_divideComplex, real part

Percentage Accurate: 61.8% → 85.0%

Time: 10.5s

Alternatives: 8

Speedup: 0.6×

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: 61.8% 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: 85.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 -2e+256)
     (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
     (if (<= t_0 INFINITY)
       (*
        (/ 1.0 (hypot y.re y.im))
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
       (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_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 (t_0 <= -2e+256) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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 (t_0 <= -2e+256)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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[t$95$0, -2e+256], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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))) < -2.0000000000000001e256

   1. Initial program 64.7%

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

   3. Taylor expanded in y.re around 0 87.5%

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

      1. associate-/l* 87.9%

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

   5. Simplified 87.9%

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

      1. pow2 87.9%

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

      2. associate-*r/ 87.5%

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

      3. associate-/r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -2.0000000000000001e256 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. add-sqr-sqrt 77.2%

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

      3. times-frac 77.3%

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

      4. hypot-define 77.3%

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

      5. fma-define 77.3%

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

      6. hypot-define 95.3%

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

   4. Applied egg-rr 95.3%

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

   if +inf.0 < (/.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 0.0%

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

   3. Taylor expanded in y.re around inf 49.5%

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

      1. associate-/l* 50.3%

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

   5. Simplified 50.3%

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

      1. associate-*r/ 49.5%

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

      2. unpow2 49.5%

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

      3. associate-/r* 50.0%

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

      4. *-commutative 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. associate-/l/ 49.5%

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

      2. times-frac 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -2 \cdot 10^{+256}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+69} \lor \neg \left(y.re \leq -130000000 \lor \neg \left(y.re \leq -1.75 \cdot 10^{-41}\right) \land y.re \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.15e+69)
         (not
          (or (<= y.re -130000000.0)
              (and (not (<= y.re -1.75e-41)) (<= y.re 8.8e+46)))))
   (/ x.re y.re)
   (+ (/ x.im y.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 <= -2.15e+69) || !((y_46_re <= -130000000.0) || (!(y_46_re <= -1.75e-41) && (y_46_re <= 8.8e+46)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_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 <= (-2.15d+69)) .or. (.not. (y_46re <= (-130000000.0d0)) .or. (.not. (y_46re <= (-1.75d-41))) .and. (y_46re <= 8.8d+46))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + (((x_46re * y_46re) / y_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 <= -2.15e+69) || !((y_46_re <= -130000000.0) || (!(y_46_re <= -1.75e-41) && (y_46_re <= 8.8e+46)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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):
	tmp = 0
	if (y_46_re <= -2.15e+69) or not ((y_46_re <= -130000000.0) or (not (y_46_re <= -1.75e-41) and (y_46_re <= 8.8e+46))):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_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 ((y_46_re <= -2.15e+69) || !((y_46_re <= -130000000.0) || (!(y_46_re <= -1.75e-41) && (y_46_re <= 8.8e+46))))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * 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)
	tmp = 0.0;
	if ((y_46_re <= -2.15e+69) || ~(((y_46_re <= -130000000.0) || (~((y_46_re <= -1.75e-41)) && (y_46_re <= 8.8e+46)))))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_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_] := If[Or[LessEqual[y$46$re, -2.15e+69], N[Not[Or[LessEqual[y$46$re, -130000000.0], And[N[Not[LessEqual[y$46$re, -1.75e-41]], $MachinePrecision], LessEqual[y$46$re, 8.8e+46]]]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.14999999999999996e69 or -1.3e8 < y.re < -1.75e-41 or 8.8000000000000001e46 < y.re

   1. Initial program 45.7%

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

   3. Taylor expanded in y.re around inf 73.7%

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

   if -2.14999999999999996e69 < y.re < -1.3e8 or -1.75e-41 < y.re < 8.8000000000000001e46

