_divideComplex, imaginary part

Percentage Accurate: 61.1% → 85.1%

Time: 12.5s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% 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: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, x.im \cdot \frac{y.re}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -2.1 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{-1}{\frac{1}{x.re}}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (fma (- x.re) (/ y.im t_0) (* x.im (/ y.re t_0))))
        (t_2 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -4.2e+153)
     t_2
     (if (<= y.im -2.1e-141)
       t_1
       (if (<= y.im 2.7e-138)
         (/ (fma (/ y.im y.re) (/ -1.0 (/ 1.0 x.re)) x.im) y.re)
         (if (<= y.im 4.5e+104) t_1 t_2))))))

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 = fma(-x_46_re, (y_46_im / t_0), (x_46_im * (y_46_re / t_0)));
	double t_2 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.2e+153) {
		tmp = t_2;
	} else if (y_46_im <= -2.1e-141) {
		tmp = t_1;
	} else if (y_46_im <= 2.7e-138) {
		tmp = fma((y_46_im / y_46_re), (-1.0 / (1.0 / x_46_re)), x_46_im) / y_46_re;
	} else if (y_46_im <= 4.5e+104) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = fma(Float64(-x_46_re), Float64(y_46_im / t_0), Float64(x_46_im * Float64(y_46_re / t_0)))
	t_2 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.2e+153)
		tmp = t_2;
	elseif (y_46_im <= -2.1e-141)
		tmp = t_1;
	elseif (y_46_im <= 2.7e-138)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), Float64(-1.0 / Float64(1.0 / x_46_re)), x_46_im) / y_46_re);
	elseif (y_46_im <= 4.5e+104)
		tmp = t_1;
	else
		tmp = t_2;
	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[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision] + N[(x$46$im * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.2e+153], t$95$2, If[LessEqual[y$46$im, -2.1e-141], t$95$1, If[LessEqual[y$46$im, 2.7e-138], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(-1.0 / N[(1.0 / x$46$re), $MachinePrecision]), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.5e+104], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.20000000000000033e153 or 4.4999999999999998e104 < y.im

   1. Initial program 35.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -4.20000000000000033e153 < y.im < -2.0999999999999999e-141 or 2.70000000000000029e-138 < y.im < 4.4999999999999998e104

   1. Initial program 78.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 84.2

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

   4. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.7

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-fma.f64 87.7

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

   6. Applied rewrites 87.7%

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

   if -2.0999999999999999e-141 < y.im < 2.70000000000000029e-138

   1. Initial program 71.7%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 92.7

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 94.0%

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

Alternative 2: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im)))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -6.2e+59)
     t_1
     (if (<= y.re -3.55e-103)
       (/ (fma y.re x.im (* y.im (- x.re))) t_0)
       (if (<= y.re 5e-161)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 7.5e+106)
           (/ (- (* x.im y.re) (* y.im x.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_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -6.2e+59) {
		tmp = t_1;
	} else if (y_46_re <= -3.55e-103) {
		tmp = fma(y_46_re, x_46_im, (y_46_im * -x_46_re)) / t_0;
	} else if (y_46_re <= 5e-161) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.5e+106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_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_re, y_46_re, Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -6.2e+59)
		tmp = t_1;
	elseif (y_46_re <= -3.55e-103)
		tmp = Float64(fma(y_46_re, x_46_im, Float64(y_46_im * Float64(-x_46_re))) / t_0);
	elseif (y_46_re <= 5e-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 <= 7.5e+106)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_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$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e+59], t$95$1, If[LessEqual[y$46$re, -3.55e-103], N[(N[(y$46$re * x$46$im + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 5e-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, 7.5e+106], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.20000000000000029e59 or 7.50000000000000058e106 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if -6.20000000000000029e59 < y.re < -3.55000000000000023e-103

