_divideComplex, imaginary part

Percentage Accurate: 61.5% → 83.9%

Time: 10.7s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_1 := \mathsf{fma}\left(\frac{y.re}{t\_0}, x.im, \frac{y.im \cdot \left(-x.re\right)}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im)))
        (t_1 (fma (/ y.re t_0) x.im (/ (* y.im (- x.re)) t_0)))
        (t_2 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -1.55e+149)
     t_2
     (if (<= y.im -1.6e-117)
       t_1
       (if (<= y.im 3.3e-131)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 8.2e+154) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = fma((y_46_re / t_0), x_46_im, ((y_46_im * -x_46_re) / t_0));
	double t_2 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.55e+149) {
		tmp = t_2;
	} else if (y_46_im <= -1.6e-117) {
		tmp = t_1;
	} else if (y_46_im <= 3.3e-131) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.2e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	t_1 = fma(Float64(y_46_re / t_0), x_46_im, Float64(Float64(y_46_im * Float64(-x_46_re)) / t_0))
	t_2 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.55e+149)
		tmp = t_2;
	elseif (y_46_im <= -1.6e-117)
		tmp = t_1;
	elseif (y_46_im <= 3.3e-131)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.2e+154)
		tmp = t_1;
	else
		tmp = t_2;
	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 * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im + N[(N[(y$46$im * (-x$46$re)), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e+149], t$95$2, If[LessEqual[y$46$im, -1.6e-117], t$95$1, If[LessEqual[y$46$im, 3.3e-131], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+154], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.54999999999999993e149 or 8.2e154 < y.im

   1. Initial program 26.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 90.1

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

   5. Simplified 90.1%

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

   if -1.54999999999999993e149 < y.im < -1.59999999999999998e-117 or 3.3000000000000002e-131 < y.im < 8.2e154

   1. Initial program 84.2%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 87.0

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

   4. Applied egg-rr 87.0%

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

   if -1.59999999999999998e-117 < y.im < 3.3000000000000002e-131

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 97.6

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

   5. Simplified 97.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}, x.im, \frac{y.im \cdot \left(-x.re\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}, x.im, \frac{y.im \cdot \left(-x.re\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-122}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-y.re, x.im, y.im \cdot x.re\right) \cdot \frac{-1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -1.55e+149)
     t_0
     (if (<= y.im -8.5e-71)
       (/ (- (* y.re x.im) (* y.im x.re)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 1.1e-122)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 8.2e+154)
           (*
            (fma (- y.re) x.im (* y.im x.re))
            (/ -1.0 (fma y.im y.im (* y.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(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.55e+149) {
		tmp = t_0;
	} else if (y_46_im <= -8.5e-71) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 1.1e-122) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.2e+154) {
		tmp = fma(-y_46_re, x_46_im, (y_46_im * x_46_re)) * (-1.0 / fma(y_46_im, y_46_im, (y_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(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.55e+149)
		tmp = t_0;
	elseif (y_46_im <= -8.5e-71)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.1e-122)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.2e+154)
		tmp = Float64(fma(Float64(-y_46_re), x_46_im, Float64(y_46_im * x_46_re)) * Float64(-1.0 / fma(y_46_im, y_46_im, Float64(y_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[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e+149], t$95$0, If[LessEqual[y$46$im, -8.5e-71], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e-122], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+154], N[(N[((-y$46$re) * x$46$im + N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.54999999999999993e149 or 8.2e154 < y.im

   1. Initial program 26.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 90.1

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

   5. Simplified 90.1%

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

   if -1.54999999999999993e149 < y.im < -8.49999999999999988e-71

   1. Initial program 85.6%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 85.6

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

   4. Applied egg-rr 85.6%

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

   if -8.49999999999999988e-71 < y.im < 1.1e-122

   1. Initial program 74.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 94.6

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

   5. Simplified 94.6%

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

   if 1.1e-122 < y.im < 8.2e154

   1. Initial program 85.6%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 85.7

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

   4. Applied egg-rr 85.7%

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-neg-out N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 85.8