   1. Initial program 70.0%

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

   3. Taylor expanded in y.re around 0 78.1%

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

      1. associate-/l* 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*r/ 78.1%

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

      3. associate-/r* 82.6%

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+69} \lor \neg \left(y.re \leq -130000000 \lor \neg \left(y.re \leq -1.75 \cdot 10^{-41}\right) \land y.re \leq 8.8 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+65} \lor \neg \left(y.re \leq -9 \cdot 10^{+15} \lor \neg \left(y.re \leq -1 \cdot 10^{-99}\right) \land y.re \leq 9.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8e+65)
         (not
          (or (<= y.re -9e+15)
              (and (not (<= y.re -1e-99)) (<= y.re 9.2e+45)))))
   (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
   (+ (/ x.im y.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 <= -8e+65) || !((y_46_re <= -9e+15) || (!(y_46_re <= -1e-99) && (y_46_re <= 9.2e+45)))) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_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 <= (-8d+65)) .or. (.not. (y_46re <= (-9d+15)) .or. (.not. (y_46re <= (-1d-99))) .and. (y_46re <= 9.2d+45))) then
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im / y_46re) / y_46re))
    else
        tmp = (x_46im / y_46im) + (((x_46re * y_46re) / y_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 <= -8e+65) || !((y_46_re <= -9e+15) || (!(y_46_re <= -1e-99) && (y_46_re <= 9.2e+45)))) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_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):
	tmp = 0
	if (y_46_re <= -8e+65) or not ((y_46_re <= -9e+15) or (not (y_46_re <= -1e-99) and (y_46_re <= 9.2e+45))):
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	else:
		tmp = (x_46_im / y_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 ((y_46_re <= -8e+65) || !((y_46_re <= -9e+15) || (!(y_46_re <= -1e-99) && (y_46_re <= 9.2e+45))))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * 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)
	tmp = 0.0;
	if ((y_46_re <= -8e+65) || ~(((y_46_re <= -9e+15) || (~((y_46_re <= -1e-99)) && (y_46_re <= 9.2e+45)))))
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	else
		tmp = (x_46_im / y_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_] := If[Or[LessEqual[y$46$re, -8e+65], N[Not[Or[LessEqual[y$46$re, -9e+15], And[N[Not[LessEqual[y$46$re, -1e-99]], $MachinePrecision], LessEqual[y$46$re, 9.2e+45]]]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.9999999999999999e65 or -9e15 < y.re < -1e-99 or 9.20000000000000049e45 < y.re

   1. Initial program 49.4%

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

   3. Taylor expanded in y.re around inf 77.5%

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

      1. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. unpow2 77.2%

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

      3. times-frac 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. associate-*l/ 77.5%

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

      2. *-lft-identity 77.5%

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

   9. Simplified 77.5%

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

   if -7.9999999999999999e65 < y.re < -9e15 or -1e-99 < y.re < 9.20000000000000049e45

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0 81.2%

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

      1. associate-/l* 80.7%

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

   5. Simplified 80.7%

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

      1. pow2 80.7%

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

      2. associate-*r/ 81.2%

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

      3. associate-/r* 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+65} \lor \neg \left(y.re \leq -9 \cdot 10^{+15} \lor \neg \left(y.re \leq -1 \cdot 10^{-99}\right) \land y.re \leq 9.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+65} \lor \neg \left(y.re \leq -4500000000000\right) \land \left(y.re \leq -1 \cdot 10^{-99} \lor \neg \left(y.re \leq 3.65 \cdot 10^{+46}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.3e+65)
         (and (not (<= y.re -4500000000000.0))
              (or (<= y.re -1e-99) (not (<= y.re 3.65e+46)))))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (+ (/ x.im y.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 <= -1.3e+65) || (!(y_46_re <= -4500000000000.0) && ((y_46_re <= -1e-99) || !(y_46_re <= 3.65e+46)))) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_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 <= (-1.3d+65)) .or. (.not. (y_46re <= (-4500000000000.0d0))) .and. (y_46re <= (-1d-99)) .or. (.not. (y_46re <= 3.65d+46))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + (((x_46re * y_46re) / y_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 <= -1.3e+65) || (!(y_46_re <= -4500000000000.0) && ((y_46_re <= -1e-99) || !(y_46_re <= 3.65e+46)))) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_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):
	tmp = 0
	if (y_46_re <= -1.3e+65) or (not (y_46_re <= -4500000000000.0) and ((y_46_re <= -1e-99) or not (y_46_re <= 3.65e+46))):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_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 ((y_46_re <= -1.3e+65) || (!(y_46_re <= -4500000000000.0) && ((y_46_re <= -1e-99) || !(y_46_re <= 3.65e+46))))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * 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)
	tmp = 0.0;
	if ((y_46_re <= -1.3e+65) || (~((y_46_re <= -4500000000000.0)) && ((y_46_re <= -1e-99) || ~((y_46_re <= 3.65e+46)))))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_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_] := If[Or[LessEqual[y$46$re, -1.3e+65], And[N[Not[LessEqual[y$46$re, -4500000000000.0]], $MachinePrecision], Or[LessEqual[y$46$re, -1e-99], N[Not[LessEqual[y$46$re, 3.65e+46]], $MachinePrecision]]]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.30000000000000001e65 or -4.5e12 < y.re < -1e-99 or 3.65000000000000014e46 < y.re

