_divideComplex, imaginary part

Percentage Accurate: 62.4% → 79.8%

Time: 14.3s

Alternatives: 12

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: 62.4% 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: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.re \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{t_1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* x.re (/ y.im y.re))))
   (if (<= y.re -2.6e+129)
     (- (/ x.im y.re) (/ t_1 y.re))
     (if (<= y.re -1.22e+51)
       (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
       (if (<= y.re -3.8e-79)
         t_0
         (if (<= y.re 4.4e-239)
           (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
           (if (<= y.re 2.5e+68) t_0 (/ (- x.im t_1) (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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.6e+129) {
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	} else if (y_46_re <= -1.22e+51) {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -3.8e-79) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e-239) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.5e+68) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.6e+129) {
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	} else if (y_46_re <= -1.22e+51) {
		tmp = (y_46_re / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -3.8e-79) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e-239) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.5e+68) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re * (y_46_im / y_46_re)
	tmp = 0
	if y_46_re <= -2.6e+129:
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re)
	elif y_46_re <= -1.22e+51:
		tmp = (y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -3.8e-79:
		tmp = t_0
	elif y_46_re <= 4.4e-239:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 2.5e+68:
		tmp = t_0
	else:
		tmp = (x_46_im - t_1) / 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(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.6e+129)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(t_1 / y_46_re));
	elseif (y_46_re <= -1.22e+51)
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -3.8e-79)
		tmp = t_0;
	elseif (y_46_re <= 4.4e-239)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.5e+68)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_re * (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.6e+129)
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	elseif (y_46_re <= -1.22e+51)
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -3.8e-79)
		tmp = t_0;
	elseif (y_46_re <= 4.4e-239)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 2.5e+68)
		tmp = t_0;
	else
		tmp = (x_46_im - t_1) / 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[(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]}, Block[{t$95$1 = N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+129], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.22e+51], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.8e-79], t$95$0, If[LessEqual[y$46$re, 4.4e-239], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+68], t$95$0, N[(N[(x$46$im - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.60000000000000012e129

   1. Initial program 35.3%

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

   3. Taylor expanded in y.re around inf 71.5%

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

      1. +-commutative 71.5%

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

      2. mul-1-neg 71.5%

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

      3. unsub-neg 71.5%

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

      4. associate-/l* 74.4%

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

      5. associate-/r/ 76.4%

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

   5. Simplified 76.4%

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

      1. associate-*l/ 71.5%

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

      2. unpow2 71.5%

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

      3. associate-/r* 83.9%

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

      4. associate-*l/ 92.9%

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

      5. associate-/r/ 92.8%

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

      6. div-inv 92.9%

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

      7. clear-num 92.8%

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

   7. Applied egg-rr 92.8%

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

   if -2.60000000000000012e129 < y.re < -1.22000000000000005e51

   1. Initial program 55.3%

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

   3. Taylor expanded in x.im around inf 55.6%

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

      1. associate-/l* 71.8%

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

      2. unpow2 71.8%

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

      3. fma-udef 71.8%

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

   5. Simplified 71.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-div 0.0%

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

      3. fma-udef 0.0%

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

      4. +-commutative 0.0%

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

      5. pow2 0.0%

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

      6. hypot-udef 0.0%

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

      7. sqrt-div 0.0%

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

      8. fma-udef 0.0%

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

      9. +-commutative 0.0%

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

      10. pow2 0.0%

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

      11. hypot-udef 0.0%

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

      12. times-frac 0.0%

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

      13. add-sqr-sqrt 71.8%

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

      14. *-un-lft-identity 71.8%

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

      15. times-frac 71.9%

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

      16. /-rgt-identity 71.9%

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

   7. Applied egg-rr 71.9%

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

      1. *-un-lft-identity 71.9%

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

      2. *-commutative 71.9%

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

      3. times-frac 89.2%

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

      4. clear-num 89.2%

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

   9. Applied egg-rr 89.2%

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

   if -1.22000000000000005e51 < y.re < -3.8000000000000001e-79 or 4.39999999999999965e-239 < y.re < 2.5000000000000002e68

   1. Initial program 83.0%

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

   if -3.8000000000000001e-79 < y.re < 4.39999999999999965e-239

   1. Initial program 78.0%

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

   3. Taylor expanded in x.im around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. +-commutative 78.0%

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

      3. *-commutative 78.0%

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

      4. fma-def 78.0%

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

      5. distribute-rgt-neg-in 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in y.re around 0 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 90.8%

