_divideComplex, imaginary part

Percentage Accurate: 61.5% → 83.2%

Time: 12.3s

Alternatives: 13

Speedup: 1.4×

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: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+152}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{0 - x.re}{t\_0}, \mathsf{fma}\left(y.re, \frac{x.im}{t\_0}, 0\right)\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (/ (fma y.re (/ x.im y.im) (- 0.0 x.re)) y.im)))
   (if (<= y.im -7.5e+152)
     t_1
     (if (<= y.im -1.15e-96)
       (fma y.im (/ (- 0.0 x.re) t_0) (fma y.re (/ x.im t_0) 0.0))
       (if (<= y.im 3.4e-165)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 7e+79)
           (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), (0.0 - x_46_re)) / y_46_im;
	double tmp;
	if (y_46_im <= -7.5e+152) {
		tmp = t_1;
	} else if (y_46_im <= -1.15e-96) {
		tmp = fma(y_46_im, ((0.0 - x_46_re) / t_0), fma(y_46_re, (x_46_im / t_0), 0.0));
	} else if (y_46_im <= 3.4e-165) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 7e+79) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(0.0 - x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.5e+152)
		tmp = t_1;
	elseif (y_46_im <= -1.15e-96)
		tmp = fma(y_46_im, Float64(Float64(0.0 - x_46_re) / t_0), fma(y_46_re, Float64(x_46_im / t_0), 0.0));
	elseif (y_46_im <= 3.4e-165)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 7e+79)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + N[(0.0 - x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+152], t$95$1, If[LessEqual[y$46$im, -1.15e-96], N[(y$46$im * N[(N[(0.0 - x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$im / t$95$0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4e-165], 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, 7e+79], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.50000000000000046e152 or 6.99999999999999961e79 < y.im

   1. Initial program 39.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

      16. --lowering--.f64 91.2

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}} \]

   if -7.50000000000000046e152 < y.im < -1.15e-96

   1. Initial program 79.6%

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

   3. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

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

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

      7. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{\color{blue}{0 - \left({y.im}^{2} + {y.re}^{2}\right)}}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

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

         \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{\color{blue}{0 - \left({y.im}^{2} + {y.re}^{2}\right)}}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \left(\color{blue}{y.im \cdot y.im} + {y.re}^{2}\right)}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)}, \frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}} + 0}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{\color{blue}{y.re \cdot x.im}}{{y.im}^{2} + {y.re}^{2}} + 0\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \color{blue}{y.re \cdot \frac{x.im}{{y.im}^{2} + {y.re}^{2}}} + 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \color{blue}{\mathsf{fma}\left(y.re, \frac{x.im}{{y.im}^{2} + {y.re}^{2}}, 0\right)}\right) \]

   5. Simplified 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \frac{x.re}{0 - \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \mathsf{fma}\left(y.re, \frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, 0\right)\right)} \]

   if -1.15e-96 < y.im < 3.4e-165

   1. Initial program 69.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 95.1

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

   5. Simplified 95.1%

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

   if 3.4e-165 < y.im < 6.99999999999999961e79

   1. Initial program 81.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{0 - x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \mathsf{fma}\left(y.re, \frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, 0\right)\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{0 - x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- 0.0 x.re)) y.im)))
   (if (<= y.im -6e+138)
     t_0
     (if (<= y.im -1.65e-65)
       (fma
        y.im
        (/ (- 0.0 x.re) (fma y.im y.im (* y.re y.re)))
        (/ (* y.re x.im) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 3.4e-165)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 1.2e+80)
           (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), (0.0 - x_46_re)) / y_46_im;
	double tmp;
	if (y_46_im <= -6e+138) {
		tmp = t_0;
	} else if (y_46_im <= -1.65e-65) {
		tmp = fma(y_46_im, ((0.0 - x_46_re) / fma(y_46_im, y_46_im, (y_46_re * y_46_re))), ((y_46_re * x_46_im) / fma(y_46_re, y_46_re, (y_46_im * y_46_im))));
	} else if (y_46_im <= 3.4e-165) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.2e+80) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(0.0 - x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6e+138)
		tmp = t_0;
	elseif (y_46_im <= -1.65e-65)
		tmp = fma(y_46_im, Float64(Float64(0.0 - x_46_re) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))), Float64(Float64(y_46_re * x_46_im) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 3.4e-165)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.2e+80)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + N[(0.0 - x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+138], t$95$0, If[LessEqual[y$46$im, -1.65e-65], N[(y$46$im * N[(N[(0.0 - x$46$re), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4e-165], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+80], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.0000000000000002e138 or 1.1999999999999999e80 < y.im

