_divideComplex, imaginary part

Percentage Accurate: 62.0% → 85.7%

Time: 13.2s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+271)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.im y.re (* x.re (- y.im))) (hypot y.re y.im)))
   (/ (- x.im (* y.im (/ x.re 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 ((((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+271) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / 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 (Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+271)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+271], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.99999999999999991e271

   1. Initial program 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. add-sqr-sqrt 81.2%

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

      3. times-frac 81.2%

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

      4. hypot-define 81.2%

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

      5. fma-neg 81.2%

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

      6. distribute-rgt-neg-in 81.2%

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

      7. hypot-define 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 1.99999999999999991e271 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 15.1%

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

   3. Taylor expanded in y.re around inf 53.7%

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

      1. +-commutative 53.7%

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

      2. mul-1-neg 53.7%

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

      3. unsub-neg 53.7%

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

   5. Simplified 53.7%

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

      1. *-un-lft-identity 53.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 53.7%

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

      3. times-frac 56.0%

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

      4. *-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

   8. Taylor expanded in x.im around 0 53.7%

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

      1. +-commutative 53.7%

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

      2. mul-1-neg 53.7%

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

      3. sub-neg 53.7%

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

      4. *-commutative 53.7%

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

      5. *-rgt-identity 53.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 53.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 53.7%

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

      8. times-frac 56.0%

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

      9. associate-/l* 56.0%

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

      10. *-commutative 56.0%

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

      11. associate-*r/ 56.0%

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

      12. *-rgt-identity 56.0%

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

      13. associate-*r/ 66.7%

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

      14. div-sub 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.7 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ x.re (- y.im)))))
   (if (<= y.im -1.05e-22)
     t_0
     (if (<= y.im 1.4e-124)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 6.7e+41)
         (*
          (fma x.im y.re (* x.re (- y.im)))
          (/ 1.0 (pow (hypot y.re y.im) 2.0)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re / -y_46_im));
	double tmp;
	if (y_46_im <= -1.05e-22) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-124) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 6.7e+41) {
		tmp = fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re / Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.05e-22)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-124)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 6.7e+41)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re / N[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] + N[(x$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e-22], t$95$0, If[LessEqual[y$46$im, 1.4e-124], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.7e+41], N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.05000000000000004e-22 or 6.6999999999999996e41 < y.im

   1. Initial program 50.5%

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

      1. div-sub 50.5%

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

      2. *-commutative 50.5%

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

      3. add-sqr-sqrt 50.5%

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

      4. times-frac 53.4%

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

      5. fma-neg 53.4%

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

      6. hypot-define 53.4%

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

      7. hypot-define 67.0%

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

      8. associate-/l* 73.4%

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

      9. add-sqr-sqrt 73.4%

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

      10. pow2 73.4%

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

      11. hypot-define 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in y.im around inf 89.7%

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

   if -1.05000000000000004e-22 < y.im < 1.39999999999999999e-124

   1. Initial program 77.1%

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

   3. Taylor expanded in y.re around inf 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

   5. Simplified 83.6%

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

      1. *-un-lft-identity 83.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 83.6%

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

      3. times-frac 85.8%

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

      4. *-commutative 85.8%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in x.im around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. sub-neg 83.6%

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

      4. *-commutative 83.6%

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

      5. *-rgt-identity 83.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 83.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 83.6%

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

      8. times-frac 85.8%

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

      9. associate-/l* 85.8%

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

      10. *-commutative 85.8%

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

      11. associate-*r/ 85.9%

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

      12. *-rgt-identity 85.9%

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

      13. associate-*r/ 82.7%

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

      14. div-sub 85.0%

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

   10. Simplified 85.0%

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

       1. associate-*r/ 89.1%

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

   12. Applied egg-rr 89.1%

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

   if 1.39999999999999999e-124 < y.im < 6.6999999999999996e41

   1. Initial program 85.6%

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

      1. div-inv 85.7%

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

      2. fma-neg 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. add-sqr-sqrt 85.7%

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

      5. pow2 85.7%

         \[\leadsto \mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}} \]

      6. hypot-define 85.7%

         \[\leadsto \mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}} \]