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

   8. Simplified 90.8%

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

   if 2.5000000000000002e68 < y.re

   1. Initial program 30.1%

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

      1. fma-neg 30.1%

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

      2. *-commutative 30.1%

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

      3. distribute-rgt-neg-out 30.1%

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

      4. *-un-lft-identity 30.1%

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

      5. fma-def 30.1%

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

      6. add-sqr-sqrt 30.1%

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

      7. times-frac 30.1%

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

      8. fma-def 30.1%

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

      9. hypot-def 30.1%

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

      10. *-commutative 30.1%

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

      11. add-sqr-sqrt 15.6%

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

      12. sqrt-unprod 20.1%

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

      13. sqr-neg 20.1%

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

      14. sqrt-unprod 12.3%

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

      15. add-sqr-sqrt 27.8%

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

      16. fma-def 27.8%

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

      17. hypot-def 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in y.re around -inf 25.4%

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

      1. mul-1-neg 25.4%

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

      2. unsub-neg 25.4%

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

      3. neg-mul-1 25.4%

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

      4. associate-/l* 30.2%

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

   7. Simplified 30.2%

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

      1. associate-*l/ 30.2%

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

      2. *-un-lft-identity 30.2%

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

      3. add-sqr-sqrt 14.8%

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

      4. sqrt-unprod 46.4%

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

      5. sqr-neg 46.4%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 91.0%

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

      8. div-inv 91.0%

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

      9. clear-num 91.0%

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

   9. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-239}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.re \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -8.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{t_1}{y.re}\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-237}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* x.re (/ y.im y.re))))
   (if (<= y.re -8.8e+89)
     (- (/ x.im y.re) (/ t_1 y.re))
     (if (<= y.re -7e-79)
       t_0
       (if (<= y.re 1.45e-237)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
         (if (<= y.re 1.12e+74) t_0 (/ (- x.im t_1) (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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -8.8e+89) {
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	} else if (y_46_re <= -7e-79) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-237) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.12e+74) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -8.8e+89) {
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	} else if (y_46_re <= -7e-79) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-237) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.12e+74) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re * (y_46_im / y_46_re)
	tmp = 0
	if y_46_re <= -8.8e+89:
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re)
	elif y_46_re <= -7e-79:
		tmp = t_0
	elif y_46_re <= 1.45e-237:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 1.12e+74:
		tmp = t_0
	else:
		tmp = (x_46_im - t_1) / 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(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -8.8e+89)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(t_1 / y_46_re));
	elseif (y_46_re <= -7e-79)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-237)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.12e+74)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - t_1) / 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_re * (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -8.8e+89)
		tmp = (x_46_im / y_46_re) - (t_1 / y_46_re);
	elseif (y_46_re <= -7e-79)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-237)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 1.12e+74)
		tmp = t_0;
	else
		tmp = (x_46_im - t_1) / 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[(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]}, Block[{t$95$1 = N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8.8e+89], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7e-79], t$95$0, If[LessEqual[y$46$re, 1.45e-237], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.12e+74], t$95$0, N[(N[(x$46$im - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.8000000000000001e89

   1. Initial program 37.4%

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

   3. Taylor expanded in y.re around inf 69.0%

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

      1. +-commutative 69.0%

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

      2. mul-1-neg 69.0%

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

      3. unsub-neg 69.0%

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

      4. associate-/l* 71.5%

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

      5. associate-/r/ 73.2%

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

   5. Simplified 73.2%

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

      1. associate-*l/ 69.0%

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

      2. unpow2 69.0%

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

      3. associate-/r* 79.3%

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

      4. associate-*l/ 86.8%

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

      5. associate-/r/ 86.8%

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

      6. div-inv 86.8%

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

      7. clear-num 86.8%

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

   7. Applied egg-rr 86.8%

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

   if -8.8000000000000001e89 < y.re < -7.00000000000000059e-79 or 1.45000000000000005e-237 < y.re < 1.12000000000000003e74

   1. Initial program 81.2%

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

   if -7.00000000000000059e-79 < y.re < 1.45000000000000005e-237

   1. Initial program 78.0%

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

   3. Taylor expanded in x.im around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. +-commutative 78.0%

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

      3. *-commutative 78.0%

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

      4. fma-def 78.0%

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

      5. distribute-rgt-neg-in 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in y.re around 0 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 90.8%