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

   6. Applied egg-rr 85.8%

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

Alternative 3: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t\_0}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(y.re, -x.im, y.im \cdot x.re\right) \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im)))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -1.65e+149)
     t_1
     (if (<= y.im -9e-71)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.im 3.4e-124)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 8.2e+154)
           (* (fma y.re (- x.im) (* y.im x.re)) (/ -1.0 t_0))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.65e+149) {
		tmp = t_1;
	} else if (y_46_im <= -9e-71) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_im <= 3.4e-124) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.2e+154) {
		tmp = fma(y_46_re, -x_46_im, (y_46_im * x_46_re)) * (-1.0 / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.65e+149)
		tmp = t_1;
	elseif (y_46_im <= -9e-71)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_im <= 3.4e-124)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.2e+154)
		tmp = Float64(fma(y_46_re, Float64(-x_46_im), Float64(y_46_im * x_46_re)) * Float64(-1.0 / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+149], t$95$1, If[LessEqual[y$46$im, -9e-71], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 3.4e-124], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+154], N[(N[(y$46$re * (-x$46$im) + N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.65e149 or 8.2e154 < y.im

   1. Initial program 26.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 90.1

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

   5. Simplified 90.1%

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

   if -1.65e149 < y.im < -9.0000000000000004e-71

   1. Initial program 85.6%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 85.6

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

   4. Applied egg-rr 85.6%

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

   if -9.0000000000000004e-71 < y.im < 3.4000000000000001e-124

   1. Initial program 74.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 94.6

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

   5. Simplified 94.6%

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

   if 3.4000000000000001e-124 < y.im < 8.2e154

   1. Initial program 85.6%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 85.7

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

   4. Applied egg-rr 85.7%

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

Alternative 4: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-118}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* y.re x.im) (* y.im x.re)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -1.65e+149)
     t_1
     (if (<= y.im -9.6e-71)
       t_0
       (if (<= y.im 4e-118)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 8.2e+154) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.65e+149) {
		tmp = t_1;
	} else if (y_46_im <= -9.6e-71) {
		tmp = t_0;
	} else if (y_46_im <= 4e-118) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.2e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.65e+149)
		tmp = t_1;
	elseif (y_46_im <= -9.6e-71)
		tmp = t_0;
	elseif (y_46_im <= 4e-118)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.2e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+149], t$95$1, If[LessEqual[y$46$im, -9.6e-71], t$95$0, If[LessEqual[y$46$im, 4e-118], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65e149 or 8.2e154 < y.im

   1. Initial program 26.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 90.1

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

   5. Simplified 90.1%

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

   if -1.65e149 < y.im < -9.6e-71 or 3.99999999999999994e-118 < y.im < 8.2e154

   1. Initial program 85.5%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 85.5

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

   4. Applied egg-rr 85.5%

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

   if -9.6e-71 < y.im < 3.99999999999999994e-118

   1. Initial program 74.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 94.6

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

   5. Simplified 94.6%

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

Alternative 5: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e-28)
   (/ (fma (- y.im) (/ x.re y.re) x.im) y.re)
   (if (<= y.re 1.12e-13)
     (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e-28) {
		tmp = fma(-y_46_im, (x_46_re / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_re <= 1.12e-13) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.1e-28)
		tmp = Float64(fma(Float64(-y_46_im), Float64(x_46_re / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_re <= 1.12e-13)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.1e-28], N[(N[((-y$46$im) * N[(x$46$re / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.12e-13], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.10000000000000006e-28

   1. Initial program 61.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 69.6

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

   5. Simplified 69.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. /-lowering-/.f64 75.5

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

   7. Applied egg-rr 75.5%

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

   if -2.10000000000000006e-28 < y.re < 1.12e-13

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 82.6

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

   5. Simplified 82.6%

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

   if 1.12e-13 < y.re

   1. Initial program 54.9%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 83.7

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

   5. Simplified 83.7%

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

Alternative 6: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))
   (if (<= y.re -2.7e-31)
     t_0
     (if (<= y.re 1.3e-13) (/ (fma y.re (/ x.im y.im) (- x.re)) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.7e-31) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-13) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.7e-31)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-13)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.7e-31], t$95$0, If[LessEqual[y$46$re, 1.3e-13], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.70000000000000014e-31 or 1.3e-13 < y.re

   1. Initial program 57.9%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 76.9

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

   5. Simplified 76.9%

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

   if -2.70000000000000014e-31 < y.re < 1.3e-13

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 82.6

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

   5. Simplified 82.6%

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

Alternative 7: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-75}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))
   (if (<= y.re -7.2e-66) t_0 (if (<= y.re 2.95e-75) (- (/ x.re y.im)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.2e-66) {
		tmp = t_0;
	} else if (y_46_re <= 2.95e-75) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    if (y_46re <= (-7.2d-66)) then
        tmp = t_0
    else if (y_46re <= 2.95d-75) then
        tmp = -(x_46re / y_46im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.2e-66) {
		tmp = t_0;
	} else if (y_46_re <= 2.95e-75) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -7.2e-66:
		tmp = t_0
	elif y_46_re <= 2.95e-75:
		tmp = -(x_46_re / y_46_im)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -7.2e-66)
		tmp = t_0;
	elseif (y_46_re <= 2.95e-75)
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -7.2e-66)
		tmp = t_0;
	elseif (y_46_re <= 2.95e-75)
		tmp = -(x_46_re / y_46_im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e-66], t$95$0, If[LessEqual[y$46$re, 2.95e-75], (-N[(x$46$re / y$46$im), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000025e-66 or 2.95e-75 < y.re

