_divideComplex, real part

Percentage Accurate: 61.6% → 86.5%

Time: 14.2s

Alternatives: 12

Speedup: 1.1×

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: 61.6% 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: 86.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.5e+17)
   (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
   (if (<= y.im 1.3e+62)
     (*
      (/ (fma x.im (/ y.im y.re) x.re) (hypot y.im y.re))
      (/ y.re (hypot y.im y.re)))
     (/ 1.0 (/ y.im (fma y.re (/ x.re y.im) x.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 <= -4.5e+17) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 1.3e+62) {
		tmp = (fma(x_46_im, (y_46_im / y_46_re), x_46_re) / hypot(y_46_im, y_46_re)) * (y_46_re / hypot(y_46_im, y_46_re));
	} else {
		tmp = 1.0 / (y_46_im / fma(y_46_re, (x_46_re / y_46_im), x_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 <= -4.5e+17)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 1.3e+62)
		tmp = Float64(Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / hypot(y_46_im, y_46_re)) * Float64(y_46_re / hypot(y_46_im, y_46_re)));
	else
		tmp = Float64(1.0 / Float64(y_46_im / fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.5e+17], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+62], N[(N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.5e17

   1. Initial program 47.8%

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

   3. Taylor expanded in x.re around 0 46.2%

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

      1. *-commutative 46.2%

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

   5. Simplified 46.2%

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

      1. add-sqr-sqrt 46.2%

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

      2. hypot-undefine 46.2%

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

      3. hypot-undefine 46.2%

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

      4. times-frac 88.9%

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

   7. Applied egg-rr 88.9%

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

   if -4.5e17 < y.im < 1.29999999999999992e62

   1. Initial program 76.4%

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

   3. Taylor expanded in y.re around inf 75.1%

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

      1. *-commutative 75.1%

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

   5. Simplified 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. add-sqr-sqrt 75.1%

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

      3. hypot-undefine 75.1%

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

      4. hypot-undefine 75.1%

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

      5. times-frac 85.7%

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

      6. +-commutative 85.7%

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

      7. associate-/l* 85.0%

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

      8. fma-define 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.2%

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

      2. *-lft-identity 85.2%

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

      3. associate-/l* 95.9%

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

      4. associate-*l/ 95.9%

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

      5. *-commutative 95.9%

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

      6. fma-undefine 95.9%

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

      7. *-commutative 95.9%

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

      8. associate-*l/ 95.7%

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

      9. associate-*r/ 96.5%

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

      10. fma-define 96.5%

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

      11. hypot-undefine 76.7%

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

      12. unpow2 76.7%

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

      13. unpow2 76.7%

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

      14. +-commutative 76.7%

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

      15. unpow2 76.7%

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

      16. unpow2 76.7%

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

      17. hypot-define 96.5%

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

      18. hypot-undefine 76.7%

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

      19. unpow2 76.7%

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

      20. unpow2 76.7%

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

      21. +-commutative 76.7%

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

   9. Simplified 96.5%

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

   if 1.29999999999999992e62 < y.im

   1. Initial program 48.0%

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

   3. Taylor expanded in y.im around inf 85.8%

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

      1. associate-/l* 89.7%

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

   5. Simplified 89.7%

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

      1. clear-num 91.0%

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

      2. inv-pow 91.0%

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

      3. +-commutative 91.0%

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

      4. fma-define 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. unpow-1 91.0%

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

      2. fma-undefine 91.0%

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

      3. associate-*r/ 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-/l* 90.9%

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

      6. fma-define 90.9%

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

   9. Simplified 90.9%

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

Alternative 2: 85.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+297)
   (/ (/ (fma y.re x.re (* y.im x.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (/ (+ x.im (* x.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) {
	double tmp;
	if ((((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+297) {
		tmp = (fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 (Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+297)
		tmp = Float64(Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+297], N[(N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999998e297

   1. Initial program 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. times-frac 82.0%

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

      4. hypot-define 82.0%

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

      5. fma-define 82.0%

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

      6. hypot-define 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.7%

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

      3. div-inv 97.8%

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

      4. fma-undefine 97.8%

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

      5. *-commutative 97.8%

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

      6. fma-define 97.8%

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

      7. *-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 4.9999999999999998e297 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 15.4%

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

   3. Taylor expanded in y.im around inf 59.6%

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

      1. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.re + y.im \cdot x.im\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re x.re) (* y.im x.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+297)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (/ (+ x.im (* x.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) {
	double t_0 = (y_46_re * x_46_re) + (y_46_im * x_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+297) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_re) + (y_46_im * x_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+297) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_re) + (y_46_im * x_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+297:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+297)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_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)
	t_0 = (y_46_re * x_46_re) + (y_46_im * x_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+297)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+297], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999998e297

