_divideComplex, real part

Percentage Accurate: 61.8% → 85.3%

Time: 11.7s

Alternatives: 8

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: 61.8% 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: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)\\ t_1 := \frac{-1}{\mathsf{fma}\left(\frac{-y.im}{t\_0}, y.im, -\frac{y.re}{x.re}\right)}\\ \mathbf{if}\;x.re \leq -4.3 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.re \leq 52000000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y.re \cdot \frac{-1}{t\_0}, y.re, -\frac{y.im}{x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma x.re y.re (* y.im x.im)))
        (t_1 (/ -1.0 (fma (/ (- y.im) t_0) y.im (- (/ y.re x.re))))))
   (if (<= x.re -4.3e-23)
     t_1
     (if (<= x.re 52000000000.0)
       (/ -1.0 (fma (* y.re (/ -1.0 t_0)) y.re (- (/ y.im x.im))))
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, y_46_re, (y_46_im * x_46_im));
	double t_1 = -1.0 / fma((-y_46_im / t_0), y_46_im, -(y_46_re / x_46_re));
	double tmp;
	if (x_46_re <= -4.3e-23) {
		tmp = t_1;
	} else if (x_46_re <= 52000000000.0) {
		tmp = -1.0 / fma((y_46_re * (-1.0 / t_0)), y_46_re, -(y_46_im / x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im))
	t_1 = Float64(-1.0 / fma(Float64(Float64(-y_46_im) / t_0), y_46_im, Float64(-Float64(y_46_re / x_46_re))))
	tmp = 0.0
	if (x_46_re <= -4.3e-23)
		tmp = t_1;
	elseif (x_46_re <= 52000000000.0)
		tmp = Float64(-1.0 / fma(Float64(y_46_re * Float64(-1.0 / t_0)), y_46_re, Float64(-Float64(y_46_im / x_46_im))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(N[((-y$46$im) / t$95$0), $MachinePrecision] * y$46$im + (-N[(y$46$re / x$46$re), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -4.3e-23], t$95$1, If[LessEqual[x$46$re, 52000000000.0], N[(-1.0 / N[(N[(y$46$re * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * y$46$re + (-N[(y$46$im / x$46$im), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.30000000000000002e-23 or 5.2e10 < x.re

   1. Initial program 56.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. distribute-frac-neg N/A

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

      17. lower-neg.f64 N/A

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in y.re around inf

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

       1. lower-/.f64 87.5

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

   11. Applied rewrites 87.5%

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

   if -4.30000000000000002e-23 < x.re < 5.2e10

   1. Initial program 68.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 77.2%

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

   7. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 91.7

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

   9. Applied rewrites 91.7%

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \left(-y.re\right), y.re, \color{blue}{\frac{-y.im}{x.im}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{-y.im}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}, y.im, -\frac{y.re}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 52000000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y.re \cdot \frac{-1}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}, y.re, -\frac{y.im}{x.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{-y.im}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}, y.im, -\frac{y.re}{x.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im + \frac{\mathsf{fma}\left(x.im, \frac{y.re \cdot \left(-y.re\right)}{y.im}, x.re \cdot y.re\right)}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.re -8e+150)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (if (<= y.re -5.5e-62)
       t_0
       (if (<= y.re 9.4e-94)
         (/
          (+ x.im (/ (fma x.im (/ (* y.re (- y.re)) y.im) (* x.re y.re)) y.im))
          y.im)
         (if (<= y.re 5.9e+119)
           t_0
           (/ (fma x.im (/ y.im y.re) 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 = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -8e+150) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -5.5e-62) {
		tmp = t_0;
	} else if (y_46_re <= 9.4e-94) {
		tmp = (x_46_im + (fma(x_46_im, ((y_46_re * -y_46_re) / y_46_im), (x_46_re * y_46_re)) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 5.9e+119) {
		tmp = t_0;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), 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(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -8e+150)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -5.5e-62)
		tmp = t_0;
	elseif (y_46_re <= 9.4e-94)
		tmp = Float64(Float64(x_46_im + Float64(fma(x_46_im, Float64(Float64(y_46_re * Float64(-y_46_re)) / y_46_im), Float64(x_46_re * y_46_re)) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 5.9e+119)
		tmp = t_0;
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), 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[(x$46$re * y$46$re + N[(y$46$im * x$46$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, -8e+150], 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, -5.5e-62], t$95$0, If[LessEqual[y$46$re, 9.4e-94], N[(N[(x$46$im + N[(N[(x$46$im * N[(N[(y$46$re * (-y$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.9e+119], t$95$0, N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.99999999999999985e150

   1. Initial program 27.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.2%

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

   if -7.99999999999999985e150 < y.re < -5.50000000000000022e-62 or 9.40000000000000007e-94 < y.re < 5.9000000000000001e119