   4. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.7 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;x.im \cdot \frac{y.re}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -6.5e+53)
     (/ (- x.im (* y.im (/ x.re y.re))) y.re)
     (if (<= y.re -2.5e-188)
       t_0
       (if (<= y.re 3.5e-100)
         (- (* x.im (/ y.re (pow y.im 2.0))) (/ x.re y.im))
         (if (<= y.re 6.5e+139)
           t_0
           (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.5e+53) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -2.5e-188) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e-100) {
		tmp = (x_46_im * (y_46_re / pow(y_46_im, 2.0))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.5e+139) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-6.5d+53)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= (-2.5d-188)) then
        tmp = t_0
    else if (y_46re <= 3.5d-100) then
        tmp = (x_46im * (y_46re / (y_46im ** 2.0d0))) - (x_46re / y_46im)
    else if (y_46re <= 6.5d+139) then
        tmp = t_0
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.5e+53) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -2.5e-188) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e-100) {
		tmp = (x_46_im * (y_46_re / Math.pow(y_46_im, 2.0))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.5e+139) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -6.5e+53:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= -2.5e-188:
		tmp = t_0
	elif y_46_re <= 3.5e-100:
		tmp = (x_46_im * (y_46_re / math.pow(y_46_im, 2.0))) - (x_46_re / y_46_im)
	elif y_46_re <= 6.5e+139:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6.5e+53)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= -2.5e-188)
		tmp = t_0;
	elseif (y_46_re <= 3.5e-100)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / (y_46_im ^ 2.0))) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 6.5e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_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)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6.5e+53)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= -2.5e-188)
		tmp = t_0;
	elseif (y_46_re <= 3.5e-100)
		tmp = (x_46_im * (y_46_re / (y_46_im ^ 2.0))) - (x_46_re / y_46_im);
	elseif (y_46_re <= 6.5e+139)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e+53], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.5e-188], t$95$0, If[LessEqual[y$46$re, 3.5e-100], N[(N[(x$46$im * N[(y$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.5e+139], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.50000000000000017e53

   1. Initial program 46.7%

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

   3. Taylor expanded in y.re around inf 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   5. Simplified 79.8%

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

      1. *-un-lft-identity 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 79.8%

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

      3. times-frac 81.7%

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

      4. *-commutative 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in x.im around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. sub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. *-rgt-identity 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 79.8%

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

      8. times-frac 81.7%

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

      9. associate-/l* 81.6%

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

      10. *-commutative 81.6%

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

      11. associate-*r/ 81.7%

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

      12. *-rgt-identity 81.7%

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

      13. associate-*r/ 83.6%

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

      14. div-sub 83.6%

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

   10. Simplified 83.6%

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

   if -6.50000000000000017e53 < y.re < -2.5e-188 or 3.5000000000000001e-100 < y.re < 6.5000000000000003e139

   1. Initial program 84.4%

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

   if -2.5e-188 < y.re < 3.5000000000000001e-100

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. associate-/l* 92.3%

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

   5. Simplified 92.3%

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

   if 6.5000000000000003e139 < y.re

   1. Initial program 30.0%

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

   3. Taylor expanded in y.re around inf 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

   5. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 89.2%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-100}:\\ \;\;\;\;x.im \cdot \frac{y.re}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.65e+54)
     (/ (- x.im (* y.im (/ x.re y.re))) y.re)
     (if (<= y.re 4.3e-234)
       t_0
       (if (<= y.re 1.25e-204)
         (/ x.re (- y.im))
         (if (<= y.re 6.5e+139)
           t_0
           (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.65e+54) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4.3e-234) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-204) {
		tmp = x_46_re / -y_46_im;
	} else if (y_46_re <= 6.5e+139) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1.65d+54)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 4.3d-234) then
        tmp = t_0
    else if (y_46re <= 1.25d-204) then
        tmp = x_46re / -y_46im
    else if (y_46re <= 6.5d+139) then
        tmp = t_0
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.65e+54) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4.3e-234) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-204) {
		tmp = x_46_re / -y_46_im;
	} else if (y_46_re <= 6.5e+139) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.65e+54:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 4.3e-234:
		tmp = t_0
	elif y_46_re <= 1.25e-204:
		tmp = x_46_re / -y_46_im
	elif y_46_re <= 6.5e+139:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.65e+54)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 4.3e-234)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-204)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	elseif (y_46_re <= 6.5e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_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)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.65e+54)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 4.3e-234)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-204)
		tmp = x_46_re / -y_46_im;
	elseif (y_46_re <= 6.5e+139)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.65e+54], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.3e-234], t$95$0, If[LessEqual[y$46$re, 1.25e-204], N[(x$46$re / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 6.5e+139], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.65e54

