_divideComplex, imaginary part

Percentage Accurate: 61.4% → 79.1%

Time: 10.1s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(x.im, \frac{y.im}{{y.re}^{3}}, \frac{x.re}{y.re \cdot y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-34)
   (fma (/ (- x.re) y.re) (/ y.im y.re) (/ x.im y.re))
   (if (<= y.re 2.4e-142)
     (/ (fma x.im (/ (- y.re) y.im) x.re) (- y.im))
     (if (<= y.re 1.05e+114)
       (/ (- (* x.im y.re) (* y.im x.re)) (fma y.im y.im (* y.re y.re)))
       (fma
        (- (fma x.im (/ y.im (pow y.re 3.0)) (/ x.re (* y.re y.re))))
        y.im
        (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = fma((-x_46_re / y_46_re), (y_46_im / y_46_re), (x_46_im / y_46_re));
	} else if (y_46_re <= 2.4e-142) {
		tmp = fma(x_46_im, (-y_46_re / y_46_im), x_46_re) / -y_46_im;
	} else if (y_46_re <= 1.05e+114) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma(-fma(x_46_im, (y_46_im / pow(y_46_re, 3.0)), (x_46_re / (y_46_re * y_46_re))), y_46_im, (x_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = fma(Float64(Float64(-x_46_re) / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re));
	elseif (y_46_re <= 2.4e-142)
		tmp = Float64(fma(x_46_im, Float64(Float64(-y_46_re) / y_46_im), x_46_re) / Float64(-y_46_im));
	elseif (y_46_re <= 1.05e+114)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = fma(Float64(-fma(x_46_im, Float64(y_46_im / (y_46_re ^ 3.0)), Float64(x_46_re / Float64(y_46_re * y_46_re)))), y_46_im, Float64(x_46_im / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.4e-34], N[(N[((-x$46$re) / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e-142], N[(N[(x$46$im * N[((-y$46$re) / y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+114], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * y$46$im + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.39999999999999991e-34

   1. Initial program 49.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in y.im around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if -2.39999999999999991e-34 < y.re < 2.39999999999999988e-142

   1. Initial program 70.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 2.39999999999999988e-142 < y.re < 1.05e114

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 79.8

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

   4. Applied rewrites 79.8%

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

   if 1.05e114 < y.re

   1. Initial program 38.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{x.im \cdot y.im}{{y.re}^{3}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x.re}{{y.re}^{2}}\right)\right)}, y.im, \frac{x.im}{y.re}\right) \]

      6. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(\frac{x.im \cdot y.im}{{y.re}^{3}} + \frac{x.re}{{y.re}^{2}}\right)\right)}, y.im, \frac{x.im}{y.re}\right) \]

      7. lower-neg.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(x.im, \frac{y.im}{{y.re}^{3}}, \frac{x.re}{{y.re}^{2}}\right)}, y.im, \frac{x.im}{y.re}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{{y.re}^{3}}}, \frac{x.re}{{y.re}^{2}}\right), y.im, \frac{x.im}{y.re}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(x.im, \frac{y.im}{\color{blue}{{y.re}^{3}}}, \frac{x.re}{{y.re}^{2}}\right), y.im, \frac{x.im}{y.re}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(x.im, \frac{y.im}{{y.re}^{3}}, \color{blue}{\frac{x.re}{{y.re}^{2}}}\right), y.im, \frac{x.im}{y.re}\right) \]