   1. Initial program 39.1%

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

   3. Taylor expanded in y.re around 0

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

      1. +-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

      16. --lowering--.f64 89.9

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

   5. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}} \]

   if -6.0000000000000002e138 < y.im < -1.6500000000000001e-65

   1. Initial program 82.5%

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

      1. 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. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-lowering-*.f64 93.0

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

   6. Applied egg-rr 93.0%

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

   if -1.6500000000000001e-65 < y.im < 3.4e-165

   1. Initial program 69.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 94.1

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

   5. Simplified 94.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 94.2

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

   7. Applied egg-rr 94.2%

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

   if 3.4e-165 < y.im < 1.1999999999999999e80

   1. Initial program 81.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{0 - x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -9.3 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma y.re (/ x.im y.im) (- 0.0 x.re)) y.im)))
   (if (<= y.im -9.3e+42)
     t_1
     (if (<= y.im -3.1e-91)
       t_0
       (if (<= y.im 3.4e-165)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 1.6e+80) 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)) / ((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), (0.0 - x_46_re)) / y_46_im;
	double tmp;
	if (y_46_im <= -9.3e+42) {
		tmp = t_1;
	} else if (y_46_im <= -3.1e-91) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e-165) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.6e+80) {
		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)) / Float64(Float64(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(0.0 - x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.3e+42)
		tmp = t_1;
	elseif (y_46_im <= -3.1e-91)
		tmp = t_0;
	elseif (y_46_im <= 3.4e-165)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.6e+80)
		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[(N[(y$46$re * y$46$re), $MachinePrecision] + 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] + N[(0.0 - x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.3e+42], t$95$1, If[LessEqual[y$46$im, -3.1e-91], t$95$0, If[LessEqual[y$46$im, 3.4e-165], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+80], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.3000000000000005e42 or 1.59999999999999995e80 < y.im

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

      16. --lowering--.f64 87.4

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}} \]

   if -9.3000000000000005e42 < y.im < -3.09999999999999981e-91 or 3.4e-165 < y.im < 1.59999999999999995e80

