_divideComplex, imaginary part

Percentage Accurate: 62.0% → 88.1%

Time: 15.9s

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: 88.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+157} \lor \neg \left(y.im \leq 1.52 \cdot 10^{+88}\right):\\ \;\;\;\;t\_0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re (/ x.im (hypot y.re y.im))) (hypot y.re y.im))))
   (if (or (<= y.im -2.55e+157) (not (<= y.im 1.52e+88)))
     (- t_0 (/ x.re y.im))
     (- t_0 (* y.im (/ x.re (pow (hypot y.re y.im) 2.0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	double tmp;
	if ((y_46_im <= -2.55e+157) || !(y_46_im <= 1.52e+88)) {
		tmp = t_0 - (x_46_re / y_46_im);
	} else {
		tmp = t_0 - (y_46_im * (x_46_re / pow(hypot(y_46_re, y_46_im), 2.0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) / Math.hypot(y_46_re, y_46_im);
	double tmp;
	if ((y_46_im <= -2.55e+157) || !(y_46_im <= 1.52e+88)) {
		tmp = t_0 - (x_46_re / y_46_im);
	} else {
		tmp = t_0 - (y_46_im * (x_46_re / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) / math.hypot(y_46_re, y_46_im)
	tmp = 0
	if (y_46_im <= -2.55e+157) or not (y_46_im <= 1.52e+88):
		tmp = t_0 - (x_46_re / y_46_im)
	else:
		tmp = t_0 - (y_46_im * (x_46_re / math.pow(math.hypot(y_46_re, y_46_im), 2.0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if ((y_46_im <= -2.55e+157) || !(y_46_im <= 1.52e+88))
		tmp = Float64(t_0 - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(t_0 - Float64(y_46_im * Float64(x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	tmp = 0.0;
	if ((y_46_im <= -2.55e+157) || ~((y_46_im <= 1.52e+88)))
		tmp = t_0 - (x_46_re / y_46_im);
	else
		tmp = t_0 - (y_46_im * (x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -2.55e+157], N[Not[LessEqual[y$46$im, 1.52e+88]], $MachinePrecision]], N[(t$95$0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(y$46$im * N[(x$46$re / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.55e157 or 1.52000000000000004e88 < y.im

   1. Initial program 36.1%

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

      1. div-sub 36.1%

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

      2. sub-neg 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. add-sqr-sqrt 36.1%

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

      5. times-frac 36.1%

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

      6. fma-def 36.1%

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

      7. hypot-def 36.1%

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

      8. hypot-def 41.3%

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

      9. associate-/l* 45.1%

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

      10. add-sqr-sqrt 45.1%

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

      11. pow2 45.1%

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

      12. hypot-def 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. fma-neg 45.1%

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

      2. *-commutative 45.1%

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

      3. associate-/l* 53.5%

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

      4. associate-/r/ 52.1%

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

      5. *-commutative 52.1%

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

   6. Simplified 52.1%

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

      1. associate-*l/ 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. div-inv 52.1%

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

      4. clear-num 52.2%

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

   8. Applied egg-rr 52.2%

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

   9. Taylor expanded in y.im around inf 89.3%

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

   if -2.55e157 < y.im < 1.52000000000000004e88

   1. Initial program 69.9%

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

      1. div-sub 68.0%

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

      2. sub-neg 68.0%

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

      3. *-un-lft-identity 68.0%

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

      4. add-sqr-sqrt 68.0%

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

      5. times-frac 68.1%

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

      6. fma-def 68.1%

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

      7. hypot-def 68.1%

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

      8. hypot-def 73.6%

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

      9. associate-/l* 76.3%

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

      10. add-sqr-sqrt 76.3%

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

      11. pow2 76.3%

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

      12. hypot-def 76.3%

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

   4. Applied egg-rr 76.3%

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

      1. fma-neg 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-/l* 92.0%

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

      4. associate-/r/ 87.9%

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

      5. *-commutative 87.9%

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

   6. Simplified 87.9%

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

      1. associate-*l/ 88.0%

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

      2. *-un-lft-identity 88.0%

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

      3. div-inv 87.6%

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

      4. clear-num 88.1%

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+157} \lor \neg \left(y.im \leq 1.52 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 2e+236)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (-
      (/ (* y.re (/ x.im (hypot y.re y.im))) (hypot y.re y.im))
      (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) / Math.hypot(y_46_re, y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) / math.hypot(y_46_re, y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+236)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+236)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im)) - (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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+236], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $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))) < 2.00000000000000011e236