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 91.9

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

   8. Applied rewrites 91.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(x.im, \frac{y.im}{{y.re}^{3}}, \frac{x.re}{y.re \cdot y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \left(-y.im\right) \cdot x.re\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, t\_1\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (* (- y.im) x.re)))
   (if (<= y.im -2.2e+57)
     t_0
     (if (<= y.im -7.5e-60)
       (/ (- (* x.im y.re) (* y.im x.re)) (* y.im y.im))
       (if (<= y.im -7.5e-149)
         (/ (fma y.re x.im t_1) (* y.re y.re))
         (if (<= y.im 5.2e-114)
           (/ x.im y.re)
           (if (<= y.im 3.9e+90)
             (/ t_1 (fma y.im y.im (* y.re y.re)))
             t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = -y_46_im * x_46_re;
	double tmp;
	if (y_46_im <= -2.2e+57) {
		tmp = t_0;
	} else if (y_46_im <= -7.5e-60) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= -7.5e-149) {
		tmp = fma(y_46_re, x_46_im, t_1) / (y_46_re * y_46_re);
	} else if (y_46_im <= 5.2e-114) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.9e+90) {
		tmp = t_1 / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(-y_46_im) * x_46_re)
	tmp = 0.0
	if (y_46_im <= -2.2e+57)
		tmp = t_0;
	elseif (y_46_im <= -7.5e-60)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= -7.5e-149)
		tmp = Float64(fma(y_46_re, x_46_im, t_1) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 5.2e-114)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.9e+90)
		tmp = Float64(t_1 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[((-y$46$im) * x$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+57], t$95$0, If[LessEqual[y$46$im, -7.5e-60], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.5e-149], N[(N[(y$46$re * x$46$im + t$95$1), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.2e-114], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.9e+90], N[(t$95$1 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.2000000000000001e57 or 3.9000000000000002e90 < y.im

   1. Initial program 39.9%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.2000000000000001e57 < y.im < -7.5000000000000002e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -7.5000000000000002e-60 < y.im < -7.49999999999999995e-149

   1. Initial program 91.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in y.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if -7.49999999999999995e-149 < y.im < 5.20000000000000026e-114

   1. Initial program 74.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 5.20000000000000026e-114 < y.im < 3.9000000000000002e90

   1. Initial program 66.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 51.0

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

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\color{blue}{\left(-x.re\right) \cdot y.im}}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, \left(-y.im\right) \cdot x.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := x.im \cdot y.re - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{t\_1}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (- (* x.im y.re) (* y.im x.re))))
   (if (<= y.im -2.2e+57)
     t_0
     (if (<= y.im -7.5e-60)
       (/ t_1 (* y.im y.im))
       (if (<= y.im -7.5e-149)
         (/ t_1 (* y.re y.re))
         (if (<= y.im 5.2e-114)
           (/ x.im y.re)
           (if (<= y.im 3.9e+90)
             (/ (* (- y.im) x.re) (fma y.im y.im (* y.re y.re)))
             t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_im <= -2.2e+57) {
		tmp = t_0;
	} else if (y_46_im <= -7.5e-60) {
		tmp = t_1 / (y_46_im * y_46_im);
	} else if (y_46_im <= -7.5e-149) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 5.2e-114) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.9e+90) {
		tmp = (-y_46_im * x_46_re) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_im <= -2.2e+57)
		tmp = t_0;
	elseif (y_46_im <= -7.5e-60)
		tmp = Float64(t_1 / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= -7.5e-149)
		tmp = Float64(t_1 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 5.2e-114)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.9e+90)
		tmp = Float64(Float64(Float64(-y_46_im) * x_46_re) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+57], t$95$0, If[LessEqual[y$46$im, -7.5e-60], N[(t$95$1 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.5e-149], N[(t$95$1 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.2e-114], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.9e+90], N[(N[((-y$46$im) * x$46$re), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.2000000000000001e57 or 3.9000000000000002e90 < y.im

   1. Initial program 39.9%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.2000000000000001e57 < y.im < -7.5000000000000002e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -7.5000000000000002e-60 < y.im < -7.49999999999999995e-149

   1. Initial program 91.1%

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

   3. Taylor expanded in y.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -7.49999999999999995e-149 < y.im < 5.20000000000000026e-114

   1. Initial program 74.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 5.20000000000000026e-114 < y.im < 3.9000000000000002e90

