_divideComplex, imaginary part

Percentage Accurate: 61.8% → 82.6%

Time: 2.6s

Alternatives: 7

Speedup: 1.6×

• Could not converge on a ground truth

  1. x.re = 4.569336682752324e-260
  2. x.im = -2.384648982040582e-58
  3. y.re = 4.614906735260092e-21
  4. y.im = +nan.0

Specification

Math

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

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)
use fmin_fmax_functions
    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: 61.8% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-y.im}{y.re}, \frac{x.re}{y.re}, \frac{x.im}{y.re}\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (/
         (- (* x.im y.re) (* x.re y.im))
         (+ (* y.re y.re) (* y.im y.im)))))
  (if (<= y.re -5.5e+66)
    (/ (- x.im (* x.re (/ y.im y.re))) y.re)
    (if (<= y.re -4.8e-93)
      t_0
      (if (<= y.re 4.9e-161)
        (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
        (if (<= y.re 1.12e+83)
          t_0
          (fma (/ (- y.im) y.re) (/ x.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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -5.5e+66) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= -4.8e-93) {
		tmp = t_0;
	} else if (y_46_re <= 4.9e-161) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.12e+83) {
		tmp = t_0;
	} else {
		tmp = fma((-y_46_im / y_46_re), (x_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)
	t_0 = 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)))
	tmp = 0.0
	if (y_46_re <= -5.5e+66)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= -4.8e-93)
		tmp = t_0;
	elseif (y_46_re <= 4.9e-161)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.12e+83)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-y_46_im) / y_46_re), Float64(x_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_] := Block[{t$95$0 = 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]}, If[LessEqual[y$46$re, -5.5e+66], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.8e-93], t$95$0, If[LessEqual[y$46$re, 4.9e-161], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.12e+83], t$95$0, N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.5e66

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 51.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.5%

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

   8. Applied rewrites 53.5%

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

   if -5.5e66 < y.re < -4.8000000000000002e-93 or 4.9000000000000004e-161 < y.re < 1.12e83

   1. Initial program 61.8%

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

   if -4.8000000000000002e-93 < y.re < 4.9000000000000004e-161

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.1%

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

   8. Applied rewrites 53.1%

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

   if 1.12e83 < y.re

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. times-frac N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-/.f64 52.2%

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

   6. Applied rewrites 52.2%

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

Alternative 2: 79.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := y.re \cdot \frac{x.im}{t\_0}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_1 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_1 - y.im \cdot \frac{x.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (fma y.im y.im (* y.re y.re)))
       (t_1 (* y.re (/ x.im t_0))))
  (if (<= y.im -3.6e+30)
    (- t_1 (/ x.re y.im))
    (if (<= y.im 4.2e-73)
      (/ (- x.im (* x.re (/ y.im y.re))) y.re)
      (if (<= y.im 2e+150)
        (- t_1 (* y.im (/ x.re t_0)))
        (/ (- (/ (* x.im y.re) y.im) x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = y_46_re * (x_46_im / t_0);
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_1 - (x_46_re / y_46_im);
	} else if (y_46_im <= 4.2e-73) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2e+150) {
		tmp = t_1 - (y_46_im * (x_46_re / t_0));
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(y_46_re * Float64(x_46_im / t_0))
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = Float64(t_1 - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 4.2e-73)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2e+150)
		tmp = Float64(t_1 - Float64(y_46_im * Float64(x_46_re / t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[(x$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], N[(t$95$1 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e-73], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2e+150], N[(t$95$1 - N[(y$46$im * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.6000000000000002e30

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-/.f64 54.0%

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

   8. Applied rewrites 54.0%

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

   if -3.6000000000000002e30 < y.im < 4.1999999999999997e-73

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 51.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.5%

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

   8. Applied rewrites 53.5%

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

   if 4.1999999999999997e-73 < y.im < 2e150

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   if 2e150 < y.im

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.1%

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

   8. Applied rewrites 53.1%

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

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 0.045:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (/ (- (/ (* x.im y.re) y.im) x.re) y.im)))
  (if (<= y.im -3.6e+30)
    t_0
    (if (<= y.im 0.045)
      (/ (- 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 = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 0.045) {
		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)
use fmin_fmax_functions
    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_46re) / y_46im) - x_46re) / y_46im
    if (y_46im <= (-3.6d+30)) then
        tmp = t_0
    else if (y_46im <= 0.045d0) 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 = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 0.045) {
		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 = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -3.6e+30:
		tmp = t_0
	elif y_46_im <= 0.045:
		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(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 0.045)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(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 = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 0.045)
		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[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], t$95$0, If[LessEqual[y$46$im, 0.045], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.6000000000000002e30 or 0.044999999999999998 < y.im

