_divideComplex, real part

Percentage Accurate: 61.1% → 82.3%

Time: 7.6s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.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)
    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: 61.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.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)
    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: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -2.15 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.25e+73)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (if (<= y.re -2.15e-76)
       t_0
       (if (<= y.re 6.7e-81)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 6.6e+122)
           t_0
           (/ (fma (/ x.im y.re) 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 t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.25e+73) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -2.15e-76) {
		tmp = t_0;
	} else if (y_46_re <= 6.7e-81) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 6.6e+122) {
		tmp = t_0;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_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)))
	tmp = 0.0
	if (y_46_re <= -1.25e+73)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -2.15e-76)
		tmp = t_0;
	elseif (y_46_re <= 6.7e-81)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 6.6e+122)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$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]}, If[LessEqual[y$46$re, -1.25e+73], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.15e-76], t$95$0, If[LessEqual[y$46$re, 6.7e-81], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+122], t$95$0, N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.24999999999999994e73

   1. Initial program 43.7%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -1.24999999999999994e73 < y.re < -2.15e-76 or 6.70000000000000021e-81 < y.re < 6.5999999999999998e122

   1. Initial program 83.7%

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

   if -2.15e-76 < y.re < 6.70000000000000021e-81

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.9

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

   7. Applied rewrites 90.9%

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

   if 6.5999999999999998e122 < y.re

   1. Initial program 26.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -2.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y.re}{x.re}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -14200.0)
   (/ x.re y.re)
   (if (<= y.re 1.16e-29) (/ x.im y.im) (pow (/ y.re x.re) -1.0))))

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 <= -14200.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.16e-29) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = pow((y_46_re / x_46_re), -1.0);
	}
	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) :: tmp
    if (y_46re <= (-14200.0d0)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.16d-29) then
        tmp = x_46im / y_46im
    else
        tmp = (y_46re / x_46re) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -14200.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.16e-29) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = Math.pow((y_46_re / x_46_re), -1.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -14200.0:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.16e-29:
		tmp = x_46_im / y_46_im
	else:
		tmp = math.pow((y_46_re / x_46_re), -1.0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -14200.0)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.16e-29)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(y_46_re / x_46_re) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -14200.0)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.16e-29)
		tmp = x_46_im / y_46_im;
	else
		tmp = (y_46_re / x_46_re) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -14200.0], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.16e-29], N[(x$46$im / y$46$im), $MachinePrecision], N[Power[N[(y$46$re / x$46$re), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -14200

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if -14200 < y.re < 1.15999999999999996e-29

   1. Initial program 70.1%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if 1.15999999999999996e-29 < y.re

   1. Initial program 52.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 65.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{y.re}{x.re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -14200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y.re}{x.re}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -14200.0)
   (/ x.re y.re)
   (if (<= y.re 1.95e-48)
     (/ x.im y.im)
     (if (<= y.re 1.65e+129)
       (* (/ x.re (fma y.im y.im (* y.re y.re))) y.re)
       (/ 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 <= -14200.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.95e-48) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1.65e+129) {
		tmp = (x_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_re;
	} 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 <= -14200.0)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.95e-48)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1.65e+129)
		tmp = Float64(Float64(x_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_re);
	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, -14200.0], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.95e-48], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+129], N[(N[(x$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -14200 or 1.64999999999999995e129 < y.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if -14200 < y.re < 1.95e-48

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 1.95e-48 < y.re < 1.64999999999999995e129

   1. Initial program 78.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -14200:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{-9} \lor \neg \left(y.im \leq 8.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.5e-9) (not (<= y.im 8.6e+26)))
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (/ (fma (/ y.im y.re) x.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_im <= -1.5e-9) || !(y_46_im <= 8.6e+26)) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, 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_im <= -1.5e-9) || !(y_46_im <= 8.6e+26))
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, 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[Or[LessEqual[y$46$im, -1.5e-9], N[Not[LessEqual[y$46$im, 8.6e+26]], $MachinePrecision]], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.49999999999999999e-9 or 8.5999999999999996e26 < y.im

   1. Initial program 52.6%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -1.49999999999999999e-9 < y.im < 8.5999999999999996e26

   1. Initial program 67.9%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{-9} \lor \neg \left(y.im \leq 8.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{-9} \lor \neg \left(y.im \leq 8.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.5e-9) (not (<= y.im 8.6e+26)))
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (/ (fma (/ x.im y.re) 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_im <= -1.5e-9) || !(y_46_im <= 8.6e+26)) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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_im <= -1.5e-9) || !(y_46_im <= 8.6e+26))
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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[Or[LessEqual[y$46$im, -1.5e-9], N[Not[LessEqual[y$46$im, 8.6e+26]], $MachinePrecision]], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.49999999999999999e-9 or 8.5999999999999996e26 < y.im

   1. Initial program 52.6%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -1.49999999999999999e-9 < y.im < 8.5999999999999996e26

   1. Initial program 67.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{-9} \lor \neg \left(y.im \leq 8.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.38 \cdot 10^{-36} \lor \neg \left(y.re \leq 3.9 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.38e-36) (not (<= y.re 3.9e-30)))
   (/ (fma (/ x.im y.re) y.im 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_re <= -1.38e-36) || !(y_46_re <= 3.9e-30)) {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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_re <= -1.38e-36) || !(y_46_re <= 3.9e-30))
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.38e-36], N[Not[LessEqual[y$46$re, 3.9e-30]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.38e-36 or 3.9000000000000003e-30 < y.re

   1. Initial program 54.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.38e-36 < y.re < 3.9000000000000003e-30

   1. Initial program 69.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.38 \cdot 10^{-36} \lor \neg \left(y.re \leq 3.9 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -15500:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -15500.0)
   (/ (fma (/ y.im y.re) x.im x.re) y.re)
   (if (<= y.re 3e-24)
     (/ (fma (/ y.re y.im) x.re x.im) y.im)
     (/ (fma (/ x.im y.re) 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 <= -15500.0) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= 3e-24) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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 <= -15500.0)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= 3e-24)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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, -15500.0], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3e-24], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -15500

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if -15500 < y.re < 2.99999999999999995e-24

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 86.4

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

   7. Applied rewrites 86.4%

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

   if 2.99999999999999995e-24 < y.re

   1. Initial program 51.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -15500:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14200 \lor \neg \left(y.re \leq 1.16 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -14200.0) (not (<= y.re 1.16e-29)))
   (/ 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_re <= -14200.0) || !(y_46_re <= 1.16e-29)) {
		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)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-14200.0d0)) .or. (.not. (y_46re <= 1.16d-29))) 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_re <= -14200.0) || !(y_46_re <= 1.16e-29)) {
		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_re <= -14200.0) or not (y_46_re <= 1.16e-29):
		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_re <= -14200.0) || !(y_46_re <= 1.16e-29))
		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_re <= -14200.0) || ~((y_46_re <= 1.16e-29)))
		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[Or[LessEqual[y$46$re, -14200.0], N[Not[LessEqual[y$46$re, 1.16e-29]], $MachinePrecision]], 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.re < -14200 or 1.15999999999999996e-29 < y.re

   1. Initial program 53.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -14200 < y.re < 1.15999999999999996e-29

   1. Initial program 70.1%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -14200 \lor \neg \left(y.re \leq 1.16 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.0% accurate, 3.2× speedup

Math

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

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)
    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 60.6%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 43.0

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

5. Applied rewrites 43.0%

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

6. Final simplification 43.0%

   \[\leadsto \frac{x.im}{y.im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(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))))