (fma (- x.re) y.im (* y.re x.im)) (fma y.im y.im (* y.re y.re)))
       (fma (/ x.im y.im) (/ y.re y.im) (/ (- x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.6e+35) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 2e-88) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.6e+86) {
		tmp = fma(-x_46_re, y_46_im, (y_46_re * x_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma((x_46_im / y_46_im), (y_46_re / y_46_im), (-x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.6e+35)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 2e-88)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.6e+86)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = fma(Float64(x_46_im / y_46_im), Float64(y_46_re / y_46_im), Float64(Float64(-x_46_re) / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.6e+35], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2e-88], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+86], N[(N[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.5999999999999996e35

   1. Initial program 44.0%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -4.5999999999999996e35 < y.im < 1.99999999999999987e-88

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if 1.99999999999999987e-88 < y.im < 1.6e86

   1. Initial program 87.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 87.9

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

   if 1.6e86 < y.im

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 85.8

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

   5. Applied rewrites 85.8%

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

Alternative 3: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 122000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -8.8e+77)
     t_0
     (if (<= y.im -1.9e+34)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im -1.2e-185)
         (/ x.im y.re)
         (if (<= y.im 122000000000.0)
           (/ (fma (- x.re) y.im (* y.re x.im)) (* y.re y.re))
           (if (<= y.im 7e+110)
             (* x.im (/ 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 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -8.8e+77) {
		tmp = t_0;
	} else if (y_46_im <= -1.9e+34) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= -1.2e-185) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 122000000000.0) {
		tmp = fma(-x_46_re, y_46_im, (y_46_re * x_46_im)) / (y_46_re * y_46_re);
	} else if (y_46_im <= 7e+110) {
		tmp = x_46_im * (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(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -8.8e+77)
		tmp = t_0;
	elseif (y_46_im <= -1.9e+34)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= -1.2e-185)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 122000000000.0)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 7e+110)
		tmp = Float64(x_46_im * 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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -8.8e+77], t$95$0, If[LessEqual[y$46$im, -1.9e+34], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.2e-185], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 122000000000.0], N[(N[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e+110], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.8000000000000002e77 or 6.9999999999999998e110 < y.im

   1. Initial program 38.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -8.8000000000000002e77 < y.im < -1.9000000000000001e34

   1. Initial program 99.6%

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

   3. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.9000000000000001e34 < y.im < -1.2000000000000001e-185

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.2000000000000001e-185 < y.im < 1.22e11

   1. Initial program 79.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.8

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in y.re around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 1.22e11 < y.im < 6.9999999999999998e110

   1. Initial program 65.7%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 122000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{t\_1}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 122000000000:\\ \;\;\;\;\frac{t\_1}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.im -8.8e+77)
     t_0
     (if (<= y.im -1.9e+34)
       (/ t_1 (* y.im y.im))
       (if (<= y.im -1.2e-185)
         (/ x.im y.re)
         (if (<= y.im 122000000000.0)
           (/ t_1 (* y.re y.re))
           (if (<= y.im 7e+110)
             (* x.im (/ 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 = -x_46_re / y_46_im;
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_im <= -8.8e+77) {
		tmp = t_0;
	} else if (y_46_im <= -1.9e+34) {
		tmp = t_1 / (y_46_im * y_46_im);
	} else if (y_46_im <= -1.2e-185) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 122000000000.0) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 7e+110) {
		tmp = x_46_im * (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(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.8e+77)
		tmp = t_0;
	elseif (y_46_im <= -1.9e+34)
		tmp = Float64(t_1 / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= -1.2e-185)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 122000000000.0)
		tmp = Float64(t_1 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 7e+110)
		tmp = Float64(x_46_im * 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[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.8e+77], t$95$0, If[LessEqual[y$46$im, -1.9e+34], N[(t$95$1 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.2e-185], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 122000000000.0], N[(t$95$1 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e+110], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.8000000000000002e77 or 6.9999999999999998e110 < y.im

   1. Initial program 38.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -8.8000000000000002e77 < y.im < -1.9000000000000001e34

   1. Initial program 99.6%

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

   3. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.9000000000000001e34 < y.im < -1.2000000000000001e-185

