_divideComplex, real part

Percentage Accurate: 62.3% → 83.7%

Time: 3.2s

Alternatives: 11

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{x.im}, y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im - \left(-x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.45e+131)
   (/ (fma x.im (/ y.im y.re) x.re) y.re)
   (if (<= y.re -5e-86)
     (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
     (if (<= y.re 6.8e-149)
       (/ (fma x.re (/ y.re y.im) x.im) y.im)
       (if (<= y.re 5.9e+91)
         (*
          (/ (fma x.re (/ y.re x.im) y.im) (fma y.im y.im (* y.re y.re)))
          x.im)
         (/ (- (* (/ x.im y.re) y.im) (- x.re)) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.45e+131) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_re <= -5e-86) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 6.8e-149) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 5.9e+91) {
		tmp = (fma(x_46_re, (y_46_re / x_46_im), y_46_im) / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else {
		tmp = (((x_46_im / y_46_re) * y_46_im) - -x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.45e+131)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_re <= -5e-86)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 6.8e-149)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_re <= 5.9e+91)
		tmp = Float64(Float64(fma(x_46_re, Float64(y_46_re / x_46_im), y_46_im) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) - Float64(-x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.45e+131], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5e-86], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.8e-149], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.9e+91], N[(N[(N[(x$46$re * N[(y$46$re / x$46$im), $MachinePrecision] + y$46$im), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] - (-x$46$re)), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.45000000000000016e131

   1. Initial program 34.5%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

   if -2.45000000000000016e131 < y.re < -4.9999999999999999e-86

   1. Initial program 74.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.5

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

   3. Applied rewrites 74.5%

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

   if -4.9999999999999999e-86 < y.re < 6.7999999999999998e-149

   1. Initial program 71.2%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 6.7999999999999998e-149 < y.re < 5.9000000000000002e91

   1. Initial program 77.5%

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

   2. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 5.9000000000000002e91 < y.re

   1. Initial program 39.6%

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

   2. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in y.re around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 84.9

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

   7. Applied rewrites 84.9%

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

Alternative 2: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im - \left(-x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.re -2.45e+131)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.re -5e-86)
       t_0
       (if (<= y.re 5.3e-145)
         (/ (fma x.re (/ y.re y.im) x.im) y.im)
         (if (<= y.re 4.9e+98)
           t_0
           (/ (- (* (/ x.im y.re) y.im) (- x.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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.45e+131) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_re <= -5e-86) {
		tmp = t_0;
	} else if (y_46_re <= 5.3e-145) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 4.9e+98) {
		tmp = t_0;
	} else {
		tmp = (((x_46_im / y_46_re) * y_46_im) - -x_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(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.45e+131)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_re <= -5e-86)
		tmp = t_0;
	elseif (y_46_re <= 5.3e-145)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_re <= 4.9e+98)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) - Float64(-x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.45e+131], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5e-86], t$95$0, If[LessEqual[y$46$re, 5.3e-145], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.9e+98], t$95$0, N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] - (-x$46$re)), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.45000000000000016e131

   1. Initial program 34.5%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

   if -2.45000000000000016e131 < y.re < -4.9999999999999999e-86 or 5.29999999999999999e-145 < y.re < 4.89999999999999979e98

   1. Initial program 76.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.0

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

   3. Applied rewrites 76.0%

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

   if -4.9999999999999999e-86 < y.re < 5.29999999999999999e-145

   1. Initial program 71.3%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.5

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

   4. Applied rewrites 90.5%

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

   if 4.89999999999999979e98 < y.re

   1. Initial program 38.5%

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

   2. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in y.re around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 85.3

