_divideComplex, imaginary part

Percentage Accurate: 61.2% → 80.5%

Time: 6.3s

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: 61.2% 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: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e+147)
   (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
   (if (<= y.im -9.2e-97)
     (/ (fma (- x.re) y.im (* x.im y.re)) (fma y.im y.im (* y.re y.re)))
     (if (<= y.im 4e-9)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (/ (fma y.re (/ x.im 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 <= -2.1e+147) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_im <= -9.2e-97) {
		tmp = fma(-x_46_re, y_46_im, (x_46_im * y_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(y_46_re, (x_46_im / 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 <= -2.1e+147)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -9.2e-97)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(x_46_im * y_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 4e-9)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), 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, -2.1e+147], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9.2e-97], N[(N[((-x$46$re) * y$46$im + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4e-9], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.10000000000000006e147

   1. Initial program 29.1%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

      12. associate-/l* N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -2.10000000000000006e147 < y.im < -9.19999999999999976e-97

   1. Initial program 83.3%

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

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 83.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.3

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

   4. Applied rewrites 83.3%

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

   if -9.19999999999999976e-97 < y.im < 4.00000000000000025e-9

   1. Initial program 73.5%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 4.00000000000000025e-9 < y.im

   1. Initial program 54.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 14.7

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

   5. Applied rewrites 14.7%

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 86.9

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

   10. Applied rewrites 86.9%

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

Alternative 2: 65.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ t_2 := \frac{t\_1}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -200000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{t\_1}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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)))
        (t_2 (/ t_1 (* y.im y.im))))
   (if (<= y.im -2e+147)
     t_0
     (if (<= y.im -200000.0)
       t_2
       (if (<= y.im -2.2e-207)
         (/ t_1 (* y.re y.re))
         (if (<= y.im 2.15e-10)
           (/ x.im y.re)
           (if (<= y.im 2.3e+135) t_2 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 t_2 = t_1 / (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -2e+147) {
		tmp = t_0;
	} else if (y_46_im <= -200000.0) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-207) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 2.15e-10) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = (x_46im * y_46re) - (x_46re * y_46im)
    t_2 = t_1 / (y_46im * y_46im)
    if (y_46im <= (-2d+147)) then
        tmp = t_0
    else if (y_46im <= (-200000.0d0)) then
        tmp = t_2
    else if (y_46im <= (-2.2d-207)) then
        tmp = t_1 / (y_46re * y_46re)
    else if (y_46im <= 2.15d-10) then
        tmp = x_46im / y_46re
    else if (y_46im <= 2.3d+135) then
        tmp = t_2
    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);
	double t_2 = t_1 / (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -2e+147) {
		tmp = t_0;
	} else if (y_46_im <= -200000.0) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-207) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 2.15e-10) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = t_2;
	} 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)
	t_2 = t_1 / (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -2e+147:
		tmp = t_0
	elif y_46_im <= -200000.0:
		tmp = t_2
	elif y_46_im <= -2.2e-207:
		tmp = t_1 / (y_46_re * y_46_re)
	elif y_46_im <= 2.15e-10:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 2.3e+135:
		tmp = t_2
	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))
	t_2 = Float64(t_1 / Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -2e+147)
		tmp = t_0;
	elseif (y_46_im <= -200000.0)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-207)
		tmp = Float64(t_1 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 2.15e-10)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 2.3e+135)
		tmp = t_2;
	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);
	t_2 = t_1 / (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2e+147)
		tmp = t_0;
	elseif (y_46_im <= -200000.0)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-207)
		tmp = t_1 / (y_46_re * y_46_re);
	elseif (y_46_im <= 2.15e-10)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 2.3e+135)
		tmp = t_2;
	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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2e+147], t$95$0, If[LessEqual[y$46$im, -200000.0], t$95$2, If[LessEqual[y$46$im, -2.2e-207], N[(t$95$1 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.15e-10], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.3e+135], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2e147 or 2.3000000000000001e135 < y.im

   1. Initial program 29.8%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -2e147 < y.im < -2e5 or 2.15000000000000007e-10 < y.im < 2.3000000000000001e135

   1. Initial program 79.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. unpow2 N/A

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

      2. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -2e5 < y.im < -2.1999999999999999e-207

   1. Initial program 87.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if -2.1999999999999999e-207 < y.im < 2.15000000000000007e-10

   1. Initial program 72.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -200000:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -4.2e+81)
     t_0
     (if (<= y.im 4e-9)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 2.3e+135)
         (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
         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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= (-4.2d+81)) then
        tmp = t_0
    else if (y_46im <= 4d-9) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else if (y_46im <= 2.3d+135) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= -4.2e+81:
		tmp = t_0
	elif y_46_im <= 4e-9:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	elif y_46_im <= 2.3e+135:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 4e-9)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.3e+135)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 4e-9)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.3e+135)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	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, -4.2e+81], t$95$0, If[LessEqual[y$46$im, 4e-9], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.3e+135], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1999999999999997e81 or 2.3000000000000001e135 < y.im

