_divideComplex, real part

Percentage Accurate: 61.8% → 83.2%

Time: 8.4s

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: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{y.re} \cdot \left(-x.im\right), y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (fma y.re x.re (* y.im x.im)) (+ (* y.im y.im) (* y.re y.re)))))
   (if (<= y.re -9.5e+113)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (if (<= y.re -1.7e-130)
       t_0
       (if (<= y.re 9e-140)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 2.25e+70)
           t_0
           (/ (fma (* (/ -1.0 y.re) (- 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 t_0 = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -9.5e+113) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -1.7e-130) {
		tmp = t_0;
	} else if (y_46_re <= 9e-140) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 2.25e+70) {
		tmp = t_0;
	} else {
		tmp = fma(((-1.0 / y_46_re) * -x_46_im), y_46_im, 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(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -9.5e+113)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -1.7e-130)
		tmp = t_0;
	elseif (y_46_re <= 9e-140)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 2.25e+70)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(-1.0 / y_46_re) * Float64(-x_46_im)), y_46_im, 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 * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e+113], 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, -1.7e-130], t$95$0, If[LessEqual[y$46$re, 9e-140], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.25e+70], t$95$0, N[(N[(N[(N[(-1.0 / y$46$re), $MachinePrecision] * (-x$46$im)), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.5000000000000001e113

   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.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 91.9

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

   5. Applied rewrites 91.9%

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

   if -9.5000000000000001e113 < y.re < -1.70000000000000003e-130 or 9.00000000000000008e-140 < y.re < 2.25e70

   1. Initial program 78.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 78.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.6

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

   4. Applied rewrites 78.6%

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

   if -1.70000000000000003e-130 < y.re < 9.00000000000000008e-140

   1. Initial program 67.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 67.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   6. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 2.25e70 < y.re

   1. Initial program 30.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 89.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{y.re} \cdot \left(-x.im\right), y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{y.re}{t\_0} \cdot x.re\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{t\_0} \cdot y.im\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.32 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))) (t_1 (* (/ y.re t_0) x.re)))
   (if (<= y.re -1.75e+88)
     (/ x.re y.re)
     (if (<= y.re -3e+29)
       (* (/ x.im t_0) y.im)
       (if (<= y.re -9e-96)
         t_1
         (if (<= y.re 2.5e-101)
           (/ x.im y.im)
           (if (<= y.re 1.32e+135) t_1 (/ 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 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = (y_46_re / t_0) * x_46_re;
	double tmp;
	if (y_46_re <= -1.75e+88) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -3e+29) {
		tmp = (x_46_im / t_0) * y_46_im;
	} else if (y_46_re <= -9e-96) {
		tmp = t_1;
	} else if (y_46_re <= 2.5e-101) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1.32e+135) {
		tmp = t_1;
	} 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 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(Float64(y_46_re / t_0) * x_46_re)
	tmp = 0.0
	if (y_46_re <= -1.75e+88)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -3e+29)
		tmp = Float64(Float64(x_46_im / t_0) * y_46_im);
	elseif (y_46_re <= -9e-96)
		tmp = t_1;
	elseif (y_46_re <= 2.5e-101)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1.32e+135)
		tmp = t_1;
	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[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e+88], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3e+29], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, -9e-96], t$95$1, If[LessEqual[y$46$re, 2.5e-101], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.32e+135], t$95$1, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.7499999999999999e88 or 1.32e135 < y.re

   1. Initial program 28.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -1.7499999999999999e88 < y.re < -2.9999999999999999e29

   1. Initial program 80.1%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if -2.9999999999999999e29 < y.re < -9e-96 or 2.5e-101 < y.re < 1.32e135

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 77.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2} + {y.re}^{2}}} \]
   6. 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

