_divideComplex, real part

Percentage Accurate: 61.6% → 79.8%

Time: 10.6s

Alternatives: 10

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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: 61.6% 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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im + \frac{\mathsf{fma}\left(x.im, -\frac{y.re \cdot y.re}{y.im}, y.re \cdot x.re\right)}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -3.7e+59)
     t_0
     (if (<= y.re 1.85e-125)
       (/
        (+ x.im (/ (fma x.im (- (/ (* y.re y.re) y.im)) (* y.re x.re)) y.im))
        y.im)
       (if (<= y.re 6.2e+48)
         (*
          (fma x.re y.re (* x.im y.im))
          (/ 1.0 (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_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -3.7e+59) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-125) {
		tmp = (x_46_im + (fma(x_46_im, -((y_46_re * y_46_re) / y_46_im), (y_46_re * x_46_re)) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 6.2e+48) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) * (1.0 / 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 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.7e+59)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-125)
		tmp = Float64(Float64(x_46_im + Float64(fma(x_46_im, Float64(-Float64(Float64(y_46_re * y_46_re) / y_46_im)), Float64(y_46_re * x_46_re)) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 6.2e+48)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) * Float64(1.0 / 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[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.7e+59], t$95$0, If[LessEqual[y$46$re, 1.85e-125], N[(N[(x$46$im + N[(N[(x$46$im * (-N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]) + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+48], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.69999999999999997e59 or 6.20000000000000011e48 < y.re

   1. Initial program 44.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 83.3%

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

   if -3.69999999999999997e59 < y.re < 1.85e-125

   1. Initial program 65.0%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 84.6%

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

   if 1.85e-125 < y.re < 6.20000000000000011e48

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 84.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 84.4

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

   4. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im + \frac{\mathsf{fma}\left(x.im, -\frac{y.re \cdot y.re}{y.im}, y.re \cdot x.re\right)}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -1.25e+60)
     t_0
     (if (<= y.re 1.85e-125)
       (/ (fma x.re (/ y.re y.im) x.im) y.im)
       (if (<= y.re 6.2e+48)
         (*
          (fma x.re y.re (* x.im y.im))
          (/ 1.0 (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_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -1.25e+60) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-125) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 6.2e+48) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) * (1.0 / 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 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.25e+60)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-125)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_re <= 6.2e+48)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) * Float64(1.0 / 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[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e+60], t$95$0, If[LessEqual[y$46$re, 1.85e-125], 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, 6.2e+48], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.24999999999999994e60 or 6.20000000000000011e48 < y.re

   1. Initial program 43.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 81.6%

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

   if -1.24999999999999994e60 < y.re < 1.85e-125

   1. Initial program 73.3%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if 1.85e-125 < y.re < 6.20000000000000011e48

   1. Initial program 76.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 76.7

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))))