   1. Initial program 38.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.1999999999999997e81 < y.im < 4.00000000000000025e-9

   1. Initial program 75.0%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if 4.00000000000000025e-9 < y.im < 2.3000000000000001e135

   1. Initial program 80.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -4.2e+81)
     t_0
     (if (<= y.im 4e-9)
       (/ (- x.im (* y.im (/ x.re y.re))) y.re)
       (if (<= y.im 2.3e+135)
         (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
         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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= (-4.2d+81)) then
        tmp = t_0
    else if (y_46im <= 4d-9) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46im <= 2.3d+135) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= -4.2e+81:
		tmp = t_0
	elif y_46_im <= 4e-9:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_im <= 2.3e+135:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 4e-9)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2.3e+135)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 4e-9)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_im <= 2.3e+135)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	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, -4.2e+81], t$95$0, If[LessEqual[y$46$im, 4e-9], 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$46$im, 2.3e+135], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1999999999999997e81 or 2.3000000000000001e135 < y.im

   1. Initial program 38.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.1999999999999997e81 < y.im < 4.00000000000000025e-9

   1. Initial program 75.0%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 79.6%

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

   if 4.00000000000000025e-9 < y.im < 2.3000000000000001e135

   1. Initial program 80.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -4.2e+81)
     t_0
     (if (<= y.im 2.15e-10)
       (/ x.im y.re)
       (if (<= y.im 2.3e+135)
         (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
         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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 2.15e-10) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= (-4.2d+81)) then
        tmp = t_0
    else if (y_46im <= 2.15d-10) then
        tmp = x_46im / y_46re
    else if (y_46im <= 2.3d+135) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    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 <= -4.2e+81) {
		tmp = t_0;
	} else if (y_46_im <= 2.15e-10) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 2.3e+135) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} 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 <= -4.2e+81:
		tmp = t_0
	elif y_46_im <= 2.15e-10:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 2.3e+135:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 2.15e-10)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 2.3e+135)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	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 <= -4.2e+81)
		tmp = t_0;
	elseif (y_46_im <= 2.15e-10)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 2.3e+135)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	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, -4.2e+81], t$95$0, If[LessEqual[y$46$im, 2.15e-10], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.3e+135], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1999999999999997e81 or 2.3000000000000001e135 < y.im

   1. Initial program 38.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -4.1999999999999997e81 < y.im < 2.15000000000000007e-10

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 2.15000000000000007e-10 < y.im < 2.3000000000000001e135

   1. Initial program 80.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.7 \cdot 10^{+68} \lor \neg \left(y.im \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.7e+68) (not (<= y.im 4e-9)))
   (/ (fma y.re (/ x.im y.im) (- 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 <= -7.7e+68) || !(y_46_im <= 4e-9)) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -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 <= -7.7e+68) || !(y_46_im <= 4e-9))
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / 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, -7.7e+68], N[Not[LessEqual[y$46$im, 4e-9]], $MachinePrecision]], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.6999999999999998e68 or 4.00000000000000025e-9 < 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 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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 13.5

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

   5. Applied rewrites 13.5%

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 86.9

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

   10. Applied rewrites 86.9%

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

   if -7.6999999999999998e68 < y.im < 4.00000000000000025e-9

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.7 \cdot 10^{+68} \lor \neg \left(y.im \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.7e+68)
   (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
   (if (<= y.im 4e-9)
     (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
     (/ (fma y.re (/ x.im 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 <= -7.7e+68) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_im <= 4e-9) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(y_46_re, (x_46_im / 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 <= -7.7e+68)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 4e-9)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), 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, -7.7e+68], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 4e-9], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.6999999999999998e68

   1. Initial program 44.6%

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

   3. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

      12. associate-/l* N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -7.6999999999999998e68 < y.im < 4.00000000000000025e-9

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if 4.00000000000000025e-9 < y.im

   1. Initial program 54.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 14.7

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

   5. Applied rewrites 14.7%

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

   7. Applied rewrites 16.5%

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

   8. Taylor expanded in y.re around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 86.9

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

   10. Applied rewrites 86.9%

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

Alternative 8: 63.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81} \lor \neg \left(y.im \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.2e+81) (not (<= y.im 2.3e-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 <= -4.2e+81) || !(y_46_im <= 2.3e-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 <= (-4.2d+81)) .or. (.not. (y_46im <= 2.3d-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 <= -4.2e+81) || !(y_46_im <= 2.3e-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 <= -4.2e+81) or not (y_46_im <= 2.3e-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 <= -4.2e+81) || !(y_46_im <= 2.3e-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 <= -4.2e+81) || ~((y_46_im <= 2.3e-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, -4.2e+81], N[Not[LessEqual[y$46$im, 2.3e-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 < -4.1999999999999997e81 or 2.30000000000000007e-10 < y.im

   1. Initial program 48.6%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -4.1999999999999997e81 < y.im < 2.30000000000000007e-10

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+81} \lor \neg \left(y.im \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.3% 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 63.9%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.6

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

5. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2024360 
(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))))