   1. Initial program 49.4%

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

   3. Taylor expanded in y.re around inf 77.5%

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

      1. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

      1. associate-*r/ 77.5%

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

      2. unpow2 77.5%

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

      3. associate-/r* 79.2%

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

      4. *-commutative 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. associate-/l/ 77.5%

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

      2. times-frac 82.4%

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

   9. Applied egg-rr 82.4%

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

   if -1.30000000000000001e65 < y.re < -4.5e12 or -1e-99 < y.re < 3.65000000000000014e46

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0 81.2%

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

      1. associate-/l* 80.7%

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

   5. Simplified 80.7%

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

      1. pow2 80.7%

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

      2. associate-*r/ 81.2%

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

      3. associate-/r* 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+65} \lor \neg \left(y.re \leq -4500000000000\right) \land \left(y.re \leq -1 \cdot 10^{-99} \lor \neg \left(y.re \leq 3.65 \cdot 10^{+46}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))
   (if (<= y.re -9.2e+134)
     t_0
     (if (<= y.re -7.2e-127)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 3e+44)
         (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -9.2e+134) {
		tmp = t_0;
	} else if (y_46_re <= -7.2e-127) {
		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));
	} else if (y_46_re <= 3e+44) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    if (y_46re <= (-9.2d+134)) then
        tmp = t_0
    else if (y_46re <= (-7.2d-127)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 3d+44) then
        tmp = (x_46im / y_46im) + (((x_46re * y_46re) / y_46im) / y_46im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -9.2e+134) {
		tmp = t_0;
	} else if (y_46_re <= -7.2e-127) {
		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));
	} else if (y_46_re <= 3e+44) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -9.2e+134:
		tmp = t_0
	elif y_46_re <= -7.2e-127:
		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))
	elif y_46_re <= 3e+44:
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -9.2e+134)
		tmp = t_0;
	elseif (y_46_re <= -7.2e-127)
		tmp = 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)));
	elseif (y_46_re <= 3e+44)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -9.2e+134)
		tmp = t_0;
	elseif (y_46_re <= -7.2e-127)
		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));
	elseif (y_46_re <= 3e+44)
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.2e+134], t$95$0, If[LessEqual[y$46$re, -7.2e-127], 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, 3e+44], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.1999999999999992e134 or 2.99999999999999987e44 < y.re