   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

   if -3.09999999999999981e-91 < y.im < 3.4e-165

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf

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

      1. /-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 94.9

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

   5. Simplified 94.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 94.9

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

   7. Applied egg-rr 94.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t\_0}{y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-158}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))) (t_1 (/ t_0 (* y.re y.re))))
   (if (<= y.re -1.95e+85)
     (/ x.im y.re)
     (if (<= y.re -4.2e-66)
       t_1
       (if (<= y.re 3.3e-158)
         (- 0.0 (/ x.re y.im))
         (if (<= y.re 1.6e-73)
           t_1
           (if (<= y.re 1.2e+46) (/ t_0 (* y.im 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 t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -1.95e+85) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.2e-66) {
		tmp = t_1;
	} else if (y_46_re <= 3.3e-158) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.6e-73) {
		tmp = t_1;
	} else if (y_46_re <= 1.2e+46) {
		tmp = t_0 / (y_46_im * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * x_46im) - (y_46im * x_46re)
    t_1 = t_0 / (y_46re * y_46re)
    if (y_46re <= (-1.95d+85)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-4.2d-66)) then
        tmp = t_1
    else if (y_46re <= 3.3d-158) then
        tmp = 0.0d0 - (x_46re / y_46im)
    else if (y_46re <= 1.6d-73) then
        tmp = t_1
    else if (y_46re <= 1.2d+46) then
        tmp = t_0 / (y_46im * 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 t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -1.95e+85) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.2e-66) {
		tmp = t_1;
	} else if (y_46_re <= 3.3e-158) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.6e-73) {
		tmp = t_1;
	} else if (y_46_re <= 1.2e+46) {
		tmp = t_0 / (y_46_im * 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):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	t_1 = t_0 / (y_46_re * y_46_re)
	tmp = 0
	if y_46_re <= -1.95e+85:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -4.2e-66:
		tmp = t_1
	elif y_46_re <= 3.3e-158:
		tmp = 0.0 - (x_46_re / y_46_im)
	elif y_46_re <= 1.6e-73:
		tmp = t_1
	elif y_46_re <= 1.2e+46:
		tmp = t_0 / (y_46_im * 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)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(t_0 / Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.95e+85)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -4.2e-66)
		tmp = t_1;
	elseif (y_46_re <= 3.3e-158)
		tmp = Float64(0.0 - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.6e-73)
		tmp = t_1;
	elseif (y_46_re <= 1.2e+46)
		tmp = Float64(t_0 / Float64(y_46_im * 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)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	t_1 = t_0 / (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.95e+85)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -4.2e-66)
		tmp = t_1;
	elseif (y_46_re <= 3.3e-158)
		tmp = 0.0 - (x_46_re / y_46_im);
	elseif (y_46_re <= 1.6e-73)
		tmp = t_1;
	elseif (y_46_re <= 1.2e+46)
		tmp = t_0 / (y_46_im * 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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.95e+85], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.2e-66], t$95$1, If[LessEqual[y$46$re, 3.3e-158], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-73], t$95$1, If[LessEqual[y$46$re, 1.2e+46], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.95000000000000017e85 or 1.20000000000000004e46 < y.re

   1. Initial program 49.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 77.4

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

   5. Simplified 77.4%

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

   if -1.95000000000000017e85 < y.re < -4.2000000000000001e-66 or 3.3000000000000002e-158 < y.re < 1.59999999999999993e-73

   1. Initial program 75.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{{y.re}^{2}}} \]
   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 60.3

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

   5. Simplified 60.3%

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

   if -4.2000000000000001e-66 < y.re < 3.3000000000000002e-158

   1. Initial program 68.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 87.0%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 76.2

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 76.2%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 76.2

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

   10. Applied egg-rr 76.2%

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

   if 1.59999999999999993e-73 < y.re < 1.20000000000000004e46

   1. Initial program 96.0%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 67.6

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

   5. Simplified 67.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-158}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-73}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.8e+105)
   (/ x.im y.re)
   (if (<= y.re -3.1e-127)
     (* x.im (/ y.re (fma y.re y.re (* y.im y.im))))
     (if (<= y.re 1e-73)
       (- 0.0 (/ x.re y.im))
       (if (<= y.re 2.5e+45)
         (/ (- (* y.re x.im) (* y.im x.re)) (* y.im 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 <= -4.8e+105) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.1e-127) {
		tmp = x_46_im * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 1e-73) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.5e+45) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_im * 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 <= -4.8e+105)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.1e-127)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 1e-73)
		tmp = Float64(0.0 - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.5e+45)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(y_46_im * 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, -4.8e+105], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-127], N[(x$46$im * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e-73], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+45], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.7999999999999995e105 or 2.5e45 < y.re

   1. Initial program 49.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 77.7

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

   5. Simplified 77.7%

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

   if -4.7999999999999995e105 < y.re < -3.1e-127

   1. Initial program 78.7%

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-lowering-*.f64 73.0

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

   4. Applied egg-rr 73.0%

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-lowering-*.f64 73.0

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

   6. Applied egg-rr 73.0%

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

   7. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 55.1

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

   9. Simplified 55.1%

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

   if -3.1e-127 < y.re < 9.99999999999999997e-74

   1. Initial program 66.2%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 80.2%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 71.1

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 71.1%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 71.1