   1. Initial program 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. add-sqr-sqrt 80.3%

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

      3. times-frac 80.3%

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

      4. hypot-def 80.3%

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

      5. hypot-def 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l/ 93.9%

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

      3. div-inv 94.0%

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

      4. *-commutative 94.0%

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

      5. *-commutative 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 2.00000000000000011e236 < (/.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 10.5%

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

      1. div-sub 8.8%

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

      2. sub-neg 8.8%

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

      3. *-un-lft-identity 8.8%

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

      4. add-sqr-sqrt 8.8%

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

      5. times-frac 8.8%

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

      6. fma-def 8.8%

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

      7. hypot-def 8.8%

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

      8. hypot-def 11.6%

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

      9. associate-/l* 18.1%

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

      10. add-sqr-sqrt 18.1%

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

      11. pow2 18.1%

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

      12. hypot-def 18.1%

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

   4. Applied egg-rr 18.1%

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

      1. fma-neg 18.1%

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

      2. *-commutative 18.1%

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

      3. associate-/l* 60.6%

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

      4. associate-/r/ 60.6%

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

      5. *-commutative 60.6%

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

   6. Simplified 60.6%

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

      1. associate-*l/ 60.6%

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

      2. *-un-lft-identity 60.6%

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

      3. div-inv 60.6%

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

      4. clear-num 60.7%

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

   8. Applied egg-rr 60.7%

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

   9. Taylor expanded in y.im around inf 67.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2} \cdot \left(y.re \cdot x.im - y.im \cdot x.re\right)\\ \mathbf{if}\;y.re \leq -7.2:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (hypot y.re y.im) -2.0) (- (* y.re x.im) (* y.im x.re)))))
   (if (<= y.re -7.2)
     (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
     (if (<= y.re -8.4e-155)
       t_0
       (if (<= y.re 1.8e-148)
         (* (/ -1.0 y.im) (- x.re (/ x.im (/ y.im y.re))))
         (if (<= y.re 5.8e+78)
           t_0
           (/ (- x.im (/ x.re (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(y_46_re, y_46_im), -2.0) * ((y_46_re * x_46_im) - (y_46_im * x_46_re));
	double tmp;
	if (y_46_re <= -7.2) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_re <= -8.4e-155) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-148) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 5.8e+78) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(y_46_re, y_46_im), -2.0) * ((y_46_re * x_46_im) - (y_46_im * x_46_re));
	double tmp;
	if (y_46_re <= -7.2) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_re <= -8.4e-155) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-148) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 5.8e+78) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(y_46_re, y_46_im), -2.0) * ((y_46_re * x_46_im) - (y_46_im * x_46_re))
	tmp = 0
	if y_46_re <= -7.2:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	elif y_46_re <= -8.4e-155:
		tmp = t_0
	elif y_46_re <= 1.8e-148:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)))
	elif y_46_re <= 5.8e+78:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((hypot(y_46_re, y_46_im) ^ -2.0) * Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)))
	tmp = 0.0
	if (y_46_re <= -7.2)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	elseif (y_46_re <= -8.4e-155)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-148)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 5.8e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (hypot(y_46_re, y_46_im) ^ -2.0) * ((y_46_re * x_46_im) - (y_46_im * x_46_re));
	tmp = 0.0;
	if (y_46_re <= -7.2)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	elseif (y_46_re <= -8.4e-155)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-148)
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	elseif (y_46_re <= 5.8e+78)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.4e-155], t$95$0, If[LessEqual[y$46$re, 1.8e-148], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+78], t$95$0, N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.20000000000000018

   1. Initial program 48.8%

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

   3. Taylor expanded in y.re around inf 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. associate-/l* 72.9%

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

      5. associate-/r/ 75.6%

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

   5. Simplified 75.6%

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

      1. *-un-lft-identity 75.6%

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

      2. unpow2 75.6%

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

      3. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-*l/ 82.9%

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

      2. *-lft-identity 82.9%

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

   9. Simplified 82.9%

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

   if -7.20000000000000018 < y.re < -8.4000000000000007e-155 or 1.7999999999999999e-148 < y.re < 5.80000000000000034e78

   1. Initial program 83.7%

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

      1. add-sqr-sqrt 52.1%

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

      2. pow2 52.1%

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

      3. sqrt-div 45.1%

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

      4. hypot-def 45.2%

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

   4. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{x.im \cdot y.re - x.re \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 43.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\sqrt{x.im \cdot y.re - x.re \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{2}\right)\right)} \]

      2. expm1-udef 34.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\sqrt{x.im \cdot y.re - x.re \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{2}\right)} - 1} \]