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -9e-96 < y.re < 2.5e-101

   1. Initial program 68.5%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

Alternative 3: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{x.re}{t\_0} \cdot y.re\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{t\_0} \cdot y.im\\ \mathbf{elif}\;y.re \leq -8.8 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))) (t_1 (* (/ x.re t_0) y.re)))
   (if (<= y.re -1.75e+88)
     (/ x.re y.re)
     (if (<= y.re -3e+29)
       (* (/ x.im t_0) y.im)
       (if (<= y.re -8.8e-95)
         t_1
         (if (<= y.re 3.4e-58)
           (/ x.im y.im)
           (if (<= y.re 2.4e+133) t_1 (/ 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 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = (x_46_re / t_0) * y_46_re;
	double tmp;
	if (y_46_re <= -1.75e+88) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -3e+29) {
		tmp = (x_46_im / t_0) * y_46_im;
	} else if (y_46_re <= -8.8e-95) {
		tmp = t_1;
	} else if (y_46_re <= 3.4e-58) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 2.4e+133) {
		tmp = t_1;
	} 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 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(Float64(x_46_re / t_0) * y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.75e+88)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -3e+29)
		tmp = Float64(Float64(x_46_im / t_0) * y_46_im);
	elseif (y_46_re <= -8.8e-95)
		tmp = t_1;
	elseif (y_46_re <= 3.4e-58)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 2.4e+133)
		tmp = t_1;
	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[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / t$95$0), $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e+88], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3e+29], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, -8.8e-95], t$95$1, If[LessEqual[y$46$re, 3.4e-58], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+133], t$95$1, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.7499999999999999e88 or 2.3999999999999999e133 < y.re

   1. Initial program 28.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -1.7499999999999999e88 < y.re < -2.9999999999999999e29

   1. Initial program 80.1%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if -2.9999999999999999e29 < y.re < -8.7999999999999995e-95 or 3.39999999999999973e-58 < y.re < 2.3999999999999999e133

   1. Initial program 78.4%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2} + {y.re}^{2}}} \]
   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 63.7

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

   5. Applied rewrites 63.7%

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

   if -8.7999999999999995e-95 < y.re < 3.39999999999999973e-58

   1. Initial program 68.7%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 4: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -5.4e+39)
     t_0
     (if (<= y.im 1.8e+90) (/ (fma (/ x.im y.re) y.im x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -5.4e+39) {
		tmp = t_0;
	} else if (y_46_im <= 1.8e+90) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.4e+39)
		tmp = t_0;
	elseif (y_46_im <= 1.8e+90)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -5.4e+39], t$95$0, If[LessEqual[y$46$im, 1.8e+90], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.40000000000000007e39 or 1.8e90 < y.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around inf

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

      1. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if -5.40000000000000007e39 < y.im < 1.8e90

   1. Initial program 71.7%

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

   3. Taylor expanded in y.re around 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 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 5: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7e+34)
   (/ x.im y.im)
   (if (<= y.im 6.2e+133)
     (/ (fma (/ x.im y.re) y.im 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 <= -7e+34) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 6.2e+133) {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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 <= -7e+34)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 6.2e+133)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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, -7e+34], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 6.2e+133], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.99999999999999996e34 or 6.2e133 < y.im

   1. Initial program 43.9%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -6.99999999999999996e34 < y.im < 6.2e133

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

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

      1. 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 77.3

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

   5. Applied rewrites 77.3%

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

Alternative 6: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.im\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.75e+88)
   (/ x.re y.re)
   (if (<= y.re -2.4e+19)
     (* (/ x.im (fma y.im y.im (* y.re y.re))) y.im)
     (if (<= y.re 2.7e-45) (/ 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 <= -1.75e+88) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -2.4e+19) {
		tmp = (x_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_im;
	} else if (y_46_re <= 2.7e-45) {
		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 <= -1.75e+88)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -2.4e+19)
		tmp = Float64(Float64(x_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_im);
	elseif (y_46_re <= 2.7e-45)
		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, -1.75e+88], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.4e+19], N[(N[(x$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.7e-45], 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 < -1.7499999999999999e88 or 2.69999999999999985e-45 < y.re

   1. Initial program 44.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -1.7499999999999999e88 < y.re < -2.4e19

   1. Initial program 82.4%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -2.4e19 < y.re < 2.69999999999999985e-45

   1. Initial program 70.7%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

Alternative 7: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -185000000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -185000000000.0)
   (/ x.re y.re)
   (if (<= y.re 2.7e-45) (/ 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 <= -185000000000.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e-45) {
		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 <= (-185000000000.0d0)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.7d-45) 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 <= -185000000000.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e-45) {
		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 <= -185000000000.0:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.7e-45:
		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 <= -185000000000.0)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.7e-45)
		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 <= -185000000000.0)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.7e-45)
		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, -185000000000.0], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.7e-45], 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 < -1.85e11 or 2.69999999999999985e-45 < 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.im around 0

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

      1. lower-/.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -1.85e11 < y.re < 2.69999999999999985e-45

   1. Initial program 71.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

Alternative 8: 43.3% 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 59.6%

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

3. Taylor expanded in y.im around inf

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

   1. lower-/.f64 44.4

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

5. Applied rewrites 44.4%

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

Reproduce

herbie shell --seed 2024255 
(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))))