_divideComplex, real part

Percentage Accurate: 61.2% → 82.6%

Time: 8.4s

Alternatives: 10

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: 61.2% 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: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.36 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.08 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.82 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.im y.im) (* y.re y.re))))
        (t_1 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -1e+80)
     t_1
     (if (<= y.re -1.36e-93)
       t_0
       (if (<= y.re 1.08e-16)
         (/ (fma x.re (/ y.re y.im) x.im) y.im)
         (if (<= y.re 1.82e+106) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -1e+80) {
		tmp = t_1;
	} else if (y_46_re <= -1.36e-93) {
		tmp = t_0;
	} else if (y_46_re <= 1.08e-16) {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_re <= 1.82e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	t_1 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1e+80)
		tmp = t_1;
	elseif (y_46_re <= -1.36e-93)
		tmp = t_0;
	elseif (y_46_re <= 1.08e-16)
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_re <= 1.82e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1e+80], t$95$1, If[LessEqual[y$46$re, -1.36e-93], t$95$0, If[LessEqual[y$46$re, 1.08e-16], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.82e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1e80 or 1.8199999999999999e106 < y.re

   1. Initial program 44.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -1e80 < y.re < -1.3599999999999999e-93 or 1.08e-16 < y.re < 1.8199999999999999e106

   1. Initial program 85.0%

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

   if -1.3599999999999999e-93 < y.re < 1.08e-16

   1. Initial program 72.8%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.36 \cdot 10^{-93}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.08 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.82 \cdot 10^{+106}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-238}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 820000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.55e+118)
   (/ x.re y.re)
   (if (<= y.re -3.5e-105)
     (* (/ y.re (fma y.re y.re (* y.im y.im))) x.re)
     (if (<= y.re 1.85e-238)
       (/ x.im y.im)
       (if (<= y.re 820000000000.0)
         (/ (fma y.im x.im (* x.re y.re)) (* y.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 <= -1.55e+118) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -3.5e-105) {
		tmp = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_re;
	} else if (y_46_re <= 1.85e-238) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 820000000000.0) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / (y_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 <= -1.55e+118)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -3.5e-105)
		tmp = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_re);
	elseif (y_46_re <= 1.85e-238)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 820000000000.0)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.55e+118], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.5e-105], N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.85e-238], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 820000000000.0], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.54999999999999993e118 or 8.2e11 < y.re

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.54999999999999993e118 < y.re < -3.5e-105

   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.re around 0

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

      1. lower-/.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if -3.5e-105 < y.re < 1.85000000000000012e-238

   1. Initial program 69.4%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 1.85000000000000012e-238 < y.re < 8.2e11

   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 y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.1

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

   7. Applied rewrites 69.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-238}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 820000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.55e+118)
   (/ x.re y.re)
   (if (<= y.re -3.5e-105)
     (* (/ y.re (fma y.re y.re (* y.im y.im))) x.re)
     (if (<= y.re 6.2e-135)
       (/ x.im y.im)
       (if (<= y.re 9e+20)
         (* (/ x.re (fma y.im y.im (* y.re y.re))) 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 <= -1.55e+118) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -3.5e-105) {
		tmp = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_re;
	} else if (y_46_re <= 6.2e-135) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 9e+20) {
		tmp = (x_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_re;
	} 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 <= -1.55e+118)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -3.5e-105)
		tmp = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_re);
	elseif (y_46_re <= 6.2e-135)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 9e+20)
		tmp = Float64(Float64(x_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_re);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.55e+118], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.5e-105], N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-135], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9e+20], N[(N[(x$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.54999999999999993e118 or 9e20 < y.re

   1. Initial program 49.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.54999999999999993e118 < y.re < -3.5e-105

   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.re around 0

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

      1. lower-/.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if -3.5e-105 < y.re < 6.2000000000000001e-135

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 6.2000000000000001e-135 < y.re < 9e20

   1. Initial program 79.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ y.re (fma y.re y.re (* y.im y.im))) x.re)))
   (if (<= y.re -1.55e+118)
     (/ x.re y.re)
     (if (<= y.re -3.5e-105)
       t_0
       (if (<= y.re 6.2e-135)
         (/ x.im y.im)
         (if (<= y.re 9e+20) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_re;
	double tmp;
	if (y_46_re <= -1.55e+118) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -3.5e-105) {
		tmp = t_0;
	} else if (y_46_re <= 6.2e-135) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 9e+20) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_re)
	tmp = 0.0
	if (y_46_re <= -1.55e+118)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -3.5e-105)
		tmp = t_0;
	elseif (y_46_re <= 6.2e-135)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 9e+20)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.55e+118], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.5e-105], t$95$0, If[LessEqual[y$46$re, 6.2e-135], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9e+20], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.54999999999999993e118 or 9e20 < y.re

