_divideComplex, real part

Percentage Accurate: 61.8% → 85.2%

Time: 10.5s

Alternatives: 8

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, \frac{y.re \cdot x.re}{t\_0}\right)\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -5.5 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.1 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

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) (/ (* y.re x.re) t_0))))
   (if (<= y.im -7e+112)
     (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))
     (if (<= y.im -5.5e-77)
       t_1
       (if (<= y.im 3.8e-145)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 6.1e+144)
           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), ((y_46_re * x_46_re) / t_0));
	double tmp;
	if (y_46_im <= -7e+112) {
		tmp = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	} else if (y_46_im <= -5.5e-77) {
		tmp = t_1;
	} else if (y_46_im <= 3.8e-145) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 6.1e+144) {
		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(Float64(y_46_re * x_46_re) / t_0))
	tmp = 0.0
	if (y_46_im <= -7e+112)
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im));
	elseif (y_46_im <= -5.5e-77)
		tmp = t_1;
	elseif (y_46_im <= 3.8e-145)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 6.1e+144)
		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[(y$46$re * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7e+112], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.5e-77], t$95$1, If[LessEqual[y$46$im, 3.8e-145], 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, 6.1e+144], 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 < -6.99999999999999994e112

   1. Initial program 33.7%

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

   3. 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 84.9

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

   5. Simplified 84.9%

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

      1. div-inv N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. /-lowering-/.f64 84.8

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

   7. Applied egg-rr 84.8%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. un-div-inv N/A

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

      11. /-lowering-/.f64 85.0

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

   9. Applied egg-rr 85.0%

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

   if -6.99999999999999994e112 < y.im < -5.49999999999999998e-77 or 3.8000000000000002e-145 < y.im < 6.09999999999999971e144

   1. Initial program 78.9%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 81.8

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

   5. Simplified 81.8%

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

   if -5.49999999999999998e-77 < y.im < 3.8000000000000002e-145

   1. Initial program 67.8%

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

   3. Taylor expanded in y.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 93.4

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

   5. Simplified 93.4%

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

   if 6.09999999999999971e144 < y.im

   1. Initial program 25.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 92.1

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

   5. Simplified 92.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -5.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))))
   (if (<= y.im -1.15e+100)
     t_0
     (if (<= y.im -4.4e-79)
       (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 6.5e-146)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.6e+104)
           (/ (fma y.re x.re (* y.im x.im)) (fma y.im y.im (* y.re y.re)))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	double tmp;
	if (y_46_im <= -1.15e+100) {
		tmp = t_0;
	} else if (y_46_im <= -4.4e-79) {
		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 <= 6.5e-146) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.6e+104) {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.15e+100)
		tmp = t_0;
	elseif (y_46_im <= -4.4e-79)
		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 <= 6.5e-146)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.6e+104)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+100], t$95$0, If[LessEqual[y$46$im, -4.4e-79], 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, 6.5e-146], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+104], N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.14999999999999995e100 or 1.6e104 < y.im

   1. Initial program 34.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 85.6

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

   5. Simplified 85.6%

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

      1. div-inv N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. /-lowering-/.f64 85.4

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

   7. Applied egg-rr 85.4%

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

      1. *-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. un-div-inv N/A

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

      11. /-lowering-/.f64 85.6

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

   9. Applied egg-rr 85.6%

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

   if -1.14999999999999995e100 < y.im < -4.3999999999999998e-79

   1. Initial program 81.6%

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

      1. /-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 81.6

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

   4. Applied egg-rr 81.6%

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

   if -4.3999999999999998e-79 < y.im < 6.4999999999999999e-146

   1. Initial program 67.8%

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

   3. Taylor expanded in y.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 93.4

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

   5. Simplified 93.4%

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

   if 6.4999999999999999e-146 < y.im < 1.6e104

   1. Initial program 80.7%

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

      1. +-commutative N/A

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

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

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

      3. *-lowering-*.f64 80.7

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

   4. Applied egg-rr 80.7%

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 80.7

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

   6. Applied egg-rr 80.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -2e+101)
     t_0
     (if (<= y.im -1.2e-76)
       (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 4.3e-150)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 5.1e+101)
           (/ (fma y.re x.re (* y.im x.im)) (fma y.im y.im (* y.re y.re)))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -2e+101) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-76) {
		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 <= 4.3e-150) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 5.1e+101) {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2e+101)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-76)
		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 <= 4.3e-150)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 5.1e+101)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2e+101], t$95$0, If[LessEqual[y$46$im, -1.2e-76], 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, 4.3e-150], 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, 5.1e+101], N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2e101 or 5.09999999999999995e101 < y.im