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-*.f64 84.1

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

   6. Applied rewrites 84.1%

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

   if -3.55000000000000023e-103 < y.re < 4.9999999999999999e-161

   1. Initial program 74.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 94.9

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

   8. Applied rewrites 94.9%

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

   if 4.9999999999999999e-161 < y.re < 7.50000000000000058e106

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.3

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

   4. Applied rewrites 86.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im y.re) (* y.im x.re)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -6.2e+59)
     t_1
     (if (<= y.re -3.55e-103)
       t_0
       (if (<= y.re 5e-161)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 7.5e+106) 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 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -6.2e+59) {
		tmp = t_1;
	} else if (y_46_re <= -3.55e-103) {
		tmp = t_0;
	} else if (y_46_re <= 5e-161) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.5e+106) {
		tmp = 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 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -6.2e+59)
		tmp = t_1;
	elseif (y_46_re <= -3.55e-103)
		tmp = t_0;
	elseif (y_46_re <= 5e-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 <= 7.5e+106)
		tmp = 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[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e+59], t$95$1, If[LessEqual[y$46$re, -3.55e-103], t$95$0, If[LessEqual[y$46$re, 5e-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, 7.5e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.20000000000000029e59 or 7.50000000000000058e106 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if -6.20000000000000029e59 < y.re < -3.55000000000000023e-103 or 4.9999999999999999e-161 < y.re < 7.50000000000000058e106

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.4

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

   4. Applied rewrites 85.4%

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

   if -3.55000000000000023e-103 < y.re < 4.9999999999999999e-161

   1. Initial program 74.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 94.9

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

   8. Applied rewrites 94.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.32e+154)
   (/ x.im y.re)
   (if (<= y.re -3.1e-86)
     (* x.im (/ y.re (fma y.re y.re (* y.im y.im))))
     (if (<= y.re 1.55e-108)
       (/ (- (* x.im y.re) (* y.im x.re)) (* y.im y.im))
       (if (<= y.re 6e+116)
         (/ (fma y.re x.im (* y.im (- x.re))) (* y.re y.re))
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.32e+154) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.1e-86) {
		tmp = x_46_im * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 1.55e-108) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 6e+116) {
		tmp = fma(y_46_re, x_46_im, (y_46_im * -x_46_re)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.32e+154)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.1e-86)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 1.55e-108)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 6e+116)
		tmp = Float64(fma(y_46_re, x_46_im, Float64(y_46_im * Float64(-x_46_re))) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.32e+154], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-86], N[(x$46$im * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.55e-108], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6e+116], N[(N[(y$46$re * x$46$im + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.31999999999999998e154 or 5.9999999999999997e116 < y.re

   1. Initial program 26.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -1.31999999999999998e154 < y.re < -3.09999999999999989e-86

   1. Initial program 75.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 44.9

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

   5. Applied rewrites 44.9%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if -3.09999999999999989e-86 < y.re < 1.55000000000000007e-108

   1. Initial program 76.6%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.55000000000000007e-108 < y.re < 5.9999999999999997e116

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   9. Applied rewrites 59.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* y.im x.re))))
   (if (<= y.re -1.32e+154)
     (/ x.im y.re)
     (if (<= y.re -3.1e-86)
       (* x.im (/ y.re (fma y.re y.re (* y.im y.im))))
       (if (<= y.re 1.55e-108)
         (/ t_0 (* y.im y.im))
         (if (<= y.re 6e+116) (/ t_0 (* 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 t_0 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -1.32e+154) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.1e-86) {
		tmp = x_46_im * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 1.55e-108) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_re <= 6e+116) {
		tmp = t_0 / (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)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_re <= -1.32e+154)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.1e-86)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 1.55e-108)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 6e+116)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.32e+154], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-86], N[(x$46$im * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.55e-108], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6e+116], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.31999999999999998e154 or 5.9999999999999997e116 < y.re

   1. Initial program 26.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -1.31999999999999998e154 < y.re < -3.09999999999999989e-86

   1. Initial program 75.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 44.9

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

   5. Applied rewrites 44.9%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if -3.09999999999999989e-86 < y.re < 1.55000000000000007e-108

   1. Initial program 76.6%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.55000000000000007e-108 < y.re < 5.9999999999999997e116

