_divideComplex, real part

Percentage Accurate: 62.0% → 83.8%

Time: 9.3s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, \frac{\mathsf{fma}\left(x.re, y.re, 0\right)}{t\_0}\right)\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 8.4 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

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 x.im (/ y.im t_0) (/ (fma x.re y.re 0.0) t_0))))
   (if (<= y.im -1.4e+115)
     (/ (fma (/ x.re y.im) y.re x.im) y.im)
     (if (<= y.im -8e-146)
       t_1
       (if (<= y.im 2.4e-52)
         (/ (fma y.im (/ x.im y.re) x.re) y.re)
         (if (<= y.im 8.4e+124)
           t_1
           (/ (fma x.re (/ y.re y.im) x.im) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(x_46_im, (y_46_im / t_0), (fma(x_46_re, y_46_re, 0.0) / t_0));
	double tmp;
	if (y_46_im <= -1.4e+115) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else if (y_46_im <= -8e-146) {
		tmp = t_1;
	} else if (y_46_im <= 2.4e-52) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 8.4e+124) {
		tmp = t_1;
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	}
	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 = fma(x_46_im, Float64(y_46_im / t_0), Float64(fma(x_46_re, y_46_re, 0.0) / t_0))
	tmp = 0.0
	if (y_46_im <= -1.4e+115)
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	elseif (y_46_im <= -8e-146)
		tmp = t_1;
	elseif (y_46_im <= 2.4e-52)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 8.4e+124)
		tmp = t_1;
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	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[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(N[(x$46$re * y$46$re + 0.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+115], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -8e-146], t$95$1, If[LessEqual[y$46$im, 2.4e-52], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.4e+124], t$95$1, N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.4e115

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 83.8

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

   5. Simplified 83.8%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 86.5

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

   7. Applied egg-rr 86.5%

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

   if -1.4e115 < y.im < -8.00000000000000021e-146 or 2.4000000000000002e-52 < y.im < 8.40000000000000046e124

   1. Initial program 74.8%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-rgt-identity N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 83.9

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

   5. Simplified 83.9%

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

   if -8.00000000000000021e-146 < y.im < 2.4000000000000002e-52

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 92.6

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

   5. Simplified 92.6%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 92.8

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

   7. Applied egg-rr 92.8%

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

   if 8.40000000000000046e124 < y.im

   1. Initial program 29.0%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 91.4

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

   5. Simplified 91.4%

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

Alternative 2: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -6.5e+42)
     t_0
     (if (<= y.im -1.2e-139)
       (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 1.1e-15) (/ (fma y.im (/ x.im y.re) x.re) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -6.5e+42) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-139) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 1.1e-15) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.5e+42)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-139)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.1e-15)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6.5e+42], t$95$0, If[LessEqual[y$46$im, -1.2e-139], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e-15], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.50000000000000052e42 or 1.09999999999999993e-15 < y.im

   1. Initial program 48.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 81.2

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

   5. Simplified 81.2%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 82.7

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

   7. Applied egg-rr 82.7%

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

   if -6.50000000000000052e42 < y.im < -1.20000000000000007e-139

   1. Initial program 78.4%

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

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

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

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

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

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

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

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

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

      5. *-lowering-*.f64 78.4

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

   4. Applied egg-rr 78.4%

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

   if -1.20000000000000007e-139 < y.im < 1.09999999999999993e-15

   1. Initial program 70.0%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 90.2

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

   5. Simplified 90.2%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 90.4

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.5e+15)
   (/ x.im y.im)
   (if (<= y.im 1.1e-15)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 6.5e+110)
       (/ (fma x.im y.im (* x.re y.re)) (* y.im y.im))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+15) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.1e-15) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 6.5e+110) {
		tmp = fma(x_46_im, y_46_im, (x_46_re * y_46_re)) / (y_46_im * y_46_im);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+15)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.1e-15)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 6.5e+110)
		tmp = Float64(fma(x_46_im, y_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.5e+15], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.1e-15], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.5e+110], N[(N[(x$46$im * y$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.5e15 or 6.4999999999999997e110 < y.im

