_divideComplex, imaginary part

Percentage Accurate: 60.3% → 81.0%

Time: 3.8s

Alternatives: 9

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: 60.3% 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: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-18}:\\ \;\;\;\;\left(-x.im\right) \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.im}{x.im}, -y.re\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{x.re}, -y.im\right)}{t\_1} \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re))
        (t_1 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.2e+116)
     t_0
     (if (<= y.re -1.05e-18)
       (* (- x.im) (/ (fma x.re (/ y.im x.im) (- y.re)) t_1))
       (if (<= y.re 2.5e-118)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 2.9e+79)
           (* (/ (fma x.im (/ y.re x.re) (- y.im)) t_1) x.re)
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double t_1 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.2e+116) {
		tmp = t_0;
	} else if (y_46_re <= -1.05e-18) {
		tmp = -x_46_im * (fma(x_46_re, (y_46_im / x_46_im), -y_46_re) / t_1);
	} else if (y_46_re <= 2.5e-118) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 2.9e+79) {
		tmp = (fma(x_46_im, (y_46_re / x_46_re), -y_46_im) / t_1) * x_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	t_1 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.2e+116)
		tmp = t_0;
	elseif (y_46_re <= -1.05e-18)
		tmp = Float64(Float64(-x_46_im) * Float64(fma(x_46_re, Float64(y_46_im / x_46_im), Float64(-y_46_re)) / t_1));
	elseif (y_46_re <= 2.5e-118)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 2.9e+79)
		tmp = Float64(Float64(fma(x_46_im, Float64(y_46_re / x_46_re), Float64(-y_46_im)) / t_1) * x_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+116], t$95$0, If[LessEqual[y$46$re, -1.05e-18], N[((-x$46$im) * N[(N[(x$46$re * N[(y$46$im / x$46$im), $MachinePrecision] + (-y$46$re)), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e-118], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.9e+79], N[(N[(N[(x$46$im * N[(y$46$re / x$46$re), $MachinePrecision] + (-y$46$im)), $MachinePrecision] / t$95$1), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.2e116 or 2.89999999999999992e79 < y.re

   1. Initial program 40.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.5

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

   7. Applied rewrites 90.5%

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

   if -2.2e116 < y.re < -1.05e-18

   1. Initial program 75.4%

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

   3. Taylor expanded in x.im around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 84.7%

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

   if -1.05e-18 < y.re < 2.50000000000000007e-118

   1. Initial program 68.9%

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

   3. Taylor expanded in y.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 89.5

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

   5. Applied rewrites 89.5%

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

   if 2.50000000000000007e-118 < y.re < 2.89999999999999992e79

   1. Initial program 80.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.7%

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

Alternative 2: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.re}{x.re}, -y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.re\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ (fma x.im (/ y.re x.re) (- y.im)) (fma y.im y.im (* y.re y.re)))
          x.re)))
   (if (<= y.re -3.2e+67)
     (fma (/ (- x.re) y.re) (/ y.im y.re) (/ x.im y.re))
     (if (<= y.re -2.3e-83)
       t_0
       (if (<= y.re 2.5e-118)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 2.9e+79)
           t_0
           (/ (- 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 t_0 = (fma(x_46_im, (y_46_re / x_46_re), -y_46_im) / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_re;
	double tmp;
	if (y_46_re <= -3.2e+67) {
		tmp = fma((-x_46_re / y_46_re), (y_46_im / y_46_re), (x_46_im / y_46_re));
	} else if (y_46_re <= -2.3e-83) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-118) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 2.9e+79) {
		tmp = t_0;
	} 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)
	t_0 = Float64(Float64(fma(x_46_im, Float64(y_46_re / x_46_re), Float64(-y_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_re)
	tmp = 0.0
	if (y_46_re <= -3.2e+67)
		tmp = fma(Float64(Float64(-x_46_re) / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re));
	elseif (y_46_re <= -2.3e-83)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-118)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 2.9e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(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_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / x$46$re), $MachinePrecision] + (-y$46$im)), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.2e+67], N[(N[((-x$46$re) / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.3e-83], t$95$0, If[LessEqual[y$46$re, 2.5e-118], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.9e+79], t$95$0, N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.19999999999999983e67

   1. Initial program 49.6%

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

   3. Taylor expanded in y.im around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -3.19999999999999983e67 < y.re < -2.2999999999999999e-83 or 2.50000000000000007e-118 < y.re < 2.89999999999999992e79