   6. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(y.re \cdot x.im - y.im \cdot x.re\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 59.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(y.re \cdot x.im - y.im \cdot x.re\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 84.7%

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

      3. *-commutative 84.7%

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

      4. *-commutative 84.7%

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

   8. Simplified 84.7%

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

   if -8.4000000000000007e-155 < y.re < 1.7999999999999999e-148

   1. Initial program 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. add-sqr-sqrt 64.4%

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

      3. times-frac 64.4%

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

      4. hypot-def 64.4%

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

      5. hypot-def 81.9%

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

   4. Applied egg-rr 81.9%

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

   5. Taylor expanded in y.im around -inf 52.4%

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

      1. mul-1-neg 52.4%

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

      2. unsub-neg 52.4%

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

      3. associate-/l* 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in y.im around -inf 91.4%

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

   if 5.80000000000000034e78 < y.re

   1. Initial program 34.1%

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

      1. *-un-lft-identity 34.1%

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

      2. add-sqr-sqrt 34.1%

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

      3. times-frac 34.1%

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

      4. hypot-def 34.1%

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

      5. hypot-def 49.3%

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

   4. Applied egg-rr 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-*l/ 49.4%

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

      3. div-inv 49.4%

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

      4. *-commutative 49.4%

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

      5. *-commutative 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in y.re around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. associate-/l* 89.0%

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

   9. Simplified 89.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.4 \cdot 10^{-155}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2} \cdot \left(y.re \cdot x.im - y.im \cdot x.re\right)\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2} \cdot \left(y.re \cdot x.im - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -9.8 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -8.4e-10)
     (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
     (if (<= y.re -9.8e-160)
       t_0
       (if (<= y.re 1.45e-122)
         (* (/ -1.0 y.im) (- x.re (/ x.im (/ y.im y.re))))
         (if (<= y.re 2.7e+77)
           t_0
           (/ (- x.im (/ x.re (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -8.4e-10) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_re <= -9.8e-160) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-122) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 2.7e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -8.4e-10) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_re <= -9.8e-160) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-122) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 2.7e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -8.4e-10:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	elif y_46_re <= -9.8e-160:
		tmp = t_0
	elif y_46_re <= 1.45e-122:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)))
	elif y_46_re <= 2.7e+77:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -8.4e-10)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	elseif (y_46_re <= -9.8e-160)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-122)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 2.7e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -8.4e-10)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	elseif (y_46_re <= -9.8e-160)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-122)
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	elseif (y_46_re <= 2.7e+77)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $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, -8.4e-10], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9.8e-160], t$95$0, If[LessEqual[y$46$re, 1.45e-122], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+77], t$95$0, N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.3999999999999999e-10

   1. Initial program 49.6%

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

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

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. associate-/l* 73.3%

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

      5. associate-/r/ 76.0%

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

   5. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 83.1%

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

   7. Applied egg-rr 83.1%

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

      1. associate-*l/ 83.1%

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

      2. *-lft-identity 83.1%

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

   9. Simplified 83.1%

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

   if -8.3999999999999999e-10 < y.re < -9.7999999999999998e-160 or 1.4500000000000001e-122 < y.re < 2.6999999999999998e77

   1. Initial program 84.2%

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

   if -9.7999999999999998e-160 < y.re < 1.4500000000000001e-122

   1. Initial program 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. add-sqr-sqrt 64.6%

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

      3. times-frac 64.5%

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

      4. hypot-def 64.5%

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

      5. hypot-def 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in y.im around -inf 53.1%

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

      1. mul-1-neg 53.1%

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

      2. unsub-neg 53.1%

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

      3. associate-/l* 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in y.im around -inf 90.3%

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

   if 2.6999999999999998e77 < y.re

   1. Initial program 34.1%

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

      1. *-un-lft-identity 34.1%

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

      2. add-sqr-sqrt 34.1%

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

      3. times-frac 34.1%

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

      4. hypot-def 34.1%

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

      5. hypot-def 49.3%

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

   4. Applied egg-rr 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-*l/ 49.4%

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

      3. div-inv 49.4%

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

      4. *-commutative 49.4%

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

      5. *-commutative 49.4%

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in y.re around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. associate-/l* 89.0%

