_divideComplex, imaginary part

Percentage Accurate: 61.6% → 80.2%

Time: 8.4s

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: 61.6% 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: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{t\_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 5.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -2.35e+113)
     (fma
      (fma
       (- (* (/ x.re (pow y.re 4.0)) y.im) (/ x.im (pow y.re 3.0)))
       y.im
       (/ (/ (- x.re) y.re) y.re))
      y.im
      (/ x.im y.re))
     (if (<= y.re -2.6e+19)
       (* (/ y.re t_0) x.im)
       (if (<= y.re 5.05e-141)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.4e+70)
           (/ (fma (- y.im) x.re (* x.im y.re)) t_0)
           (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.35e+113) {
		tmp = fma(fma((((x_46_re / pow(y_46_re, 4.0)) * y_46_im) - (x_46_im / pow(y_46_re, 3.0))), y_46_im, ((-x_46_re / y_46_re) / y_46_re)), y_46_im, (x_46_im / y_46_re));
	} else if (y_46_re <= -2.6e+19) {
		tmp = (y_46_re / t_0) * x_46_im;
	} else if (y_46_re <= 5.05e-141) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+70) {
		tmp = fma(-y_46_im, x_46_re, (x_46_im * y_46_re)) / t_0;
	} else {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.35e+113)
		tmp = fma(fma(Float64(Float64(Float64(x_46_re / (y_46_re ^ 4.0)) * y_46_im) - Float64(x_46_im / (y_46_re ^ 3.0))), y_46_im, Float64(Float64(Float64(-x_46_re) / y_46_re) / y_46_re)), y_46_im, Float64(x_46_im / y_46_re));
	elseif (y_46_re <= -2.6e+19)
		tmp = Float64(Float64(y_46_re / t_0) * x_46_im);
	elseif (y_46_re <= 5.05e-141)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.4e+70)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re)) / t_0);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.35e+113], N[(N[(N[(N[(N[(x$46$re / N[Power[y$46$re, 4.0], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision] - N[(x$46$im / N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$im + N[(N[((-x$46$re) / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] * y$46$im + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.6e+19], N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.05e-141], 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, 1.4e+70], N[(N[((-y$46$im) * x$46$re + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.3499999999999999e113

   1. Initial program 38.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(-1 \cdot \frac{x.re}{{y.re}^{2}} + y.im \cdot \left(\frac{x.re \cdot y.im}{{y.re}^{4}} - \frac{x.im}{{y.re}^{3}}\right)\right) + \frac{x.im}{y.re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{x.re}{{y.re}^{2}} + y.im \cdot \left(\frac{x.re \cdot y.im}{{y.re}^{4}} - \frac{x.im}{{y.re}^{3}}\right)\right) \cdot y.im} + \frac{x.im}{y.re} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{x.re}{{y.re}^{2}} + y.im \cdot \left(\frac{x.re \cdot y.im}{{y.re}^{4}} - \frac{x.im}{{y.re}^{3}}\right), y.im, \frac{x.im}{y.re}\right)} \]

   5. Applied rewrites 93.6%

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

   if -2.3499999999999999e113 < y.re < -2.6e19

   1. Initial program 89.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if -2.6e19 < y.re < 5.05000000000000014e-141

   1. Initial program 67.1%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if 5.05000000000000014e-141 < y.re < 1.39999999999999995e70

   1. Initial program 83.8%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 83.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 83.8

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

   4. Applied rewrites 83.8%

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

   if 1.39999999999999995e70 < y.re

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 5.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{t\_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 5.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -7.5e+113)
     t_1
     (if (<= y.re -2.6e+19)
       (* (/ y.re t_0) x.im)
       (if (<= y.re 5.05e-141)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.4e+70)
           (/ (fma (- y.im) x.re (* x.im y.re)) t_0)
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.5e+113) {
		tmp = t_1;
	} else if (y_46_re <= -2.6e+19) {
		tmp = (y_46_re / t_0) * x_46_im;
	} else if (y_46_re <= 5.05e-141) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+70) {
		tmp = fma(-y_46_im, x_46_re, (x_46_im * y_46_re)) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -7.5e+113)
		tmp = t_1;
	elseif (y_46_re <= -2.6e+19)
		tmp = Float64(Float64(y_46_re / t_0) * x_46_im);
	elseif (y_46_re <= 5.05e-141)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.4e+70)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re)) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+113], t$95$1, If[LessEqual[y$46$re, -2.6e+19], N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.05e-141], 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, 1.4e+70], N[(N[((-y$46$im) * x$46$re + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.5000000000000001e113 or 1.39999999999999995e70 < y.re

   1. Initial program 40.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   7. Applied rewrites 90.2%

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

   if -7.5000000000000001e113 < y.re < -2.6e19

   1. Initial program 89.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if -2.6e19 < y.re < 5.05000000000000014e-141

   1. Initial program 67.1%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if 5.05000000000000014e-141 < y.re < 1.39999999999999995e70

   1. Initial program 83.8%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 83.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 83.8

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

   4. Applied rewrites 83.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 5.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 8400:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -7.5e+113)
     t_0
     (if (<= y.re -2.6e+19)
       (* (/ y.re (fma y.im y.im (* y.re y.re))) x.im)
       (if (<= y.re 8400.0) (/ (- (/ (* x.im y.re) y.im) x.re) y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -7.5e+113) {
		tmp = t_0;
	} else if (y_46_re <= -2.6e+19) {
		tmp = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else if (y_46_re <= 8400.0) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -7.5e+113)
		tmp = t_0;
	elseif (y_46_re <= -2.6e+19)
		tmp = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	elseif (y_46_re <= 8400.0)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+113], t$95$0, If[LessEqual[y$46$re, -2.6e+19], N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8400.0], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.5000000000000001e113 or 8400 < y.re

