_divideComplex, real part

Percentage Accurate: 61.8% → 84.2%

Time: 2.7s

Alternatives: 9

Speedup: 1.8×

• Could not converge on a ground truth

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

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ t_3 := \frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+169}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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.im (/ y.im t_0) (* y.re (/ x.re t_0))))
       (t_2 (/ (fma y.re (/ x.re y.im) x.im) y.im))
       (t_3 (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)))
  (if (<= y.im -9.8e+169)
    t_2
    (if (<= y.im -3.6e+30)
      t_1
      (if (<= y.im -5e-22)
        t_3
        (if (<= y.im -2.9e-97)
          (/ (fma y.im x.im (* y.re x.re)) t_0)
          (if (<= y.im 1.1e-40)
            t_3
            (if (<= y.im 1.15e+136) 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_im, (y_46_im / t_0), (y_46_re * (x_46_re / t_0)));
	double t_2 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double t_3 = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_im <= -9.8e+169) {
		tmp = t_2;
	} else if (y_46_im <= -3.6e+30) {
		tmp = t_1;
	} else if (y_46_im <= -5e-22) {
		tmp = t_3;
	} else if (y_46_im <= -2.9e-97) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / t_0;
	} else if (y_46_im <= 1.1e-40) {
		tmp = t_3;
	} else if (y_46_im <= 1.15e+136) {
		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(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / t_0)))
	t_2 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	t_3 = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_im <= -9.8e+169)
		tmp = t_2;
	elseif (y_46_im <= -3.6e+30)
		tmp = t_1;
	elseif (y_46_im <= -5e-22)
		tmp = t_3;
	elseif (y_46_im <= -2.9e-97)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / t_0);
	elseif (y_46_im <= 1.1e-40)
		tmp = t_3;
	elseif (y_46_im <= 1.15e+136)
		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$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+169], t$95$2, If[LessEqual[y$46$im, -3.6e+30], t$95$1, If[LessEqual[y$46$im, -5e-22], t$95$3, If[LessEqual[y$46$im, -2.9e-97], N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.1e-40], t$95$3, If[LessEqual[y$46$im, 1.15e+136], t$95$1, t$95$2]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.8000000000000005e169 or 1.15e136 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -9.8000000000000005e169 < y.im < -3.6000000000000002e30 or 1.1e-40 < y.im < 1.15e136

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.9%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 61.9%

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

   3. Applied rewrites 61.9%

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

   if -3.6000000000000002e30 < y.im < -4.9999999999999995e-22 or -2.8999999999999999e-97 < y.im < 1.1e-40

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

   if -4.9999999999999995e-22 < y.im < -2.8999999999999999e-97

   1. Initial program 61.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.8%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 61.8%

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

   3. Applied rewrites 61.8%

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

Alternative 2: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{y.re}{t\_0}\\ t_2 := \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+169}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(x.re, t\_1, x.im \cdot \frac{y.im}{t\_0}\right)\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(x.re, t\_1, y.im \cdot \frac{x.im}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (fma y.im y.im (* y.re y.re)))
       (t_1 (/ y.re t_0))
       (t_2 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
  (if (<= y.im -9.8e+169)
    t_2
    (if (<= y.im -2.9e-97)
      (fma x.re t_1 (* x.im (/ y.im t_0)))
      (if (<= y.im 2.5e-99)
        (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
        (if (<= y.im 1.85e+123)
          (fma x.re t_1 (* y.im (/ x.im t_0)))
          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 = y_46_re / t_0;
	double t_2 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -9.8e+169) {
		tmp = t_2;
	} else if (y_46_im <= -2.9e-97) {
		tmp = fma(x_46_re, t_1, (x_46_im * (y_46_im / t_0)));
	} else if (y_46_im <= 2.5e-99) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.85e+123) {
		tmp = fma(x_46_re, t_1, (y_46_im * (x_46_im / t_0)));
	} 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 = Float64(y_46_re / t_0)
	t_2 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.8e+169)
		tmp = t_2;
	elseif (y_46_im <= -2.9e-97)
		tmp = fma(x_46_re, t_1, Float64(x_46_im * Float64(y_46_im / t_0)));
	elseif (y_46_im <= 2.5e-99)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.85e+123)
		tmp = fma(x_46_re, t_1, Float64(y_46_im * Float64(x_46_im / t_0)));
	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[(y$46$re / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+169], t$95$2, If[LessEqual[y$46$im, -2.9e-97], N[(x$46$re * t$95$1 + N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.5e-99], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.85e+123], N[(x$46$re * t$95$1 + N[(y$46$im * N[(x$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.8000000000000005e169 or 1.85e123 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -9.8000000000000005e169 < y.im < -2.8999999999999999e-97

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 61.5%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 65.0%

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

   5. Applied rewrites 65.0%

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

   if -2.8999999999999999e-97 < y.im < 2.4999999999999998e-99

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

   if 2.4999999999999998e-99 < y.im < 1.85e123

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 61.5%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 61.5%