   1. Initial program 49.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.54999999999999993e118 < y.re < -3.5e-105 or 6.2000000000000001e-135 < y.re < 9e20

   1. Initial program 81.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if -3.5e-105 < y.re < 6.2000000000000001e-135

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+20}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-238}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 600000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.im y.re) x.re) y.re)))
   (if (<= y.re -1.4e-27)
     t_0
     (if (<= y.re 1.85e-238)
       (/ x.im y.im)
       (if (<= y.re 600000000000.0)
         (/ (fma y.im x.im (* x.re y.re)) (* y.im y.im))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -1.4e-27) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-238) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 600000000000.0) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / (y_46_im * y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.4e-27)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-238)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 600000000000.0)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e-27], t$95$0, If[LessEqual[y$46$re, 1.85e-238], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 600000000000.0], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.4e-27 or 6e11 < y.re

   1. Initial program 57.9%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.1

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

   8. Applied rewrites 81.1%

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

   if -1.4e-27 < y.re < 1.85000000000000012e-238

   1. Initial program 71.8%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if 1.85000000000000012e-238 < y.re < 6e11

   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 y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.1

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

   7. Applied rewrites 69.1%

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

Alternative 6: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -2.7e-27)
     t_0
     (if (<= y.re 4.8e+20) (/ (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_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -2.7e-27) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e+20) {
		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(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.7e-27)
		tmp = t_0;
	elseif (y_46_re <= 4.8e+20)
		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[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.7e-27], t$95$0, If[LessEqual[y$46$re, 4.8e+20], 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 < -2.69999999999999989e-27 or 4.8e20 < y.re

   1. Initial program 57.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -2.69999999999999989e-27 < y.re < 4.8e20

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.5

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

   8. Applied rewrites 85.5%

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

Alternative 7: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.7e-27)
     t_0
     (if (<= y.re 4.8e+20) (/ (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 <= -2.7e-27) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e+20) {
		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 <= -2.7e-27)
		tmp = t_0;
	elseif (y_46_re <= 4.8e+20)
		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, -2.7e-27], t$95$0, If[LessEqual[y$46$re, 4.8e+20], 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 < -2.69999999999999989e-27 or 4.8e20 < y.re

   1. Initial program 57.6%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 17.7

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if -2.69999999999999989e-27 < y.re < 4.8e20

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.5

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

   8. Applied rewrites 85.5%

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

Alternative 8: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.55e+43)
   (/ x.re y.re)
   (if (<= y.re -1.15e-29)
     (* (/ y.im (fma y.re y.re (* y.im y.im))) x.im)
     (if (<= y.re 1.6e-18) (/ 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 <= -1.55e+43) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.15e-29) {
		tmp = (y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else if (y_46_re <= 1.6e-18) {
		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 <= -1.55e+43)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.15e-29)
		tmp = Float64(Float64(y_46_im / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	elseif (y_46_re <= 1.6e-18)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.55e+43], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.15e-29], N[(N[(y$46$im / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-18], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.5500000000000001e43 or 1.6e-18 < y.re

   1. Initial program 57.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -1.5500000000000001e43 < y.re < -1.14999999999999996e-29

   1. Initial program 79.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower-/.f64 40.1

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 72.1

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

   8. Applied rewrites 72.1%

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

   if -1.14999999999999996e-29 < y.re < 1.6e-18

   1. Initial program 73.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.6e+43)
   (/ x.re y.re)
   (if (<= y.re 1.6e-18) (/ 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 <= -1.6e+43) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.6e-18) {
		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 <= (-1.6d+43)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.6d-18) 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 <= -1.6e+43) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.6e-18) {
		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 <= -1.6e+43:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.6e-18:
		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 <= -1.6e+43)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.6e-18)
		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 <= -1.6e+43)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.6e-18)
		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, -1.6e+43], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-18], 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 < -1.60000000000000007e43 or 1.6e-18 < y.re

   1. Initial program 57.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -1.60000000000000007e43 < y.re < 1.6e-18

   1. Initial program 74.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

Alternative 10: 42.8% 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 66.0%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 41.6

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

5. Applied rewrites 41.6%

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

Reproduce

herbie shell --seed 2024283 
(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))))