   1. Initial program 79.2%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.3%

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

   if -2.2999999999999999e-83 < y.re < 2.50000000000000007e-118

   1. Initial program 67.3%

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

   3. Taylor expanded in y.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 91.5

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

   5. Applied rewrites 91.5%

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

   if 2.89999999999999992e79 < y.re

   1. Initial program 43.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 70.0

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

   5. Applied rewrites 70.0%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.3

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

   7. Applied rewrites 86.3%

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

Alternative 3: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -2e+116)
     t_1
     (if (<= y.re -2.35e-83)
       t_0
       (if (<= y.re 8.5e-119)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 5e+77) 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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -2e+116) {
		tmp = t_1;
	} else if (y_46_re <= -2.35e-83) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e-119) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 5e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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)))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2e+116)
		tmp = t_1;
	elseif (y_46_re <= -2.35e-83)
		tmp = t_0;
	elseif (y_46_re <= 8.5e-119)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 5e+77)
		tmp = 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[(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]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2e+116], t$95$1, If[LessEqual[y$46$re, -2.35e-83], t$95$0, If[LessEqual[y$46$re, 8.5e-119], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.00000000000000003e116 or 5.00000000000000004e77 < y.re

   1. Initial program 40.5%

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

   3. Taylor expanded in y.re around 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 76.7

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

   5. Applied rewrites 76.7%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.5

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

   7. Applied rewrites 89.5%

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

   if -2.00000000000000003e116 < y.re < -2.3500000000000002e-83 or 8.49999999999999977e-119 < y.re < 5.00000000000000004e77

   1. Initial program 79.7%

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

   if -2.3500000000000002e-83 < y.re < 8.49999999999999977e-119

   1. Initial program 67.3%

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

   3. Taylor expanded in y.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 91.5

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

   5. Applied rewrites 91.5%

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

Alternative 4: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.2e+116)
     t_0
     (if (<= y.im -3.5e+36)
       t_1
       (if (<= y.im 1.3e-7)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 3.15e+101) t_1 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -3.5e+36) {
		tmp = t_1;
	} else if (y_46_im <= 1.3e-7) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    if (y_46im <= (-5.2d+116)) then
        tmp = t_0
    else if (y_46im <= (-3.5d+36)) then
        tmp = t_1
    else if (y_46im <= 1.3d-7) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 3.15d+101) then
        tmp = t_1
    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 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -3.5e+36) {
		tmp = t_1;
	} else if (y_46_im <= 1.3e-7) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -5.2e+116:
		tmp = t_0
	elif y_46_im <= -3.5e+36:
		tmp = t_1
	elif y_46_im <= 1.3e-7:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 3.15e+101:
		tmp = t_1
	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 <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -3.5e+36)
		tmp = t_1;
	elseif (y_46_im <= 1.3e-7)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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;
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -3.5e+36)
		tmp = t_1;
	elseif (y_46_im <= 1.3e-7)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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]}, 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, -5.2e+116], t$95$0, If[LessEqual[y$46$im, -3.5e+36], t$95$1, If[LessEqual[y$46$im, 1.3e-7], 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, 3.15e+101], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.19999999999999973e116 or 3.15000000000000002e101 < y.im

   1. Initial program 37.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   if -5.19999999999999973e116 < y.im < -3.4999999999999998e36 or 1.29999999999999999e-7 < y.im < 3.15000000000000002e101

   1. Initial program 73.4%

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

   3. Taylor expanded in y.re around 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 67.2

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

   5. Applied rewrites 67.2%

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

   if -3.4999999999999998e36 < y.im < 1.29999999999999999e-7

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around 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 87.1

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

   5. Applied rewrites 87.1%

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

Alternative 5: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.2e+116)
     t_0
     (if (<= y.im -1.9e+39)
       t_1
       (if (<= y.im 1.3e-7)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 3.15e+101) t_1 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -1.9e+39) {
		tmp = t_1;
	} else if (y_46_im <= 1.3e-7) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    if (y_46im <= (-5.2d+116)) then
        tmp = t_0
    else if (y_46im <= (-1.9d+39)) then
        tmp = t_1
    else if (y_46im <= 1.3d-7) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 3.15d+101) then
        tmp = t_1
    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 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -1.9e+39) {
		tmp = t_1;
	} else if (y_46_im <= 1.3e-7) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -5.2e+116:
		tmp = t_0
	elif y_46_im <= -1.9e+39:
		tmp = t_1
	elif y_46_im <= 1.3e-7:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.15e+101:
		tmp = t_1
	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 <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -1.9e+39)
		tmp = t_1;
	elseif (y_46_im <= 1.3e-7)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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;
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -1.9e+39)
		tmp = t_1;
	elseif (y_46_im <= 1.3e-7)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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]}, 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, -5.2e+116], t$95$0, If[LessEqual[y$46$im, -1.9e+39], t$95$1, If[LessEqual[y$46$im, 1.3e-7], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.15e+101], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.19999999999999973e116 or 3.15000000000000002e101 < y.im