   1. Initial program 46.7%

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

   3. Taylor expanded in y.re around inf 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

   5. Simplified 79.8%

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

      1. *-un-lft-identity 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 79.8%

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

      3. times-frac 81.7%

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

      4. *-commutative 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in x.im around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. sub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. *-rgt-identity 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 79.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 79.8%

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

      8. times-frac 81.7%

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

      9. associate-/l* 81.6%

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

      10. *-commutative 81.6%

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

      11. associate-*r/ 81.7%

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

      12. *-rgt-identity 81.7%

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

      13. associate-*r/ 83.6%

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

      14. div-sub 83.6%

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

   10. Simplified 83.6%

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

   if -1.65e54 < y.re < 4.3000000000000001e-234 or 1.25e-204 < y.re < 6.5000000000000003e139

   1. Initial program 82.9%

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

   if 4.3000000000000001e-234 < y.re < 1.25e-204

   1. Initial program 46.4%

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

   3. Taylor expanded in y.re around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if 6.5000000000000003e139 < y.re

   1. Initial program 30.0%

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

   3. Taylor expanded in y.re around inf 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

   5. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 89.2%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.5e-109)
   (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
   (if (<= y.re 5.2e-66)
     (/ x.re (- y.im))
     (- (/ x.im y.re) (* (/ x.re y.re) (/ y.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 <= -6.5e-109) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 5.2e-66) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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 <= (-6.5d-109)) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else if (y_46re <= 5.2d-66) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_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 <= -6.5e-109) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 5.2e-66) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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 <= -6.5e-109:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	elif y_46_re <= 5.2e-66:
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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 <= -6.5e-109)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= 5.2e-66)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_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 <= -6.5e-109)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_re <= 5.2e-66)
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_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, -6.5e-109], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.2e-66], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.49999999999999959e-109

   1. Initial program 62.1%

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

   3. Taylor expanded in y.re around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

   5. Simplified 71.7%

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

      1. *-un-lft-identity 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 71.7%

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

      3. times-frac 72.8%

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

      4. *-commutative 72.8%

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

   7. Applied egg-rr 72.8%

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

   8. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. sub-neg 71.7%

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

      4. *-commutative 71.7%

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

      5. *-rgt-identity 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 71.7%

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

      8. times-frac 72.8%

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

      9. associate-/l* 72.7%

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

      10. *-commutative 72.7%

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

      11. associate-*r/ 72.8%

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

      12. *-rgt-identity 72.8%

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

      13. associate-*r/ 70.5%

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

      14. div-sub 70.5%

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

   10. Simplified 70.5%

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

       1. associate-*r/ 72.8%

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

   12. Applied egg-rr 72.8%

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

   if -6.49999999999999959e-109 < y.re < 5.1999999999999998e-66

   1. Initial program 79.8%

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

   3. Taylor expanded in y.re around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   5. Simplified 74.3%

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

   if 5.1999999999999998e-66 < y.re

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around inf 69.6%

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

      1. +-commutative 69.6%

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

      2. mul-1-neg 69.6%

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

      3. unsub-neg 69.6%

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

   5. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. pow2 69.6%

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

      3. times-frac 75.4%

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

   7. Applied egg-rr 75.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.3 \cdot 10^{-103} \lor \neg \left(y.re \leq 4.5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8.3e-103) (not (<= y.re 4.5e-61)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ x.re (- 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 <= -8.3e-103) || !(y_46_re <= 4.5e-61)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-8.3d-103)) .or. (.not. (y_46re <= 4.5d-61))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = x_46re / -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 <= -8.3e-103) || !(y_46_re <= 4.5e-61)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -8.3e-103) or not (y_46_re <= 4.5e-61):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = x_46_re / -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 <= -8.3e-103) || !(y_46_re <= 4.5e-61))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -8.3e-103) || ~((y_46_re <= 4.5e-61)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = x_46_re / -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, -8.3e-103], N[Not[LessEqual[y$46$re, 4.5e-61]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.30000000000000006e-103 or 4.5e-61 < y.re