   1. Initial program 79.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 79.4

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 79.4

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

   4. Applied rewrites 79.4%

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

   if -5.50000000000000022e-62 < y.re < 9.40000000000000007e-94

   1. Initial program 69.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 92.2%

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

   if 5.9000000000000001e119 < y.re

   1. Initial program 33.0%

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

   3. Taylor expanded in y.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 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im + \frac{\mathsf{fma}\left(x.im, \frac{y.re \cdot \left(-y.re\right)}{y.im}, x.re \cdot y.re\right)}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.re -8e+150)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (if (<= y.re -2.35e-66)
       t_0
       (if (<= y.re 9.4e-94)
         (/ (fma x.re (/ y.re y.im) x.im) y.im)
         (if (<= y.re 5.9e+119)
           t_0
           (/ (fma x.im (/ y.im y.re) 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 = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -8e+150) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -2.35e-66) {
		tmp = t_0;
	} else if (y_46_re <= 9.4e-94) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 5.9e+119) {
		tmp = t_0;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), 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(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -8e+150)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -2.35e-66)
		tmp = t_0;
	elseif (y_46_re <= 9.4e-94)
		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+119)
		tmp = t_0;
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), 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[(x$46$re * y$46$re + N[(y$46$im * x$46$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, -8e+150], 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, -2.35e-66], t$95$0, If[LessEqual[y$46$re, 9.4e-94], 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+119], t$95$0, N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.99999999999999985e150

   1. Initial program 27.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.2%

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

   if -7.99999999999999985e150 < y.re < -2.35e-66 or 9.40000000000000007e-94 < y.re < 5.9000000000000001e119

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 79.6

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -2.35e-66 < y.re < 9.40000000000000007e-94

   1. Initial program 69.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

   if 5.9000000000000001e119 < y.re

   1. Initial program 33.0%

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

   3. Taylor expanded in y.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 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.55 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (/ x.re (fma y.re y.re (* y.im y.im))))))
   (if (<= y.re -3.5e+91)
     (/ x.re y.re)
     (if (<= y.re -1.55e-64)
       t_0
       (if (<= y.re 1.4e-109)
         (/ x.im y.im)
         (if (<= y.re 1.6e+112) t_0 (/ 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 = y_46_re * (x_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -3.5e+91) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.55e-64) {
		tmp = t_0;
	} else if (y_46_re <= 1.4e-109) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1.6e+112) {
		tmp = t_0;
	} else {
		tmp = 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(y_46_re * Float64(x_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))))
	tmp = 0.0
	if (y_46_re <= -3.5e+91)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.55e-64)
		tmp = t_0;
	elseif (y_46_re <= 1.4e-109)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1.6e+112)
		tmp = t_0;
	else
		tmp = 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[(y$46$re * N[(x$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+91], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.55e-64], t$95$0, If[LessEqual[y$46$re, 1.4e-109], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.6e+112], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.50000000000000001e91 or 1.59999999999999993e112 < y.re

   1. Initial program 40.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -3.50000000000000001e91 < y.re < -1.55000000000000012e-64 or 1.39999999999999989e-109 < y.re < 1.59999999999999993e112

   1. Initial program 79.3%

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

   3. Taylor expanded in x.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 59.6%

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

   if -1.55000000000000012e-64 < y.re < 1.39999999999999989e-109

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

Alternative 5: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.45e+105)
   (/ x.im y.im)
   (if (<= y.im 5.6e+16)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 1.45e+75)
       (/ (fma x.re y.re (* y.im x.im)) (* 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 <= -1.45e+105) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.6e+16) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.45e+75) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / (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 <= -1.45e+105)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 5.6e+16)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.45e+75)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / 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, -1.45e+105], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.6e+16], 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, 1.45e+75], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.45000000000000005e105 or 1.4499999999999999e75 < y.im

   1. Initial program 44.0%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -1.45000000000000005e105 < y.im < 5.6e16

   1. Initial program 70.3%

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

   3. Taylor expanded in y.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 79.0

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

   5. Applied rewrites 79.0%

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

   if 5.6e16 < y.im < 1.4499999999999999e75

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -1.15e+99)
     t_0
     (if (<= y.im 2.25e+15) (/ (fma x.im (/ y.im y.re) x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.15e+99) {
		tmp = t_0;
	} else if (y_46_im <= 2.25e+15) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.15e+99)
		tmp = t_0;
	elseif (y_46_im <= 2.25e+15)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+99], t$95$0, If[LessEqual[y$46$im, 2.25e+15], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.1500000000000001e99 or 2.25e15 < y.im

   1. Initial program 51.0%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.1500000000000001e99 < y.im < 2.25e15

   1. Initial program 70.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 7: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.5e+14)
   (/ x.re y.re)
   (if (<= y.re 6.6e-44) (/ x.im 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 <= -5.5e+14) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.6e-44) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y_46re <= (-5.5d+14)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 6.6d-44) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.5e+14) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.6e-44) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5.5e+14:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 6.6e-44:
		tmp = x_46_im / y_46_im
	else:
		tmp = 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 <= -5.5e+14)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 6.6e-44)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5.5e+14)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 6.6e-44)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.5e+14], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.6e-44], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.5e14 or 6.60000000000000011e-44 < y.re

   1. Initial program 53.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -5.5e14 < y.re < 6.60000000000000011e-44

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

Alternative 8: 43.3% 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

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_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 62.4%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 40.6

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

5. Applied rewrites 40.6%

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

Reproduce

herbie shell --seed 2024233 
(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))))