   1. Initial program 59.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 73.0

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

   5. Simplified 73.0%

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

   if -7.20000000000000025e-66 < y.re < 2.95e-75

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 75.3

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

   5. Simplified 75.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-75}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.32 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-14}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.7e+168)
   (/ x.im y.re)
   (if (<= y.re -1.32e-67)
     (/ (- (* y.re x.im) (* y.im x.re)) (* y.re y.re))
     (if (<= y.re 7.2e-14) (- (/ x.re y.im)) (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.7e+168) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.32e-67) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_re <= 7.2e-14) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = x_46_im / 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 <= (-2.7d+168)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-1.32d-67)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / (y_46re * y_46re)
    else if (y_46re <= 7.2d-14) then
        tmp = -(x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.7e+168) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.32e-67) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_re <= 7.2e-14) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.7e+168:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -1.32e-67:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re)
	elif y_46_re <= 7.2e-14:
		tmp = -(x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.7e+168)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.32e-67)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(y_46_re * y_46_re));
	elseif (y_46_re <= 7.2e-14)
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.7e+168)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -1.32e-67)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	elseif (y_46_re <= 7.2e-14)
		tmp = -(x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.7e+168], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.32e-67], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-14], (-N[(x$46$re / y$46$im), $MachinePrecision]), N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.70000000000000016e168 or 7.1999999999999996e-14 < y.re

   1. Initial program 50.2%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 73.1

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

   5. Simplified 73.1%

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

   if -2.70000000000000016e168 < y.re < -1.3199999999999999e-67

   1. Initial program 73.5%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 61.9

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

   5. Simplified 61.9%

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

   if -1.3199999999999999e-67 < y.re < 7.1999999999999996e-14

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 71.4

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

   5. Simplified 71.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.32 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-14}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.55 \cdot 10^{-61}:\\ \;\;\;\;y.re \cdot \frac{x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.3e+47)
   (/ x.im y.re)
   (if (<= y.re -1.55e-61)
     (* y.re (/ x.im (fma y.re y.re (* y.im y.im))))
     (if (<= y.re 2e-13) (- (/ x.re y.im)) (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.3e+47) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.55e-61) {
		tmp = y_46_re * (x_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 2e-13) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.3e+47)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.55e-61)
		tmp = Float64(y_46_re * Float64(x_46_im / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 2e-13)
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.3e+47], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.55e-61], N[(y$46$re * N[(x$46$im / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e-13], (-N[(x$46$re / y$46$im), $MachinePrecision]), N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.30000000000000002e47 or 2.0000000000000001e-13 < y.re

   1. Initial program 53.7%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 70.6

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

   5. Simplified 70.6%

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

   if -1.30000000000000002e47 < y.re < -1.54999999999999997e-61

   1. Initial program 74.8%

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

   3. Taylor expanded in x.im around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 61.5

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

   7. Applied egg-rr 61.5%

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

   if -1.54999999999999997e-61 < y.re < 2.0000000000000001e-13

   1. Initial program 76.7%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 69.7

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

   5. Simplified 69.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.55 \cdot 10^{-61}:\\ \;\;\;\;y.re \cdot \frac{x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.2e-61)
   (/ x.im y.re)
   (if (<= y.re 1.1e-13) (- (/ x.re y.im)) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.2e-61) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.1e-13) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = x_46_im / 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 <= (-7.2d-61)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 1.1d-13) then
        tmp = -(x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.2e-61) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.1e-13) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.2e-61:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 1.1e-13:
		tmp = -(x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.2e-61)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.1e-13)
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.2e-61)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 1.1e-13)
		tmp = -(x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.2e-61], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.1e-13], (-N[(x$46$re / y$46$im), $MachinePrecision]), N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000028e-61 or 1.09999999999999998e-13 < y.re

   1. Initial program 57.7%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 65.6

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

   5. Simplified 65.6%

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

   if -7.20000000000000028e-61 < y.re < 1.09999999999999998e-13

   1. Initial program 76.7%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 69.7

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

   5. Simplified 69.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-13}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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_46re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 42.4

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

5. Simplified 42.4%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))