   1. Initial program 84.5%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.55 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -4.8e+153)
     t_0
     (if (<= y.im -6.6e-123)
       (* x.re (/ (- y.im) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 2.8e-138)
         (/ x.im y.re)
         (if (<= y.im 2.55e+130)
           (/ (- (* x.im y.re) (* y.im x.re)) (* y.im y.im))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -4.8e+153) {
		tmp = t_0;
	} else if (y_46_im <= -6.6e-123) {
		tmp = x_46_re * (-y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_im <= 2.8e-138) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 2.55e+130) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+153)
		tmp = t_0;
	elseif (y_46_im <= -6.6e-123)
		tmp = Float64(x_46_re * Float64(Float64(-y_46_im) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 2.8e-138)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 2.55e+130)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(y_46_im * y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+153], t$95$0, If[LessEqual[y$46$im, -6.6e-123], N[(x$46$re * N[((-y$46$im) / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8e-138], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.55e+130], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.79999999999999985e153 or 2.5499999999999998e130 < y.im

   1. Initial program 32.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -4.79999999999999985e153 < y.im < -6.6000000000000005e-123

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   if -6.6000000000000005e-123 < y.im < 2.80000000000000001e-138

   1. Initial program 72.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if 2.80000000000000001e-138 < y.im < 2.5499999999999998e130

   1. Initial program 80.4%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.55 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ t_1 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im))) (t_1 (/ x.re (- y.im))))
   (if (<= y.im -4.8e+153)
     t_1
     (if (<= y.im -6.6e-123)
       (* x.re (/ (- y.im) t_0))
       (if (<= y.im 2.15e-141)
         (/ x.im y.re)
         (if (<= y.im 1.2e+23) (* 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_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -4.8e+153) {
		tmp = t_1;
	} else if (y_46_im <= -6.6e-123) {
		tmp = x_46_re * (-y_46_im / t_0);
	} else if (y_46_im <= 2.15e-141) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.2e+23) {
		tmp = 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_re, y_46_re, Float64(y_46_im * y_46_im))
	t_1 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+153)
		tmp = t_1;
	elseif (y_46_im <= -6.6e-123)
		tmp = Float64(x_46_re * Float64(Float64(-y_46_im) / t_0));
	elseif (y_46_im <= 2.15e-141)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.2e+23)
		tmp = Float64(x_46_im * Float64(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$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+153], t$95$1, If[LessEqual[y$46$im, -6.6e-123], N[(x$46$re * N[((-y$46$im) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.15e-141], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+23], N[(x$46$im * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.79999999999999985e153 or 1.2e23 < y.im

   1. Initial program 45.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -4.79999999999999985e153 < y.im < -6.6000000000000005e-123

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   if -6.6000000000000005e-123 < y.im < 2.14999999999999987e-141

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if 2.14999999999999987e-141 < y.im < 1.2e23

   1. Initial program 81.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 34.7

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

   5. Applied rewrites 34.7%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.9

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

   8. Applied rewrites 51.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 14200000000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -4.8e+153)
     t_0
     (if (<= y.im -1.2e-41)
       (* x.re (/ (- y.im) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 14200000000000.0)
         (/ (- 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 <= -4.8e+153) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-41) {
		tmp = x_46_re * (-y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_im <= 14200000000000.0) {
		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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+153)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-41)
		tmp = Float64(x_46_re * Float64(Float64(-y_46_im) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 14200000000000.0)
		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

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, -4.8e+153], t$95$0, If[LessEqual[y$46$im, -1.2e-41], N[(x$46$re * N[((-y$46$im) / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 14200000000000.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.79999999999999985e153 or 1.42e13 < y.im

   1. Initial program 46.5%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -4.79999999999999985e153 < y.im < -1.20000000000000011e-41

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 23.1

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -1.20000000000000011e-41 < y.im < 1.42e13