   1. Initial program 37.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   if -5.19999999999999973e116 < y.im < -1.8999999999999999e39 or 1.29999999999999999e-7 < y.im < 3.15000000000000002e101

   1. Initial program 73.4%

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

   3. Taylor expanded in y.re around 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 67.2

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

   5. Applied rewrites 67.2%

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

   if -1.8999999999999999e39 < y.im < 1.29999999999999999e-7

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around 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 87.1

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

   5. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}} \]
   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 86.4

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

   7. Applied rewrites 86.4%

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

Alternative 6: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.2e+116)
     t_0
     (if (<= y.im -1.6e-20)
       t_1
       (if (<= y.im 1.05e-11)
         (/ x.im y.re)
         (if (<= y.im 3.15e+101) t_1 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -1.6e-20) {
		tmp = t_1;
	} else if (y_46_im <= 1.05e-11) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    if (y_46im <= (-5.2d+116)) then
        tmp = t_0
    else if (y_46im <= (-1.6d-20)) then
        tmp = t_1
    else if (y_46im <= 1.05d-11) then
        tmp = x_46im / y_46re
    else if (y_46im <= 3.15d+101) then
        tmp = t_1
    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 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 <= -5.2e+116) {
		tmp = t_0;
	} else if (y_46_im <= -1.6e-20) {
		tmp = t_1;
	} else if (y_46_im <= 1.05e-11) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.15e+101) {
		tmp = t_1;
	} 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
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -5.2e+116:
		tmp = t_0
	elif y_46_im <= -1.6e-20:
		tmp = t_1
	elif y_46_im <= 1.05e-11:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.15e+101:
		tmp = t_1
	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 <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -1.6e-20)
		tmp = t_1;
	elseif (y_46_im <= 1.05e-11)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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;
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5.2e+116)
		tmp = t_0;
	elseif (y_46_im <= -1.6e-20)
		tmp = t_1;
	elseif (y_46_im <= 1.05e-11)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.15e+101)
		tmp = t_1;
	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]}, 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, -5.2e+116], t$95$0, If[LessEqual[y$46$im, -1.6e-20], t$95$1, If[LessEqual[y$46$im, 1.05e-11], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.15e+101], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.19999999999999973e116 or 3.15000000000000002e101 < y.im

   1. Initial program 37.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   if -5.19999999999999973e116 < y.im < -1.59999999999999985e-20 or 1.0499999999999999e-11 < y.im < 3.15000000000000002e101

   1. Initial program 71.2%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{{y.im}^{2}}} \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -1.59999999999999985e-20 < y.im < 1.0499999999999999e-11

   1. Initial program 76.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 70.8

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

   5. Applied rewrites 70.8%

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

Alternative 7: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.5 \cdot 10^{+36} \lor \neg \left(y.im \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.5e+36) (not (<= y.im 1.3e-7)))
   (/ (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 <= -3.5e+36) || !(y_46_im <= 1.3e-7)) {
		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 <= -3.5e+36) || !(y_46_im <= 1.3e-7))
		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, -3.5e+36], N[Not[LessEqual[y$46$im, 1.3e-7]], $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 < -3.4999999999999998e36 or 1.29999999999999999e-7 < y.im

   1. Initial program 49.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

   if -3.4999999999999998e36 < y.im < 1.29999999999999999e-7

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around 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 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.5 \cdot 10^{+36} \lor \neg \left(y.im \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-20} \lor \neg \left(y.im \leq 2.05 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.1e-20) (not (<= y.im 2.05e-10)))
   (/ (- 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 <= -2.1e-20) || !(y_46_im <= 2.05e-10)) {
		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 <= (-2.1d-20)) .or. (.not. (y_46im <= 2.05d-10))) 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 <= -2.1e-20) || !(y_46_im <= 2.05e-10)) {
		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 <= -2.1e-20) or not (y_46_im <= 2.05e-10):
		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 <= -2.1e-20) || !(y_46_im <= 2.05e-10))
		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 <= -2.1e-20) || ~((y_46_im <= 2.05e-10)))
		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, -2.1e-20], N[Not[LessEqual[y$46$im, 2.05e-10]], $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 < -2.0999999999999999e-20 or 2.0499999999999999e-10 < y.im

   1. Initial program 49.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 65.1

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

   5. Applied rewrites 65.1%

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

   if -2.0999999999999999e-20 < y.im < 2.0499999999999999e-10

   1. Initial program 76.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 70.8

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

   5. Applied rewrites 70.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-20} \lor \neg \left(y.im \leq 2.05 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% 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 62.1%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.5

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

5. Applied rewrites 42.5%

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

Reproduce

herbie shell --seed 2025056 
(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))))