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

   7. Applied rewrites 85.3%

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

Alternative 3: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.re}{y.im}, \frac{y.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{if}\;y.im \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y.im, x.im, -\frac{\left(y.im \cdot y.im\right) \cdot x.re}{y.re}\right)}{y.re}, -1, -x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ x.re y.im) (/ y.re y.im) (/ x.im y.im))))
   (if (<= y.im -2.9e-11)
     t_0
     (if (<= y.im 2.7e-162)
       (-
        (/
         (fma
          (/ (fma y.im x.im (- (/ (* (* y.im y.im) x.re) y.re))) y.re)
          -1.0
          (- x.re))
         y.re))
       (if (<= y.im 3e+153)
         (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* 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 = fma((x_46_re / y_46_im), (y_46_re / y_46_im), (x_46_im / y_46_im));
	double tmp;
	if (y_46_im <= -2.9e-11) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e-162) {
		tmp = -(fma((fma(y_46_im, x_46_im, -(((y_46_im * y_46_im) * x_46_re) / y_46_re)) / y_46_re), -1.0, -x_46_re) / y_46_re);
	} else if (y_46_im <= 3e+153) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (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 = fma(Float64(x_46_re / y_46_im), Float64(y_46_re / y_46_im), Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.9e-11)
		tmp = t_0;
	elseif (y_46_im <= 2.7e-162)
		tmp = Float64(-Float64(fma(Float64(fma(y_46_im, x_46_im, Float64(-Float64(Float64(Float64(y_46_im * y_46_im) * x_46_re) / y_46_re))) / y_46_re), -1.0, Float64(-x_46_re)) / y_46_re));
	elseif (y_46_im <= 3e+153)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	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$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.9e-11], t$95$0, If[LessEqual[y$46$im, 2.7e-162], (-N[(N[(N[(N[(y$46$im * x$46$im + (-N[(N[(N[(y$46$im * y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision])), $MachinePrecision] / y$46$re), $MachinePrecision] * -1.0 + (-x$46$re)), $MachinePrecision] / y$46$re), $MachinePrecision]), If[LessEqual[y$46$im, 3e+153], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.9e-11 or 3.00000000000000019e153 < y.im

   1. Initial program 43.2%

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

   2. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if -2.9e-11 < y.im < 2.69999999999999984e-162

   1. Initial program 73.7%

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

   2. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 85.9%

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

   if 2.69999999999999984e-162 < y.im < 3.00000000000000019e153

   1. Initial program 74.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 74.4

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

   3. Applied rewrites 74.4%

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

Alternative 4: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 130000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.re} \cdot y.im - \left(-x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.45e+129)
   (/ (fma x.im (/ y.im y.re) x.re) y.re)
   (if (<= y.re 130000000.0)
     (/ (fma x.re (/ y.re y.im) x.im) y.im)
     (/ (- (* (/ x.im y.re) y.im) (- x.re)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.45e+129) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_re <= 130000000.0) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = (((x_46_im / y_46_re) * y_46_im) - -x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.45e+129)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_re <= 130000000.0)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_re) * y_46_im) - Float64(-x_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.45e+129], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 130000000.0], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision] - (-x$46$re)), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.45e129

   1. Initial program 34.7%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

   if -2.45e129 < y.re < 1.3e8

   1. Initial program 73.9%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 73.6

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

   4. Applied rewrites 73.6%

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

   if 1.3e8 < y.re

   1. Initial program 49.4%

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

   2. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y.re around inf

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

      1. div-sub N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 78.4

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

   7. Applied rewrites 78.4%

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

Alternative 5: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 130000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -2.45e+129)
     t_0
     (if (<= y.re 130000000.0) (/ (fma x.re (/ y.re y.im) x.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 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2.45e+129) {
		tmp = t_0;
	} else if (y_46_re <= 130000000.0) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_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(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.45e+129)
		tmp = t_0;
	elseif (y_46_re <= 130000000.0)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.45e+129], t$95$0, If[LessEqual[y$46$re, 130000000.0], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.45e129 or 1.3e8 < y.re