   1. Initial program 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. times-frac 82.0%

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

      4. hypot-define 82.0%

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

      5. fma-define 82.0%

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

      6. hypot-define 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.7%

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

      3. div-inv 97.8%

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

      4. fma-undefine 97.8%

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

      5. *-commutative 97.8%

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

      6. fma-define 97.8%

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

      7. *-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. fma-undefine 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 4.9999999999999998e297 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 15.4%

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

   3. Taylor expanded in y.im around inf 59.6%

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

      1. associate-/l* 65.5%

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

   5. Simplified 65.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+49}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.7e+49)
     (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
     (if (<= y.im -1.1e-150)
       t_0
       (if (<= y.im 6.2e-119)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.55e+29)
           t_0
           (/ 1.0 (/ y.im (fma y.re (/ x.re y.im) x.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.7e+49) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.1e-150) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-119) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.55e+29) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (y_46_im / fma(y_46_re, (x_46_re / y_46_im), x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.7e+49)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1.1e-150)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-119)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.55e+29)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(y_46_im / fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$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$46$im, -1.7e+49], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.1e-150], t$95$0, If[LessEqual[y$46$im, 6.2e-119], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.55e+29], t$95$0, N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7e49

   1. Initial program 43.3%

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

   3. Taylor expanded in x.re around 0 42.7%

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

      1. *-commutative 42.7%

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

   5. Simplified 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. hypot-undefine 42.7%

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

      3. hypot-undefine 42.7%

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

      4. times-frac 89.1%

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

   7. Applied egg-rr 89.1%

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

   if -1.7e49 < y.im < -1.1e-150 or 6.19999999999999956e-119 < y.im < 1.5499999999999999e29

   1. Initial program 89.1%

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

   if -1.1e-150 < y.im < 6.19999999999999956e-119

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. +-commutative 94.7%

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

      2. associate-/l* 94.9%

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

      3. fma-define 94.9%

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

   5. Simplified 94.9%

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

   if 1.5499999999999999e29 < y.im

   1. Initial program 49.1%

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

   3. Taylor expanded in y.im around inf 84.8%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

      1. clear-num 89.6%

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

      2. inv-pow 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

   7. Applied egg-rr 89.6%

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

      1. unpow-1 89.6%

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

      2. fma-undefine 89.6%

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

      3. associate-*r/ 85.9%

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

      4. *-commutative 85.9%

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

      5. associate-/l* 89.6%

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

      6. fma-define 89.6%

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

   9. Simplified 89.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+49}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \mathbf{if}\;y.im \leq -1.8 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ 1.0 (/ y.im (fma y.re (/ x.re y.im) x.im)))))
   (if (<= y.im -1.8e+149)
     t_1
     (if (<= y.im -7.5e-150)
       t_0
       (if (<= y.im 1.5e-117)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.8e+29) 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 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = 1.0 / (y_46_im / fma(y_46_re, (x_46_re / y_46_im), x_46_im));
	double tmp;
	if (y_46_im <= -1.8e+149) {
		tmp = t_1;
	} else if (y_46_im <= -7.5e-150) {
		tmp = t_0;
	} else if (y_46_im <= 1.5e-117) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(1.0 / Float64(y_46_im / fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.8e+149)
		tmp = t_1;
	elseif (y_46_im <= -7.5e-150)
		tmp = t_0;
	elseif (y_46_im <= 1.5e-117)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.8e+149], t$95$1, If[LessEqual[y$46$im, -7.5e-150], t$95$0, If[LessEqual[y$46$im, 1.5e-117], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.79999999999999997e149 or 1.79999999999999988e29 < y.im

   1. Initial program 37.9%

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

   3. Taylor expanded in y.im around inf 85.7%

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

      1. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

      1. clear-num 89.8%

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

      2. inv-pow 89.8%

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

      3. +-commutative 89.8%

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

      4. fma-define 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. unpow-1 89.8%

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

      2. fma-undefine 89.8%

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

      3. associate-*r/ 86.4%

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

      4. *-commutative 86.4%

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

      5. associate-/l* 89.7%

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

      6. fma-define 89.7%

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

   9. Simplified 89.7%

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

   if -1.79999999999999997e149 < y.im < -7.5000000000000004e-150 or 1.49999999999999996e-117 < y.im < 1.79999999999999988e29