   1. Initial program 59.1%

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

   3. Taylor expanded in y.re around inf 70.8%

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

      1. +-commutative 70.8%

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

      2. mul-1-neg 70.8%

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

      3. unsub-neg 70.8%

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

   5. Simplified 70.8%

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

      1. *-un-lft-identity 70.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 70.8%

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

      3. times-frac 70.8%

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

      4. *-commutative 70.8%

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

   7. Applied egg-rr 70.8%

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

   8. Taylor expanded in x.im around 0 70.8%

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

      1. +-commutative 70.8%

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

      2. mul-1-neg 70.8%

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

      3. sub-neg 70.8%

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

      4. *-commutative 70.8%

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

      5. *-rgt-identity 70.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 70.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 70.8%

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

      8. times-frac 70.8%

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

      9. associate-/l* 70.8%

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

      10. *-commutative 70.8%

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

      11. associate-*r/ 70.9%

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

      12. *-rgt-identity 70.9%

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

      13. associate-*r/ 72.7%

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

      14. div-sub 72.7%

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

   10. Simplified 72.7%

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

   if -8.30000000000000006e-103 < y.re < 4.5e-61

   1. Initial program 79.8%

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

   3. Taylor expanded in y.re around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   5. Simplified 74.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.3 \cdot 10^{-103} \lor \neg \left(y.re \leq 4.5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-103)
   (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
   (if (<= y.re 2.2e-60)
     (/ x.re (- y.im))
     (/ (- x.im (* y.im (/ x.re 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_re <= -2.4e-103) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 2.2e-60) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / 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_46re <= (-2.4d-103)) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else if (y_46re <= 2.2d-60) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (y_46im * (x_46re / 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_re <= -2.4e-103) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 2.2e-60) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / 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_re <= -2.4e-103:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	elif y_46_re <= 2.2e-60:
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / 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_re <= -2.4e-103)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= 2.2e-60)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / 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_re <= -2.4e-103)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_re <= 2.2e-60)
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / 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[LessEqual[y$46$re, -2.4e-103], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.2e-60], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.4000000000000002e-103

   1. Initial program 62.1%

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

   3. Taylor expanded in y.re around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

   5. Simplified 71.7%

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

      1. *-un-lft-identity 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 71.7%

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

      3. times-frac 72.8%

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

      4. *-commutative 72.8%

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

   7. Applied egg-rr 72.8%

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

   8. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. sub-neg 71.7%

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

      4. *-commutative 71.7%

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

      5. *-rgt-identity 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 71.7%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 71.7%

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

      8. times-frac 72.8%

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

      9. associate-/l* 72.7%

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

      10. *-commutative 72.7%

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

      11. associate-*r/ 72.8%

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

      12. *-rgt-identity 72.8%

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

      13. associate-*r/ 70.5%

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

      14. div-sub 70.5%

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

   10. Simplified 70.5%

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

       1. associate-*r/ 72.8%

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

   12. Applied egg-rr 72.8%

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

   if -2.4000000000000002e-103 < y.re < 2.1999999999999999e-60

   1. Initial program 79.8%

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

   3. Taylor expanded in y.re around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   5. Simplified 74.3%

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

   if 2.1999999999999999e-60 < y.re

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around inf 69.6%

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

      1. +-commutative 69.6%

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

      2. mul-1-neg 69.6%

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

      3. unsub-neg 69.6%

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

   5. Simplified 69.6%

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

      1. *-un-lft-identity 69.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(x.re \cdot y.im\right)}}{{y.re}^{2}} \]

