_divideComplex, imaginary part

Percentage Accurate: 61.4% → 80.6%

Time: 10.4s

Alternatives: 9

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: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -0.0001107)
     t_0
     (if (<= y.im 2.3e-57)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 4.3e+91)
         (*
          (/ 1.0 (fma y.im y.im (* y.re y.re)))
          (fma (- y.im) x.re (* y.re x.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_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-57) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.3e+91) {
		tmp = (1.0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * fma(-y_46_im, x_46_re, (y_46_re * x_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(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-57)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4.3e+91)
		tmp = Float64(Float64(1.0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_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[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -0.0001107], t$95$0, If[LessEqual[y$46$im, 2.3e-57], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.3e+91], N[(N[(1.0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.10699999999999994e-4 or 4.3000000000000001e91 < y.im

   1. Initial program 54.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 17.8

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

   7. Applied rewrites 17.8%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.0

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

   10. Applied rewrites 87.0%

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

   if -1.10699999999999994e-4 < y.im < 2.3e-57

   1. Initial program 73.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 2.3e-57 < y.im < 4.3000000000000001e91

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. clear-num N/A

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

      16. flip-+ N/A

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

      17. lift-+.f64 N/A

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

   4. Applied rewrites 83.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(y.re, \frac{y.re}{x.re \cdot y.im}, \frac{y.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -0.0001107)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 2.6e-50)
     (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
     (/ 1.0 (- (fma y.re (/ y.re (* x.re y.im)) (/ y.im x.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_im <= -0.0001107) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 2.6e-50) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = 1.0 / -fma(y_46_re, (y_46_re / (x_46_re * y_46_im)), (y_46_im / x_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_im <= -0.0001107)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 2.6e-50)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(1.0 / Float64(-fma(y_46_re, Float64(y_46_re / Float64(x_46_re * y_46_im)), Float64(y_46_im / x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -0.0001107], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.6e-50], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(1.0 / (-N[(y$46$re * N[(y$46$re / N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.10699999999999994e-4

   1. Initial program 59.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 24.0

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

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 83.4

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

   10. Applied rewrites 83.4%

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

   if -1.10699999999999994e-4 < y.im < 2.6000000000000001e-50

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 2.6000000000000001e-50 < y.im

   1. Initial program 58.3%

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

   3. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in y.re around inf

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

   8. Applied rewrites 13.7%

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

   10. Applied rewrites 12.0%

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

   11. Taylor expanded in y.re around 0

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

   13. Applied rewrites 85.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(y.re, \frac{y.re}{x.re \cdot y.im}, \frac{y.im}{x.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -0.0001107)
     t_0
     (if (<= y.im 1.25e-61)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 4.3e+91)
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.im y.im) (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-61) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.3e+91) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-61)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4.3e+91)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -0.0001107], t$95$0, If[LessEqual[y$46$im, 1.25e-61], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.3e+91], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.10699999999999994e-4 or 4.3000000000000001e91 < y.im

   1. Initial program 54.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 17.8

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

   7. Applied rewrites 17.8%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 87.0

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

   10. Applied rewrites 87.0%

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

   if -1.10699999999999994e-4 < y.im < 1.25e-61

   1. Initial program 73.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 1.25e-61 < y.im < 4.3000000000000001e91

   1. Initial program 83.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -0.0001107)
     t_0
     (if (<= y.im 3.9e-50)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 8.5e+135)
         (* (/ x.re (fma y.re y.re (* y.im y.im))) (- y.im))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-50) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.5e+135) {
		tmp = (x_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * -y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-50)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.5e+135)
		tmp = Float64(Float64(x_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * 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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -0.0001107], t$95$0, If[LessEqual[y$46$im, 3.9e-50], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.5e+135], N[(N[(x$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-y$46$im)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.10699999999999994e-4 or 8.49999999999999992e135 < y.im