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

   9. Simplified 89.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -9.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))
   (if (<= y.re -8.4e-10)
     t_1
     (if (<= y.re -3.5e-156)
       t_0
       (if (<= y.re 5e-123)
         (* (/ -1.0 y.im) (- x.re (/ x.im (/ y.im y.re))))
         (if (<= y.re 1.25e+79) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	double tmp;
	if (y_46_re <= -8.4e-10) {
		tmp = t_1;
	} else if (y_46_re <= -3.5e-156) {
		tmp = t_0;
	} else if (y_46_re <= 5e-123) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.25e+79) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    if (y_46re <= (-8.4d-10)) then
        tmp = t_1
    else if (y_46re <= (-3.5d-156)) then
        tmp = t_0
    else if (y_46re <= 5d-123) then
        tmp = ((-1.0d0) / y_46im) * (x_46re - (x_46im / (y_46im / y_46re)))
    else if (y_46re <= 1.25d+79) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	double tmp;
	if (y_46_re <= -8.4e-10) {
		tmp = t_1;
	} else if (y_46_re <= -3.5e-156) {
		tmp = t_0;
	} else if (y_46_re <= 5e-123) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.25e+79) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	tmp = 0
	if y_46_re <= -8.4e-10:
		tmp = t_1
	elif y_46_re <= -3.5e-156:
		tmp = t_0
	elif y_46_re <= 5e-123:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)))
	elif y_46_re <= 1.25e+79:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -8.4e-10)
		tmp = t_1;
	elseif (y_46_re <= -3.5e-156)
		tmp = t_0;
	elseif (y_46_re <= 5e-123)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 1.25e+79)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -8.4e-10)
		tmp = t_1;
	elseif (y_46_re <= -3.5e-156)
		tmp = t_0;
	elseif (y_46_re <= 5e-123)
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (y_46_im / y_46_re)));
	elseif (y_46_re <= 1.25e+79)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8.4e-10], t$95$1, If[LessEqual[y$46$re, -3.5e-156], t$95$0, If[LessEqual[y$46$re, 5e-123], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+79], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.3999999999999999e-10 or 1.25e79 < y.re

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around inf 76.2%

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

      1. +-commutative 76.2%

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

      2. mul-1-neg 76.2%

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

      3. unsub-neg 76.2%

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

      4. associate-/l* 77.4%

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

      5. associate-/r/ 79.0%

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

   5. Simplified 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.3%

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

      2. *-lft-identity 85.3%

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

   9. Simplified 85.3%

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

   if -8.3999999999999999e-10 < y.re < -3.4999999999999999e-156 or 5.0000000000000003e-123 < y.re < 1.25e79

   1. Initial program 84.2%

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

   if -3.4999999999999999e-156 < y.re < 5.0000000000000003e-123

   1. Initial program 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. add-sqr-sqrt 64.6%

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

      3. times-frac 64.5%

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

      4. hypot-def 64.5%

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

      5. hypot-def 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in y.im around -inf 53.1%

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

      1. mul-1-neg 53.1%

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

      2. unsub-neg 53.1%

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

      3. associate-/l* 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in y.im around -inf 90.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-10} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8e-10) (not (<= y.re 2.6e+80)))
   (/ x.im y.re)
   (* (/ -1.0 y.im) (- x.re (/ x.im (/ 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 <= -8e-10) || !(y_46_re <= 2.6e+80)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= (-8d-10)) .or. (.not. (y_46re <= 2.6d+80))) then
        tmp = x_46im / y_46re
    else
        tmp = ((-1.0d0) / y_46im) * (x_46re - (x_46im / (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 <= -8e-10) || !(y_46_re <= 2.6e+80)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= -8e-10) or not (y_46_re <= 2.6e+80):
		tmp = x_46_im / y_46_re
	else:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= -8e-10) || !(y_46_re <= 2.6e+80))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im / 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 <= -8e-10) || ~((y_46_re <= 2.6e+80)))
		tmp = x_46_im / y_46_re;
	else
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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[Or[LessEqual[y$46$re, -8e-10], N[Not[LessEqual[y$46$re, 2.6e+80]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.00000000000000029e-10 or 2.59999999999999982e80 < y.re

   1. Initial program 42.5%

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

   3. Taylor expanded in y.re around inf 73.2%

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

   if -8.00000000000000029e-10 < y.re < 2.59999999999999982e80

   1. Initial program 75.4%

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

      1. *-un-lft-identity 75.4%

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

      2. add-sqr-sqrt 75.4%

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

      3. times-frac 75.4%

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

      4. hypot-def 75.4%

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

      5. hypot-def 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in y.im around -inf 47.2%