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

   10. Applied egg-rr 71.1%

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

   if 9.99999999999999997e-74 < y.re < 2.5e45

   1. Initial program 96.2%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 65.3

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

   5. Simplified 65.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-73}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -6200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 42:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- 0.0 x.re)) y.im)))
   (if (<= y.im -6200000.0)
     t_0
     (if (<= y.im 42.0) (/ (- x.im (/ (* y.im x.re) 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), (0.0 - x_46_re)) / y_46_im;
	double tmp;
	if (y_46_im <= -6200000.0) {
		tmp = t_0;
	} else if (y_46_im <= 42.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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(0.0 - x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6200000.0)
		tmp = t_0;
	elseif (y_46_im <= 42.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / 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] + N[(0.0 - x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6200000.0], t$95$0, If[LessEqual[y$46$im, 42.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.2e6 or 42 < y.im

   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 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-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

      16. --lowering--.f64 82.4

          \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

   5. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, 0 - x.re\right)}{y.im}} \]

   if -6.2e6 < y.im < 42

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 86.5

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

   5. Simplified 86.5%

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

Alternative 7: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, 0 - x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -64000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 20:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.re y.im) (- 0.0 x.re)) y.im)))
   (if (<= y.im -64000.0)
     t_0
     (if (<= y.im 20.0) (/ (- x.im (/ (* y.im x.re) 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(x_46_im, (y_46_re / y_46_im), (0.0 - x_46_re)) / y_46_im;
	double tmp;
	if (y_46_im <= -64000.0) {
		tmp = t_0;
	} else if (y_46_im <= 20.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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(x_46_im, Float64(y_46_re / y_46_im), Float64(0.0 - x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -64000.0)
		tmp = t_0;
	elseif (y_46_im <= 20.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / 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[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + N[(0.0 - x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -64000.0], t$95$0, If[LessEqual[y$46$im, 20.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -64000 or 20 < y.im

   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. Step-by-step derivation

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}} \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}}} \cdot \frac{1}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{\frac{-1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)}} \]

   4. Applied egg-rr 53.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{-1}{\mathsf{fma}\left(y.re, -x.im, y.im \cdot x.re\right)} \cdot \left(0 - \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\right)}} \]

   5. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

      8. --lowering--.f64 81.3

         \[\leadsto \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, \color{blue}{0 - x.re}\right)}{y.im} \]

   7. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, 0 - x.re\right)}{y.im}} \]

   if -64000 < y.im < 20

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 86.5

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

   5. Simplified 86.5%

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

Alternative 8: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -55000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 12:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* y.re x.im) y.im) x.re) y.im)))
   (if (<= y.im -55000000.0)
     t_0
     (if (<= y.im 12.0) (/ (- x.im (/ (* y.im x.re) 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -55000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 12.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} 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 = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    if (y_46im <= (-55000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 12.0d0) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -55000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 12.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -55000000.0:
		tmp = t_0
	elif y_46_im <= 12.0:
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -55000000.0)
		tmp = t_0;
	elseif (y_46_im <= 12.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -55000000.0)
		tmp = t_0;
	elseif (y_46_im <= 12.0)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	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[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -55000000.0], t$95$0, If[LessEqual[y$46$im, 12.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.5e7 or 12 < y.im

   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 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 73.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 76.7

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

   8. Simplified 76.7%

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

   if -5.5e7 < y.im < 12

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 86.5

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

   5. Simplified 86.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -55000000:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 12:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.35e+42)
     t_0
     (if (<= y.im 5.8e+39) (/ (- x.im (/ (* y.im x.re) 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 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.35e+42) {
		tmp = t_0;
	} else if (y_46_im <= 5.8e+39) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} 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 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-1.35d+42)) then
        tmp = t_0
    else if (y_46im <= 5.8d+39) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    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 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.35e+42) {
		tmp = t_0;
	} else if (y_46_im <= 5.8e+39) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.35e+42:
		tmp = t_0
	elif y_46_im <= 5.8e+39:
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.35e+42)
		tmp = t_0;
	elseif (y_46_im <= 5.8e+39)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	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 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.35e+42)
		tmp = t_0;
	elseif (y_46_im <= 5.8e+39)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.35e+42], t$95$0, If[LessEqual[y$46$im, 5.8e+39], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.35e42 or 5.80000000000000059e39 < y.im