   1. Initial program 43.2%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 74.9

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

   5. Simplified 74.9%

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

   if -2.5e15 < y.im < 1.09999999999999993e-15

   1. Initial program 71.6%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

   if 1.09999999999999993e-15 < y.im < 6.4999999999999997e110

   1. Initial program 80.0%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 75.7

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

   5. Simplified 75.7%

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

   6. Taylor expanded in y.im around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 71.6

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

   8. Simplified 71.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.5e+15)
   (/ x.im y.im)
   (if (<= y.im 3.5e-30)
     (/ x.re y.re)
     (if (<= y.im 7.5e+112)
       (/ (fma x.im y.im (* x.re y.re)) (* y.im y.im))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+15) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.5e-30) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 7.5e+112) {
		tmp = fma(x_46_im, y_46_im, (x_46_re * y_46_re)) / (y_46_im * y_46_im);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+15)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 3.5e-30)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 7.5e+112)
		tmp = Float64(fma(x_46_im, y_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.5e+15], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-30], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+112], N[(N[(x$46$im * y$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.5e15 or 7.5e112 < y.im

   1. Initial program 43.2%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 74.9

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

   5. Simplified 74.9%

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

   if -2.5e15 < y.im < 3.5000000000000003e-30

   1. Initial program 70.9%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 69.7

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

   5. Simplified 69.7%

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

   if 3.5000000000000003e-30 < y.im < 7.5e112

   1. Initial program 82.2%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 71.3

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

   5. Simplified 71.3%

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

   6. Taylor expanded in y.im around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 67.6

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

   8. Simplified 67.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -75000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -75000000000.0)
     t_0
     (if (<= y.im 1.4e-14) (/ (fma y.im (/ x.im y.re) x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -75000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-14) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -75000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-14)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -75000000000.0], t$95$0, If[LessEqual[y$46$im, 1.4e-14], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5e10 or 1.4e-14 < y.im

   1. Initial program 50.0%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 81.1

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

   5. Simplified 81.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 82.5

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

   7. Applied egg-rr 82.5%

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

   if -7.5e10 < y.im < 1.4e-14

   1. Initial program 71.6%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 83.1

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

   7. Applied egg-rr 83.1%

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

Alternative 6: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.6e+72)
   (/ (fma y.im (/ x.im y.re) x.re) y.re)
   (if (<= y.re 5.7e-11)
     (/ (fma x.re (/ y.re y.im) x.im) y.im)
     (/ (fma x.im (/ y.im y.re) x.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.6e+72) {
		tmp = fma(y_46_im, (x_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_re <= 5.7e-11) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.6e+72)
		tmp = Float64(fma(y_46_im, Float64(x_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_re <= 5.7e-11)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.6e+72], N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.7e-11], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.6000000000000001e72

   1. Initial program 54.4%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 87.5

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

   5. Simplified 87.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 87.8

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

   7. Applied egg-rr 87.8%

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

   if -1.6000000000000001e72 < y.re < 5.6999999999999997e-11

   1. Initial program 68.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.1

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

   5. Simplified 82.1%

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

   if 5.6999999999999997e-11 < y.re

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 78.4

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

   5. Simplified 78.4%

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

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -1.6e+72)
     t_0
     (if (<= y.re 5.7e-11) (/ (fma x.re (/ y.re y.im) x.im) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -1.6e+72) {
		tmp = t_0;
	} else if (y_46_re <= 5.7e-11) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.6e+72)
		tmp = t_0;
	elseif (y_46_re <= 5.7e-11)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e+72], t$95$0, If[LessEqual[y$46$re, 5.7e-11], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.6000000000000001e72 or 5.6999999999999997e-11 < y.re

   1. Initial program 49.3%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.2

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

   5. Simplified 82.2%

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

   if -1.6000000000000001e72 < y.re < 5.6999999999999997e-11

   1. Initial program 68.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.1

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

   5. Simplified 82.1%

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

Alternative 8: 64.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.5e+15)
   (/ x.im y.im)
   (if (<= y.im 9.2e-33) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+15) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 9.2e-33) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	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_46im <= (-2.5d+15)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 9.2d-33) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+15) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 9.2e-33) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.5e+15:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 9.2e-33:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+15)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 9.2e-33)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.5e+15)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 9.2e-33)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.5e+15], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 9.2e-33], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5e15 or 9.19999999999999942e-33 < y.im

   1. Initial program 51.1%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 69.6

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

   5. Simplified 69.6%

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

   if -2.5e15 < y.im < 9.19999999999999942e-33

   1. Initial program 70.9%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 69.7

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

   5. Simplified 69.7%

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

Alternative 9: 43.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 45.1

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

5. Simplified 45.1%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))