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 14200000000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 14200000000000:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -4.8e+153)
     t_0
     (if (<= y.im -2.2e-28)
       (* x.re (/ (- y.im) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 14200000000000.0)
         (/ (- 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 <= -4.8e+153) {
		tmp = t_0;
	} else if (y_46_im <= -2.2e-28) {
		tmp = x_46_re * (-y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_im <= 14200000000000.0) {
		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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+153)
		tmp = t_0;
	elseif (y_46_im <= -2.2e-28)
		tmp = Float64(x_46_re * Float64(Float64(-y_46_im) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 14200000000000.0)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+153], t$95$0, If[LessEqual[y$46$im, -2.2e-28], N[(x$46$re * N[((-y$46$im) / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 14200000000000.0], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.79999999999999985e153 or 1.42e13 < y.im

   1. Initial program 46.5%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -4.79999999999999985e153 < y.im < -2.19999999999999996e-28

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -2.19999999999999996e-28 < y.im < 1.42e13

   1. Initial program 74.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 77.2

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

   8. Applied rewrites 77.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 14200000000000:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -6.8e-7)
     t_0
     (if (<= y.im 2.15e-141)
       (/ x.im y.re)
       (if (<= y.im 1.2e+23)
         (* x.im (/ y.re (fma y.re y.re (* y.im y.im))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -6.8e-7) {
		tmp = t_0;
	} else if (y_46_im <= 2.15e-141) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.2e+23) {
		tmp = x_46_im * (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -6.8e-7)
		tmp = t_0;
	elseif (y_46_im <= 2.15e-141)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.2e+23)
		tmp = Float64(x_46_im * Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -6.8e-7], t$95$0, If[LessEqual[y$46$im, 2.15e-141], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+23], N[(x$46$im * N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.79999999999999948e-7 or 1.2e23 < y.im

   1. Initial program 55.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -6.79999999999999948e-7 < y.im < 2.14999999999999987e-141

   1. Initial program 72.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 2.14999999999999987e-141 < y.im < 1.2e23

   1. Initial program 81.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 34.7

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

   5. Applied rewrites 34.7%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.9

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

   8. Applied rewrites 51.9%

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

Alternative 11: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -6.7 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -6.7e+24)
     t_0
     (if (<= y.re 1.45e-30) (/ (- (/ (* x.im y.re) y.im) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -6.7e+24) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-30) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    if (y_46re <= (-6.7d+24)) then
        tmp = t_0
    else if (y_46re <= 1.45d-30) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -6.7e+24) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-30) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -6.7e+24:
		tmp = t_0
	elif y_46_re <= 1.45e-30:
		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(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -6.7e+24)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-30)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -6.7e+24)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-30)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -6.7e+24], t$95$0, If[LessEqual[y$46$re, 1.45e-30], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.6999999999999999e24 or 1.44999999999999995e-30 < y.re

   1. Initial program 52.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 76.4

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

   8. Applied rewrites 76.4%

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

   if -6.6999999999999999e24 < y.re < 1.44999999999999995e-30

   1. Initial program 79.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

Alternative 12: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 14200000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -6.8e-7)
     t_0
     (if (<= y.im 14200000000000.0) (/ 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 <= -6.8e-7) {
		tmp = t_0;
	} else if (y_46_im <= 14200000000000.0) {
		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)
    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 <= (-6.8d-7)) then
        tmp = t_0
    else if (y_46im <= 14200000000000.0d0) 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 <= -6.8e-7) {
		tmp = t_0;
	} else if (y_46_im <= 14200000000000.0) {
		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 <= -6.8e-7:
		tmp = t_0
	elif y_46_im <= 14200000000000.0:
		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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -6.8e-7)
		tmp = t_0;
	elseif (y_46_im <= 14200000000000.0)
		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 <= -6.8e-7)
		tmp = t_0;
	elseif (y_46_im <= 14200000000000.0)
		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, -6.8e-7], t$95$0, If[LessEqual[y$46$im, 14200000000000.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.79999999999999948e-7 or 1.42e13 < y.im

   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.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -6.79999999999999948e-7 < y.im < 1.42e13

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

Alternative 13: 42.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 40.6

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

5. Applied rewrites 40.6%

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

Reproduce

herbie shell --seed 2024221 
(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))))