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

   8. Simplified 90.8%

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

   if 1.12000000000000003e74 < y.re

   1. Initial program 30.1%

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

      1. fma-neg 30.1%

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

      2. *-commutative 30.1%

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

      3. distribute-rgt-neg-out 30.1%

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

      4. *-un-lft-identity 30.1%

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

      5. fma-def 30.1%

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

      6. add-sqr-sqrt 30.1%

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

      7. times-frac 30.1%

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

      8. fma-def 30.1%

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

      9. hypot-def 30.1%

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

      10. *-commutative 30.1%

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

      11. add-sqr-sqrt 15.6%

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

      12. sqrt-unprod 20.1%

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

      13. sqr-neg 20.1%

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

      14. sqrt-unprod 12.3%

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

      15. add-sqr-sqrt 27.8%

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

      16. fma-def 27.8%

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

      17. hypot-def 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in y.re around -inf 25.4%

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

      1. mul-1-neg 25.4%

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

      2. unsub-neg 25.4%

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

      3. neg-mul-1 25.4%

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

      4. associate-/l* 30.2%

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

   7. Simplified 30.2%

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

      1. associate-*l/ 30.2%

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

      2. *-un-lft-identity 30.2%

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

      3. add-sqr-sqrt 14.8%

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

      4. sqrt-unprod 46.4%

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

      5. sqr-neg 46.4%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 91.0%

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

      8. div-inv 91.0%

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

      9. clear-num 91.0%

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

   9. Applied egg-rr 91.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-79}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-237}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-306}:\\ \;\;\;\;y.re \cdot \frac{x.im}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ (* x.re (/ y.im y.re)) y.re))))
   (if (<= y.re -3.1e+88)
     t_1
     (if (<= y.re -1.1e-78)
       t_0
       (if (<= y.re 1.95e-306)
         (- (* y.re (/ x.im (pow y.im 2.0))) (/ x.re y.im))
         (if (<= y.re 6.1e+74) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -3.1e+88) {
		tmp = t_1;
	} else if (y_46_re <= -1.1e-78) {
		tmp = t_0;
	} else if (y_46_re <= 1.95e-306) {
		tmp = (y_46_re * (x_46_im / pow(y_46_im, 2.0))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.1e+74) {
		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) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((x_46re * (y_46im / y_46re)) / y_46re)
    if (y_46re <= (-3.1d+88)) then
        tmp = t_1
    else if (y_46re <= (-1.1d-78)) then
        tmp = t_0
    else if (y_46re <= 1.95d-306) then
        tmp = (y_46re * (x_46im / (y_46im ** 2.0d0))) - (x_46re / y_46im)
    else if (y_46re <= 6.1d+74) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -3.1e+88) {
		tmp = t_1;
	} else if (y_46_re <= -1.1e-78) {
		tmp = t_0;
	} else if (y_46_re <= 1.95e-306) {
		tmp = (y_46_re * (x_46_im / Math.pow(y_46_im, 2.0))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.1e+74) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -3.1e+88:
		tmp = t_1
	elif y_46_re <= -1.1e-78:
		tmp = t_0
	elif y_46_re <= 1.95e-306:
		tmp = (y_46_re * (x_46_im / math.pow(y_46_im, 2.0))) - (x_46_re / y_46_im)
	elif y_46_re <= 6.1e+74:
		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(x_46_re * y_46_im)) / 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(Float64(x_46_re * Float64(y_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.1e+88)
		tmp = t_1;
	elseif (y_46_re <= -1.1e-78)
		tmp = t_0;
	elseif (y_46_re <= 1.95e-306)
		tmp = Float64(Float64(y_46_re * Float64(x_46_im / (y_46_im ^ 2.0))) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 6.1e+74)
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.1e+88)
		tmp = t_1;
	elseif (y_46_re <= -1.1e-78)
		tmp = t_0;
	elseif (y_46_re <= 1.95e-306)
		tmp = (y_46_re * (x_46_im / (y_46_im ^ 2.0))) - (x_46_re / y_46_im);
	elseif (y_46_re <= 6.1e+74)
		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[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e+88], t$95$1, If[LessEqual[y$46$re, -1.1e-78], t$95$0, If[LessEqual[y$46$re, 1.95e-306], N[(N[(y$46$re * N[(x$46$im / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.1e+74], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.1000000000000001e88 or 6.0999999999999997e74 < y.re