   1. Initial program 66.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 51.0

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

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\color{blue}{\left(-x.re\right) \cdot y.im}}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im \cdot y.re}{t\_0}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -7.8e-35)
     (/ x.im y.re)
     (if (<= y.re 1.95e-115)
       (/ (- x.re) y.im)
       (if (<= y.re 7.4e-32)
         (/ (* x.im y.re) t_0)
         (if (<= y.re 2.9e+64) (/ (* (- y.im) x.re) t_0) (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -7.8e-35) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.95e-115) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 7.4e-32) {
		tmp = (x_46_im * y_46_re) / t_0;
	} else if (y_46_re <= 2.9e+64) {
		tmp = (-y_46_im * x_46_re) / t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -7.8e-35)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.95e-115)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 7.4e-32)
		tmp = Float64(Float64(x_46_im * y_46_re) / t_0);
	elseif (y_46_re <= 2.9e+64)
		tmp = Float64(Float64(Float64(-y_46_im) * x_46_re) / t_0);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.8e-35], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.95e-115], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.4e-32], N[(N[(x$46$im * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2.9e+64], N[(N[((-y$46$im) * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.79999999999999961e-35 or 2.89999999999999993e64 < y.re

   1. Initial program 47.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -7.79999999999999961e-35 < y.re < 1.9499999999999999e-115

   1. Initial program 71.8%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if 1.9499999999999999e-115 < y.re < 7.4e-32

   1. Initial program 85.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 7.4e-32 < y.re < 2.89999999999999993e64

   1. Initial program 68.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 50.5

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

   7. Applied rewrites 50.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im \cdot y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{y.re \cdot y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-34)
   (fma (/ (- x.re) y.re) (/ y.im y.re) (/ x.im y.re))
   (if (<= y.re 2.4e-142)
     (/ (fma x.im (/ (- y.re) y.im) x.re) (- y.im))
     (if (<= y.re 1.85e+139)
       (/ (- (* x.im y.re) (* y.im x.re)) (fma y.im y.im (* y.re y.re)))
       (fma (- x.re) (/ y.im (* y.re y.re)) (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = fma((-x_46_re / y_46_re), (y_46_im / y_46_re), (x_46_im / y_46_re));
	} else if (y_46_re <= 2.4e-142) {
		tmp = fma(x_46_im, (-y_46_re / y_46_im), x_46_re) / -y_46_im;
	} else if (y_46_re <= 1.85e+139) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma(-x_46_re, (y_46_im / (y_46_re * y_46_re)), (x_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = fma(Float64(Float64(-x_46_re) / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re));
	elseif (y_46_re <= 2.4e-142)
		tmp = Float64(fma(x_46_im, Float64(Float64(-y_46_re) / y_46_im), x_46_re) / Float64(-y_46_im));
	elseif (y_46_re <= 1.85e+139)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = fma(Float64(-x_46_re), Float64(y_46_im / Float64(y_46_re * y_46_re)), Float64(x_46_im / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.4e-34], N[(N[((-x$46$re) / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e-142], N[(N[(x$46$im * N[((-y$46$re) / y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+139], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x$46$re) * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.39999999999999991e-34

   1. Initial program 49.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in y.im around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if -2.39999999999999991e-34 < y.re < 2.39999999999999988e-142

   1. Initial program 70.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 2.39999999999999988e-142 < y.re < 1.84999999999999996e139