   1. Initial program 54.0%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if -1.10699999999999994e-4 < y.im < 3.90000000000000021e-50

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 3.90000000000000021e-50 < y.im < 8.49999999999999992e135

   1. Initial program 77.2%

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

   3. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -1.95e+18)
     t_0
     (if (<= y.im 3.6e-51)
       (/ x.im y.re)
       (if (<= y.im 8.5e+135)
         (* (/ x.re (fma y.re y.re (* y.im y.im))) (- y.im))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -1.95e+18) {
		tmp = t_0;
	} else if (y_46_im <= 3.6e-51) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 8.5e+135) {
		tmp = (x_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * -y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.95e+18)
		tmp = t_0;
	elseif (y_46_im <= 3.6e-51)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 8.5e+135)
		tmp = Float64(Float64(x_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * 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[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.95e+18], t$95$0, If[LessEqual[y$46$im, 3.6e-51], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.5e+135], N[(N[(x$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-y$46$im)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.95e18 or 8.49999999999999992e135 < y.im

   1. Initial program 50.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   if -1.95e18 < y.im < 3.6e-51

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if 3.6e-51 < y.im < 8.49999999999999992e135

   1. Initial program 77.2%

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

   3. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -0.0001107)
     t_0
     (if (<= y.im 1.7e-45) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e-45) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 1.7e-45)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -0.0001107], t$95$0, If[LessEqual[y$46$im, 1.7e-45], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.10699999999999994e-4 or 1.70000000000000002e-45 < y.im

   1. Initial program 59.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 21.3

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

   7. Applied rewrites 21.3%

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

   8. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 83.6

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

   10. Applied rewrites 83.6%

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

   if -1.10699999999999994e-4 < y.im < 1.70000000000000002e-45

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 7: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* y.re x.im) y.im) x.re) y.im)))
   (if (<= y.im -0.0001107)
     t_0
     (if (<= y.im 1.7e-45) (/ (- x.im (/ (* x.re 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e-45) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    if (y_46im <= (-0.0001107d0)) then
        tmp = t_0
    else if (y_46im <= 1.7d-45) then
        tmp = (x_46im - ((x_46re * y_46im) / 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -0.0001107) {
		tmp = t_0;
	} else if (y_46_im <= 1.7e-45) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -0.0001107:
		tmp = t_0
	elif y_46_im <= 1.7e-45:
		tmp = (x_46_im - ((x_46_re * 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(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 1.7e-45)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -0.0001107)
		tmp = t_0;
	elseif (y_46_im <= 1.7e-45)
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -0.0001107], t$95$0, If[LessEqual[y$46$im, 1.7e-45], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.10699999999999994e-4 or 1.70000000000000002e-45 < y.im

   1. Initial program 59.0%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -1.10699999999999994e-4 < y.im < 1.70000000000000002e-45

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0001107:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -1.95e+18) t_0 (if (<= y.im 3.4e-45) (/ 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 <= -1.95e+18) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e-45) {
		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 <= (-1.95d+18)) then
        tmp = t_0
    else if (y_46im <= 3.4d-45) 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 <= -1.95e+18) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e-45) {
		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 <= -1.95e+18:
		tmp = t_0
	elif y_46_im <= 3.4e-45:
		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 <= -1.95e+18)
		tmp = t_0;
	elseif (y_46_im <= 3.4e-45)
		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 <= -1.95e+18)
		tmp = t_0;
	elseif (y_46_im <= 3.4e-45)
		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, -1.95e+18], t$95$0, If[LessEqual[y$46$im, 3.4e-45], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.95e18 or 3.40000000000000004e-45 < y.im

   1. Initial program 56.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

   if -1.95e18 < y.im < 3.40000000000000004e-45

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

Alternative 9: 43.0% accurate, 3.2× speedup

Math

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

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 65.8%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 44.3

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

5. Applied rewrites 44.3%

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

Reproduce

herbie shell --seed 2024235 
(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))))