   1. Initial program 88.1%

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

   if -7.5000000000000004e-150 < y.im < 1.49999999999999996e-117

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. +-commutative 94.7%

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

      2. associate-/l* 94.9%

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

      3. fma-define 94.9%

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

   5. Simplified 94.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)))
   (if (<= y.im -3.9e+136)
     t_1
     (if (<= y.im -2e-149)
       t_0
       (if (<= y.im 4.2e-114)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.8e+29) 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 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	double tmp;
	if (y_46_im <= -3.9e+136) {
		tmp = t_1;
	} else if (y_46_im <= -2e-149) {
		tmp = t_0;
	} else if (y_46_im <= 4.2e-114) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.9e+136)
		tmp = t_1;
	elseif (y_46_im <= -2e-149)
		tmp = t_0;
	elseif (y_46_im <= 4.2e-114)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.9e+136], t$95$1, If[LessEqual[y$46$im, -2e-149], t$95$0, If[LessEqual[y$46$im, 4.2e-114], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.90000000000000019e136 or 1.79999999999999988e29 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 86.3%

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

      1. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

      1. clear-num 89.4%

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

      2. un-div-inv 89.5%

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

   7. Applied egg-rr 89.5%

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

   if -3.90000000000000019e136 < y.im < -1.99999999999999996e-149 or 4.19999999999999985e-114 < y.im < 1.79999999999999988e29

   1. Initial program 87.5%

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

   if -1.99999999999999996e-149 < y.im < 4.19999999999999985e-114

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. +-commutative 94.7%

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

      2. associate-/l* 94.9%

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

      3. fma-define 94.9%

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

   5. Simplified 94.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-149}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -5.7 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)))
   (if (<= y.im -1.65e+124)
     t_1
     (if (<= y.im -5.7e-154)
       t_0
       (if (<= y.im 3.2e-114)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 1.55e+29) 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 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.65e+124) {
		tmp = t_1;
	} else if (y_46_im <= -5.7e-154) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-114) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.55e+29) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    if (y_46im <= (-1.65d+124)) then
        tmp = t_1
    else if (y_46im <= (-5.7d-154)) then
        tmp = t_0
    else if (y_46im <= 3.2d-114) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 1.55d+29) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.65e+124) {
		tmp = t_1;
	} else if (y_46_im <= -5.7e-154) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-114) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.55e+29) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	tmp = 0
	if y_46_im <= -1.65e+124:
		tmp = t_1
	elif y_46_im <= -5.7e-154:
		tmp = t_0
	elif y_46_im <= 3.2e-114:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 1.55e+29:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.65e+124)
		tmp = t_1;
	elseif (y_46_im <= -5.7e-154)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-114)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.55e+29)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.65e+124)
		tmp = t_1;
	elseif (y_46_im <= -5.7e-154)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-114)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 1.55e+29)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+124], t$95$1, If[LessEqual[y$46$im, -5.7e-154], t$95$0, If[LessEqual[y$46$im, 3.2e-114], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.55e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65000000000000007e124 or 1.5499999999999999e29 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 86.3%

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

      1. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

      1. clear-num 89.4%

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

      2. un-div-inv 89.5%

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

   7. Applied egg-rr 89.5%

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

   if -1.65000000000000007e124 < y.im < -5.6999999999999998e-154 or 3.2000000000000002e-114 < y.im < 1.5499999999999999e29

   1. Initial program 87.5%

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

   if -5.6999999999999998e-154 < y.im < 3.2000000000000002e-114

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. *-commutative 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in y.im around 0 94.7%

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

      1. associate-*r/ 94.9%

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

   8. Simplified 94.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -5.7 \cdot 10^{-154}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-11} \lor \neg \left(y.im \leq 1.95 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.2e-11) (not (<= y.im 1.95e-7)))
   (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
   (/ (+ x.re (* x.im (/ y.im y.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 <= -3.2e-11) || !(y_46_im <= 1.95e-7)) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.2d-11)) .or. (.not. (y_46im <= 1.95d-7))) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.2e-11) || !(y_46_im <= 1.95e-7)) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.2e-11) or not (y_46_im <= 1.95e-7):
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_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 <= -3.2e-11) || !(y_46_im <= 1.95e-7))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.2e-11) || ~((y_46_im <= 1.95e-7)))
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.2e-11], N[Not[LessEqual[y$46$im, 1.95e-7]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.19999999999999994e-11 or 1.95000000000000012e-7 < y.im