   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.re around inf 77.1%

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

      1. associate-/l* 77.5%

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

   5. Simplified 77.5%

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

      1. associate-*r/ 77.1%

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

      2. unpow2 77.1%

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

      3. associate-/r* 79.3%

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

      4. *-commutative 79.3%

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

   7. Applied egg-rr 79.3%

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

      1. associate-/l/ 77.1%

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

      2. times-frac 83.4%

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

   9. Applied egg-rr 83.4%

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

   if -9.1999999999999992e134 < y.re < -7.1999999999999999e-127

   1. Initial program 79.0%

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

   if -7.1999999999999999e-127 < y.re < 2.99999999999999987e44

   1. Initial program 69.4%

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

   3. Taylor expanded in y.re around 0 83.4%

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

      1. associate-/l* 82.8%

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

   5. Simplified 82.8%

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

      1. pow2 82.8%

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

      2. associate-*r/ 83.4%

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

      3. associate-/r* 89.4%

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

   7. Applied egg-rr 89.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+64} \lor \neg \left(y.re \leq -3.6 \cdot 10^{+15} \lor \neg \left(y.re \leq -8 \cdot 10^{-126}\right) \land y.re \leq 7.4 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.8e+64)
         (not
          (or (<= y.re -3.6e+15)
              (and (not (<= y.re -8e-126)) (<= y.re 7.4e+48)))))
   (/ 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 <= -3.8e+64) || !((y_46_re <= -3.6e+15) || (!(y_46_re <= -8e-126) && (y_46_re <= 7.4e+48)))) {
		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 <= (-3.8d+64)) .or. (.not. (y_46re <= (-3.6d+15)) .or. (.not. (y_46re <= (-8d-126))) .and. (y_46re <= 7.4d+48))) 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 <= -3.8e+64) || !((y_46_re <= -3.6e+15) || (!(y_46_re <= -8e-126) && (y_46_re <= 7.4e+48)))) {
		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 <= -3.8e+64) or not ((y_46_re <= -3.6e+15) or (not (y_46_re <= -8e-126) and (y_46_re <= 7.4e+48))):
		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 <= -3.8e+64) || !((y_46_re <= -3.6e+15) || (!(y_46_re <= -8e-126) && (y_46_re <= 7.4e+48))))
		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 <= -3.8e+64) || ~(((y_46_re <= -3.6e+15) || (~((y_46_re <= -8e-126)) && (y_46_re <= 7.4e+48)))))
		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, -3.8e+64], N[Not[Or[LessEqual[y$46$re, -3.6e+15], And[N[Not[LessEqual[y$46$re, -8e-126]], $MachinePrecision], LessEqual[y$46$re, 7.4e+48]]]], $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 < -3.8000000000000001e64 or -3.6e15 < y.re < -7.9999999999999996e-126 or 7.3999999999999998e48 < y.re

   1. Initial program 50.8%

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

   3. Taylor expanded in y.re around inf 68.5%

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

   if -3.8000000000000001e64 < y.re < -3.6e15 or -7.9999999999999996e-126 < y.re < 7.3999999999999998e48

   1. Initial program 67.7%

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

   3. Taylor expanded in y.re around 0 71.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+64} \lor \neg \left(y.re \leq -3.6 \cdot 10^{+15} \lor \neg \left(y.re \leq -8 \cdot 10^{-126}\right) \land y.re \leq 7.4 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 7.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 7.2e+198) (/ x.im y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 7.2e+198) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= 7.2d+198) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 7.2e+198) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 7.2e+198:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 7.2e+198)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 7.2e+198)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 7.2e+198], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 7.2000000000000004e198

   1. Initial program 59.1%

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

   3. Taylor expanded in y.re around 0 45.6%

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

   if 7.2000000000000004e198 < y.re

   1. Initial program 51.0%

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

      1. *-un-lft-identity 51.0%

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

      2. add-sqr-sqrt 51.0%

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

      3. times-frac 51.0%

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

      4. hypot-define 51.0%

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

      5. fma-define 51.0%

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

      6. hypot-define 61.7%

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

   4. Applied egg-rr 61.7%

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

   5. Taylor expanded in y.im around -inf 11.6%

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

      1. distribute-lft-out 11.6%

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

      2. associate-/l* 11.5%

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

   7. Simplified 11.5%

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

   8. Taylor expanded in y.re around -inf 12.4%

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

   9. Taylor expanded in x.im around inf 29.6%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 7.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 58.4%

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

3. Taylor expanded in y.re around 0 42.6%

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

4. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2024053 
(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))))