   1. Initial program 34.2%

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

   3. Taylor expanded in y.re around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. associate-/l* 76.2%

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

      5. associate-/r/ 78.0%

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

   5. Simplified 78.0%

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

      1. associate-*l/ 74.6%

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

      2. unpow2 74.6%

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

      3. associate-/r* 82.4%

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

      4. associate-*l/ 88.6%

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

      5. associate-/r/ 88.6%

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

      6. div-inv 88.6%

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

      7. clear-num 88.6%

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

   7. Applied egg-rr 88.6%

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

   if -3.1000000000000001e88 < y.re < -1.0999999999999999e-78 or 1.95e-306 < y.re < 6.0999999999999997e74

   1. Initial program 81.8%

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

   if -1.0999999999999999e-78 < y.re < 1.95e-306

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around 0 92.8%

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

      1. +-commutative 92.8%

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

      2. mul-1-neg 92.8%

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

      3. unsub-neg 92.8%

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

      4. associate-/l* 91.1%

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

      5. associate-/r/ 89.5%

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

   5. Simplified 89.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-306}:\\ \;\;\;\;y.re \cdot \frac{x.im}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ (* x.re (/ y.im y.re)) y.re))))
   (if (<= y.re -1.65e+89)
     t_1
     (if (<= y.re -1.2e-78)
       t_0
       (if (<= y.re 9.2e-240)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
         (if (<= y.re 6.1e+74) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -1.65e+89) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-78) {
		tmp = t_0;
	} else if (y_46_re <= 9.2e-240) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.1e+74) {
		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) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((x_46re * (y_46im / y_46re)) / y_46re)
    if (y_46re <= (-1.65d+89)) then
        tmp = t_1
    else if (y_46re <= (-1.2d-78)) then
        tmp = t_0
    else if (y_46re <= 9.2d-240) then
        tmp = (x_46im / ((y_46im ** 2.0d0) / y_46re)) - (x_46re / y_46im)
    else if (y_46re <= 6.1d+74) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -1.65e+89) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-78) {
		tmp = t_0;
	} else if (y_46_re <= 9.2e-240) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.1e+74) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -1.65e+89:
		tmp = t_1
	elif y_46_re <= -1.2e-78:
		tmp = t_0
	elif y_46_re <= 9.2e-240:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 6.1e+74:
		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(x_46_re * y_46_im)) / 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(Float64(x_46_re * Float64(y_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.65e+89)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-78)
		tmp = t_0;
	elseif (y_46_re <= 9.2e-240)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 6.1e+74)
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.65e+89)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-78)
		tmp = t_0;
	elseif (y_46_re <= 9.2e-240)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 6.1e+74)
		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[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.65e+89], t$95$1, If[LessEqual[y$46$re, -1.2e-78], t$95$0, If[LessEqual[y$46$re, 9.2e-240], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.1e+74], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.64999999999999987e89 or 6.0999999999999997e74 < y.re

   1. Initial program 34.2%

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

   3. Taylor expanded in y.re around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. associate-/l* 76.2%

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

      5. associate-/r/ 78.0%

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

   5. Simplified 78.0%

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

      1. associate-*l/ 74.6%

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

      2. unpow2 74.6%

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

      3. associate-/r* 82.4%

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

      4. associate-*l/ 88.6%

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

      5. associate-/r/ 88.6%

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

      6. div-inv 88.6%

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

      7. clear-num 88.6%

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

   7. Applied egg-rr 88.6%

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

   if -1.64999999999999987e89 < y.re < -1.2e-78 or 9.19999999999999972e-240 < y.re < 6.0999999999999997e74

   1. Initial program 81.2%

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

   if -1.2e-78 < y.re < 9.19999999999999972e-240

   1. Initial program 78.0%

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

   3. Taylor expanded in x.im around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. +-commutative 78.0%

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

      3. *-commutative 78.0%

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

      4. fma-def 78.0%

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

      5. distribute-rgt-neg-in 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in y.re around 0 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 90.8%