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 76.6%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 73.1

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 73.1%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 73.1

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

   10. Applied egg-rr 73.1%

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

   if -1.35e42 < y.im < 5.80000000000000059e39

   1. Initial program 75.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -2.55e+42)
     t_0
     (if (<= y.im 9e+41) (/ (- x.im (* x.re (/ 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 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.55e+42) {
		tmp = t_0;
	} else if (y_46_im <= 9e+41) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} 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 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-2.55d+42)) then
        tmp = t_0
    else if (y_46im <= 9d+41) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    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 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.55e+42) {
		tmp = t_0;
	} else if (y_46_im <= 9e+41) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.55e+42:
		tmp = t_0
	elif y_46_im <= 9e+41:
		tmp = (x_46_im - (x_46_re * (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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.55e+42)
		tmp = t_0;
	elseif (y_46_im <= 9e+41)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	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 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.55e+42)
		tmp = t_0;
	elseif (y_46_im <= 9e+41)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.55e+42], t$95$0, If[LessEqual[y$46$im, 9e+41], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.55e42 or 9.0000000000000002e41 < y.im

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 76.6%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 73.1

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 73.1%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 73.1

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

   10. Applied egg-rr 73.1%

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

   if -2.55e42 < y.im < 9.0000000000000002e41

   1. Initial program 75.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.4

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

   5. Simplified 81.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 81.4

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

   7. Applied egg-rr 81.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+42}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.9e+106)
   (/ x.im y.re)
   (if (<= y.re -3.2e-127)
     (* x.im (/ y.re (fma y.re y.re (* y.im y.im))))
     (if (<= y.re 1.9e+45) (- 0.0 (/ 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 <= -1.9e+106) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-127) {
		tmp = x_46_im * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 1.9e+45) {
		tmp = 0.0 - (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 <= -1.9e+106)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.2e-127)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 1.9e+45)
		tmp = Float64(0.0 - 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, -1.9e+106], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.2e-127], N[(x$46$im * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+45], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.8999999999999999e106 or 1.9000000000000001e45 < y.re

   1. Initial program 49.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 77.7

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

   5. Simplified 77.7%

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

   if -1.8999999999999999e106 < y.re < -3.20000000000000017e-127

   1. Initial program 78.7%

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

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-lowering-*.f64 73.0

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

   4. Applied egg-rr 73.0%

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-lowering-*.f64 73.0

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

   6. Applied egg-rr 73.0%

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

   7. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 55.1

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

   9. Simplified 55.1%

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

   if -3.20000000000000017e-127 < y.re < 1.9000000000000001e45

   1. Initial program 73.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 77.5%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 65.6

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 65.6%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 65.6

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

   10. Applied egg-rr 65.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -125000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+46}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -125000.0)
   (/ x.im y.re)
   (if (<= y.re 2.4e+46) (- 0.0 (/ 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 <= -125000.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+46) {
		tmp = 0.0 - (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 <= (-125000.0d0)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.4d+46) then
        tmp = 0.0d0 - (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 <= -125000.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+46) {
		tmp = 0.0 - (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 <= -125000.0:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.4e+46:
		tmp = 0.0 - (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 <= -125000.0)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.4e+46)
		tmp = Float64(0.0 - 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 <= -125000.0)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.4e+46)
		tmp = 0.0 - (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, -125000.0], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+46], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -125000 or 2.40000000000000008e46 < y.re

   1. Initial program 55.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 74.4

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

   5. Simplified 74.4%

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

   if -125000 < y.re < 2.40000000000000008e46

   1. Initial program 73.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + y.re \cdot \left(\frac{x.im}{{y.im}^{2}} + \frac{x.re \cdot y.re}{{y.im}^{3}}\right)} \]

   5. Simplified 73.2%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      3. --lowering--.f64 60.3

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

   8. Simplified 60.3%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   10. Applied egg-rr 60.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -125000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+46}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.6% 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 65.6%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 42.7

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

5. Simplified 42.7%

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

Reproduce

herbie shell --seed 2024196 
(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))))