   1. Initial program 43.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.7

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

   4. Applied rewrites 80.7%

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

   if -2.45e129 < y.re < 1.3e8

   1. Initial program 73.9%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 6: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9.2e-11)
   (/ x.im y.im)
   (if (<= y.im 2e-21)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 3.5e+153)
       (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9.2e-11) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-21) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 3.5e+153) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.2e-11)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2e-21)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 3.5e+153)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9.2e-11], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2e-21], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.5e+153], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.20000000000000054e-11 or 3.4999999999999999e153 < y.im

   1. Initial program 43.2%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.9

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

   4. Applied rewrites 68.9%

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

   if -9.20000000000000054e-11 < y.im < 1.99999999999999982e-21

   1. Initial program 75.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.9

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

   4. Applied rewrites 82.9%

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

   if 1.99999999999999982e-21 < y.im < 3.4999999999999999e153

   1. Initial program 70.5%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.2

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

   4. Applied rewrites 58.2%

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

Alternative 7: 64.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e-12)
   (/ x.im y.im)
   (if (<= y.im 2.95e-117)
     (/ (fma y.im x.im (* y.re x.re)) (* y.re y.re))
     (if (<= y.im 3.5e+153)
       (* x.im (/ y.im (fma y.re y.re (* y.im y.im))))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e-12) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.95e-117) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_im <= 3.5e+153) {
		tmp = x_46_im * (y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e-12)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.95e-117)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 3.5e+153)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e-12], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.95e-117], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e+153], N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.24999999999999992e-12 or 3.4999999999999999e153 < y.im

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if -1.24999999999999992e-12 < y.im < 2.9500000000000002e-117

   1. Initial program 74.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 64.6

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

   7. Applied rewrites 64.6%

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

   if 2.9500000000000002e-117 < y.im < 3.4999999999999999e153

   1. Initial program 73.9%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.9

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

   4. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 54.9

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

   6. Applied rewrites 54.9%

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

Alternative 8: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e-12)
   (/ x.im y.im)
   (if (<= y.im 2.95e-117)
     (/ (fma y.im x.im (* y.re x.re)) (* y.re y.re))
     (if (<= y.im 3.5e+153)
       (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e-12) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.95e-117) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_im <= 3.5e+153) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e-12)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.95e-117)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 3.5e+153)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e-12], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.95e-117], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e+153], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.24999999999999992e-12 or 3.4999999999999999e153 < y.im

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.7

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

   4. Applied rewrites 68.7%

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

   if -1.24999999999999992e-12 < y.im < 2.9500000000000002e-117

   1. Initial program 74.1%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 64.6

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

   7. Applied rewrites 64.6%

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

   if 2.9500000000000002e-117 < y.im < 3.4999999999999999e153

   1. Initial program 73.9%

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

   2. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.9

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

   4. Applied rewrites 54.9%

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

Alternative 9: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e-12)
   (/ x.im y.im)
   (if (<= y.im 1.6e-21)
     (/ (fma y.im x.im (* y.re x.re)) (* y.re y.re))
     (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e-12) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.6e-21) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e-12)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.6e-21)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e-12], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.6e-21], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.24999999999999992e-12 or 1.6000000000000001e-21 < y.im

   1. Initial program 50.4%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 63.2

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

   4. Applied rewrites 63.2%

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

   if -1.24999999999999992e-12 < y.im < 1.6000000000000001e-21

   1. Initial program 75.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.9

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 62.1

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

   7. Applied rewrites 62.1%

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

Alternative 10: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e-11)
   (/ x.im y.im)
   (if (<= y.im 2e-21) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e-11) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-21) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.25d-11)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2d-21) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e-11) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-21) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.25e-11:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2e-21:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e-11)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2e-21)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.25e-11)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2e-21)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e-11], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2e-21], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.25000000000000005e-11 or 1.99999999999999982e-21 < y.im

   1. Initial program 50.4%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

   if -1.25000000000000005e-11 < y.im < 1.99999999999999982e-21

   1. Initial program 75.0%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 65.4

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

   4. Applied rewrites 65.4%

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

Alternative 11: 42.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.3%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 42.1

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

4. Applied rewrites 42.1%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))