   1. Initial program 78.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   if 1.84999999999999996e139 < y.re

   1. Initial program 36.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 96.9

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

   8. Applied rewrites 96.9%

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

   10. Applied rewrites 96.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{y.re \cdot y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{y.re \cdot y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-34)
   (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)
   (if (<= y.re 2.4e-142)
     (/ (fma x.im (/ (- y.re) y.im) x.re) (- y.im))
     (if (<= y.re 1.85e+139)
       (/ (- (* x.im y.re) (* y.im x.re)) (fma y.im y.im (* y.re y.re)))
       (fma (- x.re) (/ y.im (* y.re y.re)) (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	} else if (y_46_re <= 2.4e-142) {
		tmp = fma(x_46_im, (-y_46_re / y_46_im), x_46_re) / -y_46_im;
	} else if (y_46_re <= 1.85e+139) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = fma(-x_46_re, (y_46_im / (y_46_re * y_46_re)), (x_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re);
	elseif (y_46_re <= 2.4e-142)
		tmp = Float64(fma(x_46_im, Float64(Float64(-y_46_re) / y_46_im), x_46_re) / Float64(-y_46_im));
	elseif (y_46_re <= 1.85e+139)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = fma(Float64(-x_46_re), Float64(y_46_im / Float64(y_46_re * y_46_re)), Float64(x_46_im / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.4e-34], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e-142], N[(N[(x$46$im * N[((-y$46$re) / y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+139], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x$46$re) * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.39999999999999991e-34

   1. Initial program 49.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 76.5%

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

   if -2.39999999999999991e-34 < y.re < 2.39999999999999988e-142

   1. Initial program 70.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 2.39999999999999988e-142 < y.re < 1.84999999999999996e139

   1. Initial program 78.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   if 1.84999999999999996e139 < y.re

   1. Initial program 36.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 96.9

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

   8. Applied rewrites 96.9%

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

   10. Applied rewrites 96.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{y.re \cdot y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.2e+57)
     t_0
     (if (<= y.im -1e-59)
       (/ (- (* x.im y.re) (* y.im x.re)) (* y.im y.im))
       (if (<= y.im 2.1e+122) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.2e+57) {
		tmp = t_0;
	} else if (y_46_im <= -1e-59) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.1e+122) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-2.2d+57)) then
        tmp = t_0
    else if (y_46im <= (-1d-59)) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / (y_46im * y_46im)
    else if (y_46im <= 2.1d+122) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.2e+57) {
		tmp = t_0;
	} else if (y_46_im <= -1e-59) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.1e+122) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -2.2e+57:
		tmp = t_0
	elif y_46_im <= -1e-59:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im)
	elif y_46_im <= 2.1e+122:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.2e+57)
		tmp = t_0;
	elseif (y_46_im <= -1e-59)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.1e+122)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.2e+57)
		tmp = t_0;
	elseif (y_46_im <= -1e-59)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	elseif (y_46_im <= 2.1e+122)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+57], t$95$0, If[LessEqual[y$46$im, -1e-59], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.1e+122], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.2000000000000001e57 or 2.10000000000000016e122 < y.im

   1. Initial program 40.5%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -2.2000000000000001e57 < y.im < -1e-59

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -1e-59 < y.im < 2.10000000000000016e122

   1. Initial program 72.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im \cdot y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.8e-35)
   (/ x.im y.re)
   (if (<= y.re 1.95e-115)
     (/ (- x.re) y.im)
     (if (<= y.re 7.4e-32)
       (/ (* x.im y.re) (fma y.im y.im (* y.re y.re)))
       (if (<= y.re 1.15e+39)
         (/ (* (- y.im) x.re) (* y.re y.re))
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.8e-35) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.95e-115) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 7.4e-32) {
		tmp = (x_46_im * y_46_re) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_re <= 1.15e+39) {
		tmp = (-y_46_im * x_46_re) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.8e-35)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.95e-115)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 7.4e-32)
		tmp = Float64(Float64(x_46_im * y_46_re) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 1.15e+39)
		tmp = Float64(Float64(Float64(-y_46_im) * x_46_re) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.8e-35], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.95e-115], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.4e-32], N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+39], N[(N[((-y$46$im) * x$46$re), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.79999999999999961e-35 or 1.15000000000000006e39 < y.re

   1. Initial program 48.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -7.79999999999999961e-35 < y.re < 1.9499999999999999e-115