   1. Initial program 46.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   7. Applied rewrites 86.5%

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

   if -7.5000000000000001e113 < y.re < -2.6e19

   1. Initial program 89.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

   if -2.6e19 < y.re < 8400

   1. Initial program 70.8%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 8400:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e+19)
   (/ x.im y.re)
   (if (<= y.re 2.2e-136)
     (/ (- x.re) y.im)
     (if (<= y.re 2.7e+132)
       (* (/ y.re (fma y.im y.im (* y.re y.re))) x.im)
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.6e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.2e-136) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.7e+132) {
		tmp = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_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 <= -2.6e+19)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.2e-136)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 2.7e+132)
		tmp = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.6e+19], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.2e-136], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+132], N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.6e19 or 2.7e132 < y.re

   1. Initial program 46.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.6e19 < y.re < 2.2000000000000001e-136

   1. Initial program 67.1%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if 2.2000000000000001e-136 < y.re < 2.7e132

   1. Initial program 82.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

Alternative 5: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-121}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e+19)
   (/ x.im y.re)
   (if (<= y.re 7e-121)
     (/ (- x.re) y.im)
     (if (<= y.re 9e+121)
       (* (/ x.im (fma y.im y.im (* y.re y.re))) y.re)
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.6e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 7e-121) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 9e+121) {
		tmp = (x_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_re;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.6e+19)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 7e-121)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 9e+121)
		tmp = Float64(Float64(x_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_re);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.6e+19], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7e-121], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9e+121], N[(N[(x$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.6e19 or 9.0000000000000007e121 < y.re

   1. Initial program 46.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.6e19 < y.re < 6.99999999999999985e-121

   1. Initial program 67.6%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if 6.99999999999999985e-121 < y.re < 9.0000000000000007e121

   1. Initial program 83.1%

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

   3. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

Alternative 6: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -7.6e+39)
     t_0
     (if (<= y.im 3.05e+133) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -7.6e+39) {
		tmp = t_0;
	} else if (y_46_im <= 3.05e+133) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-7.6d+39)) then
        tmp = t_0
    else if (y_46im <= 3.05d+133) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -7.6e+39) {
		tmp = t_0;
	} else if (y_46_im <= 3.05e+133) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -7.6e+39:
		tmp = t_0
	elif y_46_im <= 3.05e+133:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.6e+39)
		tmp = t_0;
	elseif (y_46_im <= 3.05e+133)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.6e+39)
		tmp = t_0;
	elseif (y_46_im <= 3.05e+133)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.6e+39], t$95$0, If[LessEqual[y$46$im, 3.05e+133], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5999999999999996e39 or 3.04999999999999979e133 < y.im

   1. Initial program 48.2%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   if -7.5999999999999996e39 < y.im < 3.04999999999999979e133

   1. Initial program 72.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -8e+39)
     t_0
     (if (<= y.im 3.05e+133) (/ (- x.im (* (/ x.re y.re) y.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 <= -8e+39) {
		tmp = t_0;
	} else if (y_46_im <= 3.05e+133) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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 <= (-8d+39)) then
        tmp = t_0
    else if (y_46im <= 3.05d+133) then
        tmp = (x_46im - ((x_46re / y_46re) * y_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 <= -8e+39) {
		tmp = t_0;
	} else if (y_46_im <= 3.05e+133) {
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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 <= -8e+39:
		tmp = t_0
	elif y_46_im <= 3.05e+133:
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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 <= -8e+39)
		tmp = t_0;
	elseif (y_46_im <= 3.05e+133)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_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 <= -8e+39)
		tmp = t_0;
	elseif (y_46_im <= 3.05e+133)
		tmp = (x_46_im - ((x_46_re / y_46_re) * y_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, -8e+39], t$95$0, If[LessEqual[y$46$im, 3.05e+133], N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.99999999999999952e39 or 3.04999999999999979e133 < y.im

   1. Initial program 48.2%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   if -7.99999999999999952e39 < y.im < 3.04999999999999979e133

   1. Initial program 72.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 76.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+39}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e+19)
   (/ x.im y.re)
   (if (<= y.re 2.1e-51) (/ (- 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 <= -2.6e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.1e-51) {
		tmp = -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)
    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 <= (-2.6d+19)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.1d-51) then
        tmp = -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 <= -2.6e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.1e-51) {
		tmp = -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 <= -2.6e+19:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.1e-51:
		tmp = -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 <= -2.6e+19)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.1e-51)
		tmp = Float64(Float64(-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 <= -2.6e+19)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.1e-51)
		tmp = -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, -2.6e+19], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-51], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.6e19 or 2.10000000000000002e-51 < y.re

   1. Initial program 55.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -2.6e19 < y.re < 2.10000000000000002e-51

   1. Initial program 70.7%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 63.3

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

   5. Applied rewrites 63.3%

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

Alternative 9: 42.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in y.im around 0

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

   1. lower-/.f64 45.0

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

5. Applied rewrites 45.0%

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

Reproduce

herbie shell --seed 2024255 
(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))))