   1. Initial program 53.3%

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

   3. Taylor expanded in y.im around inf 82.9%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 85.3%

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

   7. Applied egg-rr 85.3%

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

   if -3.19999999999999994e-11 < y.im < 1.95000000000000012e-7

   1. Initial program 74.8%

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

   3. Taylor expanded in y.re around inf 87.4%

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

      1. *-commutative 87.4%

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

   5. Simplified 87.4%

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

   6. Taylor expanded in y.im around 0 87.4%

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

      1. associate-*r/ 87.5%

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

   8. Simplified 87.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-11} \lor \neg \left(y.im \leq 1.95 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 560000000000:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.5e+36)
   (/ 1.0 (/ y.re x.re))
   (if (<= y.re 560000000000.0)
     (/ (+ x.im (/ x.re (/ y.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 <= -7.5e+36) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 560000000000.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-7.5d+36)) then
        tmp = 1.0d0 / (y_46re / x_46re)
    else if (y_46re <= 560000000000.0d0) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.5e+36) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 560000000000.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.5e+36:
		tmp = 1.0 / (y_46_re / x_46_re)
	elif y_46_re <= 560000000000.0:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -7.5e+36)
		tmp = Float64(1.0 / Float64(y_46_re / x_46_re));
	elseif (y_46_re <= 560000000000.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.5e+36)
		tmp = 1.0 / (y_46_re / x_46_re);
	elseif (y_46_re <= 560000000000.0)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.5e+36], N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 560000000000.0], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000054e36

   1. Initial program 50.1%

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

   3. Taylor expanded in y.re around inf 63.7%

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

      1. clear-num 65.0%

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

      2. inv-pow 65.0%

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

   5. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{{\left(\frac{y.re}{x.re}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 65.0%

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

   7. Simplified 65.0%

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

   if -7.50000000000000054e36 < y.re < 5.6e11

   1. Initial program 68.9%

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

   3. Taylor expanded in y.im around inf 84.6%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 85.2%

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

   7. Applied egg-rr 85.2%

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

   if 5.6e11 < y.re

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 72.3%

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

Alternative 10: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 800000000000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.35e+37)
   (/ 1.0 (/ y.re x.re))
   (if (<= y.re 800000000000.0)
     (/ (+ x.im (* x.re (/ y.re y.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 <= -1.35e+37) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 800000000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.35d+37)) then
        tmp = 1.0d0 / (y_46re / x_46re)
    else if (y_46re <= 800000000000.0d0) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.35e+37) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 800000000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.35e+37:
		tmp = 1.0 / (y_46_re / x_46_re)
	elif y_46_re <= 800000000000.0:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -1.35e+37)
		tmp = Float64(1.0 / Float64(y_46_re / x_46_re));
	elseif (y_46_re <= 800000000000.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.35e+37)
		tmp = 1.0 / (y_46_re / x_46_re);
	elseif (y_46_re <= 800000000000.0)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.35e+37], N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 800000000000.0], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.34999999999999993e37

   1. Initial program 50.1%

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

   3. Taylor expanded in y.re around inf 63.7%

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

      1. clear-num 65.0%

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

      2. inv-pow 65.0%

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

   5. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{{\left(\frac{y.re}{x.re}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 65.0%

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

   7. Simplified 65.0%

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

   if -1.34999999999999993e37 < y.re < 8e11

   1. Initial program 68.9%

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

   3. Taylor expanded in y.im around inf 84.6%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   if 8e11 < y.re

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 72.3%

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

Alternative 11: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{-47} \lor \neg \left(y.im \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1e-47) (not (<= y.im 4e-8))) (/ 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_im <= -1e-47) || !(y_46_im <= 4e-8)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1d-47)) .or. (.not. (y_46im <= 4d-8))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1e-47) || !(y_46_im <= 4e-8)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1e-47) or not (y_46_im <= 4e-8):
		tmp = 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_im <= -1e-47) || !(y_46_im <= 4e-8))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1e-47) || ~((y_46_im <= 4e-8)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1e-47], N[Not[LessEqual[y$46$im, 4e-8]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.9999999999999997e-48 or 4.0000000000000001e-8 < y.im

   1. Initial program 55.0%

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

   3. Taylor expanded in y.re around 0 73.1%

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

   if -9.9999999999999997e-48 < y.im < 4.0000000000000001e-8

   1. Initial program 74.2%

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

   3. Taylor expanded in y.re around inf 67.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{-47} \lor \neg \left(y.im \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 5.0× 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 64.1%

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

3. Taylor expanded in y.re around 0 47.6%

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

Reproduce

herbie shell --seed 2024132 
(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))))