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

   3. Applied rewrites 61.5%

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

Alternative 3: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.re, \frac{y.re}{t\_0}, x.im \cdot \frac{y.im}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+169}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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.re t_0) (* x.im (/ y.im t_0))))
       (t_2 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
  (if (<= y.im -9.8e+169)
    t_2
    (if (<= y.im -2.9e-97)
      t_1
      (if (<= y.im 4e-131)
        (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
        (if (<= y.im 1.15e+136) 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_re / t_0), (x_46_im * (y_46_im / t_0)));
	double t_2 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -9.8e+169) {
		tmp = t_2;
	} else if (y_46_im <= -2.9e-97) {
		tmp = t_1;
	} else if (y_46_im <= 4e-131) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.15e+136) {
		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(x_46_re, Float64(y_46_re / t_0), Float64(x_46_im * Float64(y_46_im / t_0)))
	t_2 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.8e+169)
		tmp = t_2;
	elseif (y_46_im <= -2.9e-97)
		tmp = t_1;
	elseif (y_46_im <= 4e-131)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.15e+136)
		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$46$re / t$95$0), $MachinePrecision] + N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+169], t$95$2, If[LessEqual[y$46$im, -2.9e-97], t$95$1, If[LessEqual[y$46$im, 4e-131], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.15e+136], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.8000000000000005e169 or 1.15e136 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -9.8000000000000005e169 < y.im < -2.8999999999999999e-97 or 3.9999999999999999e-131 < y.im < 1.15e136

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 61.5%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 61.5%

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

   3. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 65.0%

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

   5. Applied rewrites 65.0%

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

   if -2.8999999999999999e-97 < y.im < 3.9999999999999999e-131

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

Alternative 4: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (/
         (fma y.im x.im (* y.re x.re))
         (fma y.im y.im (* y.re y.re))))
       (t_1 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
  (if (<= y.im -6e+86)
    t_1
    (if (<= y.im -2.9e-97)
      t_0
      (if (<= y.im 4e-131)
        (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
        (if (<= y.im 1.1e+109) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -6e+86) {
		tmp = t_1;
	} else if (y_46_im <= -2.9e-97) {
		tmp = t_0;
	} else if (y_46_im <= 4e-131) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.1e+109) {
		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(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	t_1 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6e+86)
		tmp = t_1;
	elseif (y_46_im <= -2.9e-97)
		tmp = t_0;
	elseif (y_46_im <= 4e-131)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.1e+109)
		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[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+86], t$95$1, If[LessEqual[y$46$im, -2.9e-97], t$95$0, If[LessEqual[y$46$im, 4e-131], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+109], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.9999999999999995e86 or 1.1e109 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -5.9999999999999995e86 < y.im < -2.8999999999999999e-97 or 3.9999999999999999e-131 < y.im < 1.1e109

   1. Initial program 61.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 61.8%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 61.8%

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

   3. Applied rewrites 61.8%

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

   if -2.8999999999999999e-97 < y.im < 3.9999999999999999e-131

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

Alternative 5: 77.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
  (if (<= y.im -3.6e+30)
    t_0
    (if (<= y.im 0.045)
      (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
      t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 0.045) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 0.045)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], t$95$0, If[LessEqual[y$46$im, 0.045], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -3.6000000000000002e30 < y.im < 0.044999999999999998

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

Alternative 6: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= y.re -4e-26)
  (/ x.re y.re)
  (if (<= y.re 2.4e+69)
    (/ (fma (/ y.re y.im) x.re x.im) y.im)
    (/ x.re 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 <= -4e-26) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.4e+69) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = x_46_re / 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 <= -4e-26)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.4e+69)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(x_46_re / 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, -4e-26], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+69], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.0000000000000002e-26 or 2.4000000000000002e69 < y.re

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

   if -4.0000000000000002e-26 < y.re < 2.4000000000000002e69

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 54.9%

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

   8. Applied rewrites 54.9%

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

Alternative 7: 69.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (/ (fma y.re (/ x.re y.im) x.im) y.im)))
  (if (<= y.im -3.6e+30) t_0 (if (<= y.im 0.045) (/ x.re y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 0.045) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 0.045)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], t$95$0, If[LessEqual[y$46$im, 0.045], N[(x$46$re / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.8%

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

   4. Applied rewrites 52.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.1%

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

   6. Applied rewrites 54.1%

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

   if -3.6000000000000002e30 < y.im < 0.044999999999999998

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

Alternative 8: 63.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 0.045:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= y.im -3.7e+33)
  (/ x.im y.im)
  (if (<= y.im 0.045) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.7e+33) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 0.045) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-3.7d+33)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 0.045d0) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.7e+33) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 0.045) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.7e+33:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 0.045:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.7e+33)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 0.045)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.7e+33)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 0.045)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.7e+33], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 0.045], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.6999999999999999e33 or 0.044999999999999998 < y.im

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   if -3.6999999999999999e33 < y.im < 0.044999999999999998

   1. Initial program 61.8%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-/.f64 43.2%

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 42.0%

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

   7. Applied rewrites 42.0%

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

Alternative 9: 43.2% accurate, 4.6× speedup

Math

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

FPCore

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

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 61.8%

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

2. Taylor expanded in y.re around 0

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

   1. lower-/.f64 43.2%

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

4. Applied rewrites 43.2%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))