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.1%

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

   8. Applied rewrites 53.1%

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

   if -3.6000000000000002e30 < y.im < 0.044999999999999998

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 51.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.5%

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

   8. Applied rewrites 53.5%

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

Alternative 4: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= y.re -6.1e-52)
  (/ x.im y.re)
  (if (<= y.re 2.4e+69)
    (/ (- (/ (* x.im y.re) y.im) x.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 <= -6.1e-52) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+69) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-6.1d-52)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.4d+69) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.1e-52) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+69) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -6.1e-52:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.4e+69:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	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 <= -6.1e-52)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.4e+69)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -6.1e-52)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.4e+69)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.1e-52], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+69], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.0999999999999999e-52 or 2.4000000000000002e69 < y.re

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.6%

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

   4. Applied rewrites 42.6%

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

   if -6.0999999999999999e-52 < y.re < 2.4000000000000002e69

   1. Initial program 61.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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. mult-flip 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 61.5%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 61.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.1%

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

   8. Applied rewrites 53.1%

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

Alternative 5: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 280:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (/ (- x.re) y.im)))
  (if (<= y.im -3.6e+30) t_0 (if (<= y.im 280.0) (/ 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 <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 280.0) {
		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)
use fmin_fmax_functions
    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 <= (-3.6d+30)) then
        tmp = t_0
    else if (y_46im <= 280.0d0) 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 <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 280.0) {
		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 <= -3.6e+30:
		tmp = t_0
	elif y_46_im <= 280.0:
		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 <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 280.0)
		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 <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 280.0)
		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, -3.6e+30], t$95$0, If[LessEqual[y$46$im, 280.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.6000000000000002e30 or 280 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 43.3%

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

   4. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 43.3%

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

   6. Applied rewrites 43.3%

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

   if -3.6000000000000002e30 < y.im < 280

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.6%

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

   4. Applied rewrites 42.6%

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

Alternative 6: 46.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{0}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= y.im -4.8e+184)
  (/ 0.0 y.re)
  (if (<= y.im 3.8e+145) (/ x.im y.re) (/ 0.0 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_im <= -4.8e+184) {
		tmp = 0.0 / y_46_re;
	} else if (y_46_im <= 3.8e+145) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = 0.0 / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-4.8d+184)) then
        tmp = 0.0d0 / y_46re
    else if (y_46im <= 3.8d+145) then
        tmp = x_46im / y_46re
    else
        tmp = 0.0d0 / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.8e+184) {
		tmp = 0.0 / y_46_re;
	} else if (y_46_im <= 3.8e+145) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = 0.0 / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.8e+184:
		tmp = 0.0 / y_46_re
	elif y_46_im <= 3.8e+145:
		tmp = x_46_im / y_46_re
	else:
		tmp = 0.0 / 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_im <= -4.8e+184)
		tmp = Float64(0.0 / y_46_re);
	elseif (y_46_im <= 3.8e+145)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(0.0 / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.8e+184)
		tmp = 0.0 / y_46_re;
	elseif (y_46_im <= 3.8e+145)
		tmp = x_46_im / y_46_re;
	else
		tmp = 0.0 / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.8e+184], N[(0.0 / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+145], N[(x$46$im / y$46$re), $MachinePrecision], N[(0.0 / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.7999999999999999e184 or 3.8000000000000001e145 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.6%

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{0}{y.re} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.1%

      \[\leadsto \frac{0}{y.re} \]

   if -4.7999999999999999e184 < y.im < 3.8000000000000001e145

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.6%

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

   4. Applied rewrites 42.6%

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

Alternative 7: 42.6% accurate, 4.6× speedup

Math

\[\frac{x.im}{y.re} \]

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)
use fmin_fmax_functions
    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 61.8%

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

2. Taylor expanded in y.re around inf

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

   1. lower-/.f64 42.6%

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

4. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(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))))