   1. Initial program 71.8%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if 1.9499999999999999e-115 < y.re < 7.4e-32

   1. Initial program 85.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 7.4e-32 < y.re < 1.15000000000000006e39

   1. Initial program 68.1%

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

   3. Taylor expanded in y.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 51.4

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

   8. Applied rewrites 51.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im \cdot y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot x.re}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{-y.re}{y.im}, x.re\right)}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)))
   (if (<= y.re -2.4e-34)
     t_0
     (if (<= y.re 1.25e-51)
       (/ (fma x.im (/ (- y.re) y.im) x.re) (- y.im))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-51) {
		tmp = fma(x_46_im, (-y_46_re / y_46_im), x_46_re) / -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(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-51)
		tmp = Float64(fma(x_46_im, Float64(Float64(-y_46_re) / y_46_im), x_46_re) / Float64(-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[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.4e-34], t$95$0, If[LessEqual[y$46$re, 1.25e-51], N[(N[(x$46$im * N[((-y$46$re) / y$46$im), $MachinePrecision] + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.39999999999999991e-34 or 1.25000000000000001e-51 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 77.9%

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

   if -2.39999999999999991e-34 < y.re < 1.25000000000000001e-51

   1. Initial program 73.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 23.4

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

   5. Applied rewrites 23.4%

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

   6. Taylor expanded in y.im around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 83.9

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

   8. Applied rewrites 83.9%

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

Alternative 10: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ y.im y.re) (- x.re) x.im) y.re)))
   (if (<= y.re -2.4e-34)
     t_0
     (if (<= y.re 1.25e-51) (/ (- (/ (* x.im y.re) y.im) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_im / y_46_re), -x_46_re, x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-51) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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(y_46_im / y_46_re), Float64(-x_46_re), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-51)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / 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[(y$46$im / y$46$re), $MachinePrecision] * (-x$46$re) + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.4e-34], t$95$0, If[LessEqual[y$46$re, 1.25e-51], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.39999999999999991e-34 or 1.25000000000000001e-51 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 77.9%

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

   if -2.39999999999999991e-34 < y.re < 1.25000000000000001e-51

   1. Initial program 73.2%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, -x.re, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))
   (if (<= y.re -2.4e-34)
     t_0
     (if (<= y.re 1.25e-51) (/ (- (/ (* x.im y.re) y.im) x.re) 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 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-51) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    if (y_46re <= (-2.4d-34)) then
        tmp = t_0
    else if (y_46re <= 1.25d-51) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.4e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-51) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -2.4e-34:
		tmp = t_0
	elif y_46_re <= 1.25e-51:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.4e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-51)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.4e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-51)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.4e-34], t$95$0, If[LessEqual[y$46$re, 1.25e-51], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.39999999999999991e-34 or 1.25000000000000001e-51 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -2.39999999999999991e-34 < y.re < 1.25000000000000001e-51

   1. Initial program 73.2%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.5e-39) t_0 (if (<= y.im 7e-30) (/ x.im 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 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.5e-39) {
		tmp = t_0;
	} else if (y_46_im <= 7e-30) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-2.5d-39)) then
        tmp = t_0
    else if (y_46im <= 7d-30) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.5e-39) {
		tmp = t_0;
	} else if (y_46_im <= 7e-30) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -2.5e-39:
		tmp = t_0
	elif y_46_im <= 7e-30:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e-39)
		tmp = t_0;
	elseif (y_46_im <= 7e-30)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.5e-39)
		tmp = t_0;
	elseif (y_46_im <= 7e-30)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e-39], t$95$0, If[LessEqual[y$46$im, 7e-30], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.4999999999999999e-39 or 7.0000000000000006e-30 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if -2.4999999999999999e-39 < y.im < 7.0000000000000006e-30

   1. Initial program 75.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 63.9

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

   5. Applied rewrites 63.9%

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

Alternative 13: 42.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in y.im around 0

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

   1. lower-/.f64 42.4

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

5. Applied rewrites 42.4%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))