   1. Initial program 33.5%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 85.4

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

   5. Simplified 85.4%

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

   if -2e101 < y.im < -1.20000000000000007e-76

   1. Initial program 82.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 82.1

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

   4. Applied egg-rr 82.1%

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

   if -1.20000000000000007e-76 < y.im < 4.30000000000000004e-150

   1. Initial program 67.8%

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

   3. Taylor expanded in y.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 93.4

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

   5. Simplified 93.4%

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

   if 4.30000000000000004e-150 < y.im < 5.09999999999999995e101

   1. Initial program 80.7%

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

      1. +-commutative N/A

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

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

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

      3. *-lowering-*.f64 80.7

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

   4. Applied egg-rr 80.7%

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 80.7

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

   6. Applied egg-rr 80.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -4e+100)
     t_1
     (if (<= y.im -9e-79)
       t_0
       (if (<= y.im 8.5e-148)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 5.6e+102) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -4e+100) {
		tmp = t_1;
	} else if (y_46_im <= -9e-79) {
		tmp = t_0;
	} else if (y_46_im <= 8.5e-148) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 5.6e+102) {
		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(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)))
	t_1 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4e+100)
		tmp = t_1;
	elseif (y_46_im <= -9e-79)
		tmp = t_0;
	elseif (y_46_im <= 8.5e-148)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 5.6e+102)
		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[(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]}, Block[{t$95$1 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+100], t$95$1, If[LessEqual[y$46$im, -9e-79], t$95$0, If[LessEqual[y$46$im, 8.5e-148], 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, 5.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.00000000000000006e100 or 5.60000000000000037e102 < y.im

   1. Initial program 33.5%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 85.4

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

   5. Simplified 85.4%

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

   if -4.00000000000000006e100 < y.im < -9.0000000000000006e-79 or 8.49999999999999989e-148 < y.im < 5.60000000000000037e102

   1. Initial program 81.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 81.2

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

   4. Applied egg-rr 81.2%

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

   if -9.0000000000000006e-79 < y.im < 8.49999999999999989e-148

   1. Initial program 67.8%

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

   3. Taylor expanded in y.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 93.4

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

   5. Simplified 93.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.8e+104)
   (/ x.im y.im)
   (if (<= y.im -1.2e-24)
     (/ (fma x.im y.im (* y.re x.re)) (* y.im y.im))
     (if (<= y.im 1.35e-70)
       (/ (fma x.im (/ y.im y.re) 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 <= -6.8e+104) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.2e-24) {
		tmp = fma(x_46_im, y_46_im, (y_46_re * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 1.35e-70) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), 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 <= -6.8e+104)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.2e-24)
		tmp = Float64(fma(x_46_im, y_46_im, Float64(y_46_re * x_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 1.35e-70)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	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, -6.8e+104], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.2e-24], N[(N[(x$46$im * y$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.35e-70], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.7999999999999994e104 or 1.3500000000000001e-70 < y.im

   1. Initial program 45.7%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 68.9

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

   5. Simplified 68.9%

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

   if -6.7999999999999994e104 < y.im < -1.1999999999999999e-24

   1. Initial program 88.9%

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

   3. Taylor expanded in y.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 72.5

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

   5. Simplified 72.5%

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

   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. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 72.2

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

   8. Simplified 72.2%

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

   if -1.1999999999999999e-24 < y.im < 1.3500000000000001e-70

   1. Initial program 70.5%

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

   3. Taylor expanded in y.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 85.8

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

   5. Simplified 85.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -9e-28)
     t_0
     (if (<= y.im 1.7e-95) (/ (fma x.im (/ y.im y.re) x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -9e-28) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e-95) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e-28)
		tmp = t_0;
	elseif (y_46_im <= 1.7e-95)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9e-28], t$95$0, If[LessEqual[y$46$im, 1.7e-95], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.9999999999999996e-28 or 1.69999999999999997e-95 < y.im

   1. Initial program 53.4%

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

   3. Taylor expanded in y.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 77.3

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

   5. Simplified 77.3%

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

   if -8.9999999999999996e-28 < y.im < 1.69999999999999997e-95

   1. Initial program 71.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 88.4

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

   5. Simplified 88.4%

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

Alternative 7: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.65e+23)
   (/ x.re y.re)
   (if (<= y.re 8.5e+64) (/ x.im y.im) (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.65e+23) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 8.5e+64) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.65d+23)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 8.5d+64) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.65e+23) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 8.5e+64) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.65e+23:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 8.5e+64:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.65e+23)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 8.5e+64)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.65e+23)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 8.5e+64)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.65e+23], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 8.5e+64], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.6500000000000001e23 or 8.4999999999999998e64 < y.re

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 76.3

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

   5. Simplified 76.3%

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

   if -2.6500000000000001e23 < y.re < 8.4999999999999998e64

   1. Initial program 73.4%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 66.7

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

   5. Simplified 66.7%

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

Alternative 8: 43.6% 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 61.1%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 46.4

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

5. Simplified 46.4%

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

Reproduce

herbie shell --seed 2024198 
(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))))