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

      1. mul-1-neg 47.2%

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

      2. unsub-neg 47.2%

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

      3. associate-/l* 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in y.im around -inf 78.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-10} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.6e-29) (not (<= y.re 2.05e+77)))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
   (* (/ -1.0 y.im) (- x.re (/ x.im (/ 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 <= -3.6e-29) || !(y_46_re <= 2.05e+77)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= (-3.6d-29)) .or. (.not. (y_46re <= 2.05d+77))) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else
        tmp = ((-1.0d0) / y_46im) * (x_46re - (x_46im / (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 <= -3.6e-29) || !(y_46_re <= 2.05e+77)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= -3.6e-29) or not (y_46_re <= 2.05e+77):
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	else:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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 <= -3.6e-29) || !(y_46_re <= 2.05e+77))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im / 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 <= -3.6e-29) || ~((y_46_re <= 2.05e+77)))
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	else
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im / (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[Or[LessEqual[y$46$re, -3.6e-29], N[Not[LessEqual[y$46$re, 2.05e+77]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.59999999999999974e-29 or 2.05e77 < y.re

   1. Initial program 44.9%

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

   3. Taylor expanded in y.re around inf 76.1%

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

      1. +-commutative 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. associate-/l* 77.4%

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

      5. associate-/r/ 78.9%

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

   5. Simplified 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. unpow2 78.9%

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

      3. times-frac 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.0%

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

      2. *-lft-identity 85.0%

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

   9. Simplified 85.0%

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

   if -3.59999999999999974e-29 < y.re < 2.05e77

   1. Initial program 74.5%

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

      1. *-un-lft-identity 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 74.5%

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

      4. hypot-def 74.5%

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

      5. hypot-def 85.2%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in y.im around -inf 47.4%

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

      1. mul-1-neg 47.4%

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

      2. unsub-neg 47.4%

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

      3. associate-/l* 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in y.im around -inf 79.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-35} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.7e-35) (not (<= y.im 7.6e+68)))
   (/ (- 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 <= -1.7e-35) || !(y_46_im <= 7.6e+68)) {
		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 <= (-1.7d-35)) .or. (.not. (y_46im <= 7.6d+68))) 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 <= -1.7e-35) || !(y_46_im <= 7.6e+68)) {
		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 <= -1.7e-35) or not (y_46_im <= 7.6e+68):
		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 <= -1.7e-35) || !(y_46_im <= 7.6e+68))
		tmp = Float64(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 <= -1.7e-35) || ~((y_46_im <= 7.6e+68)))
		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, -1.7e-35], N[Not[LessEqual[y$46$im, 7.6e+68]], $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 < -1.7000000000000001e-35 or 7.6000000000000002e68 < y.im

   1. Initial program 49.7%

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

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

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

   5. Simplified 73.1%

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

   if -1.7000000000000001e-35 < y.im < 7.6000000000000002e68

   1. Initial program 69.4%

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

   3. Taylor expanded in y.re around inf 65.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-35} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 3.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 3.6e+105) (/ 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_im <= 3.6e+105) {
		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_46im <= 3.6d+105) 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_im <= 3.6e+105) {
		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_im <= 3.6e+105:
		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_im <= 3.6e+105)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 3.6e+105)
		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[LessEqual[y$46$im, 3.6e+105], 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.im < 3.5999999999999999e105

   1. Initial program 64.1%

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

   3. Taylor expanded in y.re around inf 50.7%

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

   if 3.5999999999999999e105 < y.im

   1. Initial program 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. add-sqr-sqrt 39.8%

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

      3. times-frac 39.8%

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

      4. hypot-def 39.8%

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

      5. hypot-def 59.9%

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in y.im around -inf 30.2%

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

      1. mul-1-neg 30.2%

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

      2. unsub-neg 30.2%

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

      3. associate-/l* 30.2%

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

   7. Simplified 30.2%

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

   8. Taylor expanded in y.re around 0 30.4%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 3.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]
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 60.4%

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

   1. *-un-lft-identity 60.4%

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

   2. add-sqr-sqrt 60.4%

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

   3. times-frac 60.4%

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

   4. hypot-def 60.4%

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

   5. hypot-def 72.1%

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

4. Applied egg-rr 72.1%

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

5. Taylor expanded in y.im around -inf 30.4%

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

   1. mul-1-neg 30.4%

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

   2. unsub-neg 30.4%

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

   3. associate-/l* 31.7%

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

7. Simplified 31.7%

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

8. Taylor expanded in y.re around -inf 10.5%

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

9. Final simplification 10.5%

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

Alternative 11: 43.1% 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 60.4%

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

3. Taylor expanded in y.re around inf 45.1%

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

4. Final simplification 45.1%

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

Reproduce

herbie shell --seed 2024031 
(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))))