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.2000000000000001e-185 < y.im < 1.22e11

   1. Initial program 79.8%

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

   3. Taylor expanded in y.re around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if 1.22e11 < y.im < 6.9999999999999998e110

   1. Initial program 65.7%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

Alternative 5: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1 (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))))
   (if (<= y.im -8.8e+77)
     t_0
     (if (<= y.im -1.9e+34)
       t_1
       (if (<= y.im 5.8e-77)
         (/ x.im y.re)
         (if (<= y.im 9.2e-24)
           t_1
           (if (<= y.im 7e+110)
             (* x.im (/ 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 = -x_46_re / y_46_im;
	double t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -8.8e+77) {
		tmp = t_0;
	} else if (y_46_im <= -1.9e+34) {
		tmp = t_1;
	} else if (y_46_im <= 5.8e-77) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 9.2e-24) {
		tmp = t_1;
	} else if (y_46_im <= 7e+110) {
		tmp = x_46_im * (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(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.8e+77)
		tmp = t_0;
	elseif (y_46_im <= -1.9e+34)
		tmp = t_1;
	elseif (y_46_im <= 5.8e-77)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 9.2e-24)
		tmp = t_1;
	elseif (y_46_im <= 7e+110)
		tmp = Float64(x_46_im * 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[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.8e+77], t$95$0, If[LessEqual[y$46$im, -1.9e+34], t$95$1, If[LessEqual[y$46$im, 5.8e-77], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 9.2e-24], t$95$1, If[LessEqual[y$46$im, 7e+110], N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.8000000000000002e77 or 6.9999999999999998e110 < y.im

   1. Initial program 38.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -8.8000000000000002e77 < y.im < -1.9000000000000001e34 or 5.7999999999999997e-77 < y.im < 9.2000000000000004e-24

   1. Initial program 99.3%

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

   3. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.9000000000000001e34 < y.im < 5.7999999999999997e-77

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if 9.2000000000000004e-24 < y.im < 6.9999999999999998e110

   1. Initial program 71.0%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

Alternative 6: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)))
   (if (<= y.im -4.6e+35)
     t_0
     (if (<= y.im 2e-88)
       (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
       (if (<= y.im 1.6e+86)
         (/ (fma (- x.re) y.im (* y.re x.im)) (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_re / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.6e+35) {
		tmp = t_0;
	} else if (y_46_im <= 2e-88) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.6e+86) {
		tmp = fma(-x_46_re, y_46_im, (y_46_re * x_46_im)) / 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_re / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.6e+35)
		tmp = t_0;
	elseif (y_46_im <= 2e-88)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.6e+86)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(y_46_re * x_46_im)) / 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$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.6e+35], t$95$0, If[LessEqual[y$46$im, 2e-88], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+86], N[(N[((-x$46$re) * y$46$im + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.5999999999999996e35 or 1.6e86 < y.im

   1. Initial program 44.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -4.5999999999999996e35 < y.im < 1.99999999999999987e-88

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if 1.99999999999999987e-88 < y.im < 1.6e86

   1. Initial program 87.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 87.9

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

Alternative 7: 63.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\left(-x.re\right) \cdot \frac{y.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ (- x.re) y.im)))
   (if (<= y.im -1.4e+35)
     t_1
     (if (<= y.im 5.2e-86)
       (/ x.im y.re)
       (if (<= y.im 1.3e-16)
         (* (- x.re) (/ y.im t_0))
         (if (<= y.im 7e+110) (* x.im (/ y.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 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -1.4e+35) {
		tmp = t_1;
	} else if (y_46_im <= 5.2e-86) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.3e-16) {
		tmp = -x_46_re * (y_46_im / t_0);
	} else if (y_46_im <= 7e+110) {
		tmp = x_46_im * (y_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 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.4e+35)
		tmp = t_1;
	elseif (y_46_im <= 5.2e-86)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.3e-16)
		tmp = Float64(Float64(-x_46_re) * Float64(y_46_im / t_0));
	elseif (y_46_im <= 7e+110)
		tmp = Float64(x_46_im * Float64(y_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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+35], t$95$1, If[LessEqual[y$46$im, 5.2e-86], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.3e-16], N[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e+110], N[(x$46$im * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.39999999999999999e35 or 6.9999999999999998e110 < y.im