      2. pow2 69.6%

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

      3. times-frac 68.4%

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

      4. *-commutative 68.4%

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

   7. Applied egg-rr 68.4%

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

   8. Taylor expanded in x.im around 0 69.6%

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

      1. +-commutative 69.6%

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

      2. mul-1-neg 69.6%

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

      3. sub-neg 69.6%

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

      4. *-commutative 69.6%

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

      5. *-rgt-identity 69.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{\left(y.im \cdot x.re\right) \cdot 1}}{{y.re}^{2}} \]

      6. *-commutative 69.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{\color{blue}{1 \cdot \left(y.im \cdot x.re\right)}}{{y.re}^{2}} \]

      7. unpow2 69.6%

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

      8. times-frac 68.4%

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

      9. associate-/l* 68.4%

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

      10. *-commutative 68.4%

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

      11. associate-*r/ 68.4%

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

      12. *-rgt-identity 68.4%

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

      13. associate-*r/ 75.4%

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

      14. div-sub 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{-82} \lor \neg \left(y.re \leq 0.0035\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3e-82) (not (<= y.re 0.0035)))
   (/ x.im y.re)
   (/ x.re (- 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 <= -3e-82) || !(y_46_re <= 0.0035)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-3d-82)) .or. (.not. (y_46re <= 0.0035d0))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -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 <= -3e-82) || !(y_46_re <= 0.0035)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -3e-82) or not (y_46_re <= 0.0035):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -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 <= -3e-82) || !(y_46_re <= 0.0035))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -3e-82) || ~((y_46_re <= 0.0035)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -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, -3e-82], N[Not[LessEqual[y$46$re, 0.0035]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.9999999999999999e-82 or 0.00350000000000000007 < y.re

   1. Initial program 55.2%

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

   3. Taylor expanded in y.re around inf 65.6%

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

   if -2.9999999999999999e-82 < y.re < 0.00350000000000000007

   1. Initial program 81.7%

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

   3. Taylor expanded in y.re around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   5. Simplified 69.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{-82} \lor \neg \left(y.re \leq 0.0035\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+162} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4e+162) (not (<= y.im 1.2e+167)))
   (/ x.re 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_im <= -4e+162) || !(y_46_im <= 1.2e+167)) {
		tmp = x_46_re / 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_46im <= (-4d+162)) .or. (.not. (y_46im <= 1.2d+167))) then
        tmp = x_46re / 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_im <= -4e+162) || !(y_46_im <= 1.2e+167)) {
		tmp = x_46_re / 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_im <= -4e+162) or not (y_46_im <= 1.2e+167):
		tmp = x_46_re / 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_im <= -4e+162) || !(y_46_im <= 1.2e+167))
		tmp = Float64(x_46_re / 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_im <= -4e+162) || ~((y_46_im <= 1.2e+167)))
		tmp = x_46_re / 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[Or[LessEqual[y$46$im, -4e+162], N[Not[LessEqual[y$46$im, 1.2e+167]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.9999999999999998e162 or 1.19999999999999999e167 < y.im

   1. Initial program 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. add-sqr-sqrt 37.4%

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

      3. times-frac 37.4%

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

      4. hypot-define 37.4%

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

      5. fma-neg 37.4%

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

      6. distribute-rgt-neg-in 37.4%

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

      7. hypot-define 68.4%

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

   4. Applied egg-rr 68.4%

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

   5. Taylor expanded in y.re around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. *-commutative 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in y.im around -inf 36.4%

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

   if -3.9999999999999998e162 < y.im < 1.19999999999999999e167

   1. Initial program 76.5%

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

   3. Taylor expanded in y.re around inf 52.4%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+162} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.6% 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 67.0%

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

   1. *-un-lft-identity 67.0%

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

   2. add-sqr-sqrt 67.0%

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

   3. times-frac 67.0%

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

   4. hypot-define 67.0%

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

   5. fma-neg 67.0%

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

   6. distribute-rgt-neg-in 67.0%

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

   7. hypot-define 80.0%

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

4. Applied egg-rr 80.0%

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

5. Taylor expanded in y.re around -inf 31.6%

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

   1. neg-mul-1 31.6%

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

7. Simplified 31.6%

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

8. Taylor expanded in y.im around -inf 12.0%

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

9. Final simplification 12.0%

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

Alternative 11: 41.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in y.re around inf 44.2%

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

4. Final simplification 44.2%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))