   1. Initial program 44.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.39999999999999999e35 < y.im < 5.2000000000000002e-86

   1. Initial program 74.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 5.2000000000000002e-86 < y.im < 1.2999999999999999e-16

   1. Initial program 99.2%

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

   3. Taylor expanded in x.re around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if 1.2999999999999999e-16 < y.im < 6.9999999999999998e110

   1. Initial program 71.0%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

Alternative 8: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -8.8e+77)
     t_0
     (if (<= y.im -4.6e+35)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 8.2e+110) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -8.8e+77) {
		tmp = t_0;
	} else if (y_46_im <= -4.6e+35) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 8.2e+110) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-8.8d+77)) then
        tmp = t_0
    else if (y_46im <= (-4.6d+35)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    else if (y_46im <= 8.2d+110) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -8.8e+77) {
		tmp = t_0;
	} else if (y_46_im <= -4.6e+35) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 8.2e+110) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -8.8e+77:
		tmp = t_0
	elif y_46_im <= -4.6e+35:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	elif y_46_im <= 8.2e+110:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -8.8e+77)
		tmp = t_0;
	elseif (y_46_im <= -4.6e+35)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 8.2e+110)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -8.8e+77)
		tmp = t_0;
	elseif (y_46_im <= -4.6e+35)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	elseif (y_46_im <= 8.2e+110)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -8.8e+77], t$95$0, If[LessEqual[y$46$im, -4.6e+35], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+110], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.8000000000000002e77 or 8.1999999999999997e110 < y.im

   1. Initial program 38.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -8.8000000000000002e77 < y.im < -4.5999999999999996e35

   1. Initial program 99.5%

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

   3. Taylor expanded in y.re around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -4.5999999999999996e35 < y.im < 8.1999999999999997e110

   1. Initial program 76.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 9: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+35} \lor \neg \left(y.im \leq 1.16 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.6e+35) (not (<= y.im 1.16e+69)))
   (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
   (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.6e+35) || !(y_46_im <= 1.16e+69)) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.6e+35) || !(y_46_im <= 1.16e+69))
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.6e+35], N[Not[LessEqual[y$46$im, 1.16e+69]], $MachinePrecision]], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.5999999999999996e35 or 1.16000000000000005e69 < y.im

   1. Initial program 44.8%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -4.5999999999999996e35 < y.im < 1.16000000000000005e69

   1. Initial program 77.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+35} \lor \neg \left(y.im \leq 1.16 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+126} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.25e+126) (not (<= y.im 1.2e+111)))
   (/ (- x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.25e+126) || !(y_46_im <= 1.2e+111)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.25d+126)) .or. (.not. (y_46im <= 1.2d+111))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.25e+126) || !(y_46_im <= 1.2e+111)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.25e+126) or not (y_46_im <= 1.2e+111):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.25e+126) || !(y_46_im <= 1.2e+111))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.25e+126) || ~((y_46_im <= 1.2e+111)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.25e+126], N[Not[LessEqual[y$46$im, 1.2e+111]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.24999999999999987e126 or 1.20000000000000003e111 < y.im

   1. Initial program 37.2%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if -2.24999999999999987e126 < y.im < 1.20000000000000003e111

   1. Initial program 74.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.4

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

   7. Applied rewrites 75.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+126} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+35} \lor \neg \left(y.im \leq 7 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.4e+35) (not (<= y.im 7e+110)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e+35) || !(y_46_im <= 7e+110)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.4d+35)) .or. (.not. (y_46im <= 7d+110))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e+35) || !(y_46_im <= 7e+110)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.4e+35) or not (y_46_im <= 7e+110):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.4e+35) || !(y_46_im <= 7e+110))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.4e+35) || ~((y_46_im <= 7e+110)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.4e+35], N[Not[LessEqual[y$46$im, 7e+110]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.39999999999999999e35 or 6.9999999999999998e110 < y.im

   1. Initial program 44.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -1.39999999999999999e35 < y.im < 6.9999999999999998e110

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+35} \lor \neg \left(y.im \leq 7 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46re
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_re;
}

Python

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

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_re;
end

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 43.7

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

5. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2025040 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))