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

   8. Simplified 90.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ (* x.re (/ y.im y.re)) y.re))))
   (if (<= y.re -5.4e+93)
     t_1
     (if (<= y.re -3.4e-79)
       t_0
       (if (<= y.re -2.6e-205)
         (- (/ x.re y.im))
         (if (<= y.re 2.5e+70) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -5.4e+93) {
		tmp = t_1;
	} else if (y_46_re <= -3.4e-79) {
		tmp = t_0;
	} else if (y_46_re <= -2.6e-205) {
		tmp = -(x_46_re / y_46_im);
	} else if (y_46_re <= 2.5e+70) {
		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) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((x_46re * (y_46im / y_46re)) / y_46re)
    if (y_46re <= (-5.4d+93)) then
        tmp = t_1
    else if (y_46re <= (-3.4d-79)) then
        tmp = t_0
    else if (y_46re <= (-2.6d-205)) then
        tmp = -(x_46re / y_46im)
    else if (y_46re <= 2.5d+70) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -5.4e+93) {
		tmp = t_1;
	} else if (y_46_re <= -3.4e-79) {
		tmp = t_0;
	} else if (y_46_re <= -2.6e-205) {
		tmp = -(x_46_re / y_46_im);
	} else if (y_46_re <= 2.5e+70) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -5.4e+93:
		tmp = t_1
	elif y_46_re <= -3.4e-79:
		tmp = t_0
	elif y_46_re <= -2.6e-205:
		tmp = -(x_46_re / y_46_im)
	elif y_46_re <= 2.5e+70:
		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(x_46_re * y_46_im)) / 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(Float64(x_46_re * Float64(y_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -5.4e+93)
		tmp = t_1;
	elseif (y_46_re <= -3.4e-79)
		tmp = t_0;
	elseif (y_46_re <= -2.6e-205)
		tmp = Float64(-Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.5e+70)
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.4e+93)
		tmp = t_1;
	elseif (y_46_re <= -3.4e-79)
		tmp = t_0;
	elseif (y_46_re <= -2.6e-205)
		tmp = -(x_46_re / y_46_im);
	elseif (y_46_re <= 2.5e+70)
		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[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.4e+93], t$95$1, If[LessEqual[y$46$re, -3.4e-79], t$95$0, If[LessEqual[y$46$re, -2.6e-205], (-N[(x$46$re / y$46$im), $MachinePrecision]), If[LessEqual[y$46$re, 2.5e+70], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.3999999999999999e93 or 2.5000000000000001e70 < y.re

   1. Initial program 34.2%

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

   3. Taylor expanded in y.re around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. associate-/l* 76.2%

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

      5. associate-/r/ 78.0%

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

   5. Simplified 78.0%

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

      1. associate-*l/ 74.6%

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

      2. unpow2 74.6%

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

      3. associate-/r* 82.4%

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

      4. associate-*l/ 88.6%

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

      5. associate-/r/ 88.6%

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

      6. div-inv 88.6%

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

      7. clear-num 88.6%

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

   7. Applied egg-rr 88.6%

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

   if -5.3999999999999999e93 < y.re < -3.39999999999999976e-79 or -2.5999999999999998e-205 < y.re < 2.5000000000000001e70

   1. Initial program 81.8%

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

   if -3.39999999999999976e-79 < y.re < -2.5999999999999998e-205

   1. Initial program 72.4%

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

   3. Taylor expanded in y.re around 0 90.9%

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

      1. associate-*r/ 90.9%

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

      2. neg-mul-1 90.9%

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

   5. Simplified 90.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-78} \lor \neg \left(y.re \leq 0.12\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (/ (* x.re (/ y.im y.re)) y.re))))
   (if (<= y.re -2.35e+86)
     t_0
     (if (<= y.re -2.5e+65)
       (/ x.im (* y.im (/ y.im y.re)))
       (if (or (<= y.re -3.5e-78) (not (<= y.re 0.12)))
         t_0
         (- (/ 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 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.35e+86) {
		tmp = t_0;
	} else if (y_46_re <= -2.5e+65) {
		tmp = x_46_im / (y_46_im * (y_46_im / y_46_re));
	} else if ((y_46_re <= -3.5e-78) || !(y_46_re <= 0.12)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - ((x_46re * (y_46im / y_46re)) / y_46re)
    if (y_46re <= (-2.35d+86)) then
        tmp = t_0
    else if (y_46re <= (-2.5d+65)) then
        tmp = x_46im / (y_46im * (y_46im / y_46re))
    else if ((y_46re <= (-3.5d-78)) .or. (.not. (y_46re <= 0.12d0))) then
        tmp = t_0
    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 t_0 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.35e+86) {
		tmp = t_0;
	} else if (y_46_re <= -2.5e+65) {
		tmp = x_46_im / (y_46_im * (y_46_im / y_46_re));
	} else if ((y_46_re <= -3.5e-78) || !(y_46_re <= 0.12)) {
		tmp = t_0;
	} 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):
	t_0 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -2.35e+86:
		tmp = t_0
	elif y_46_re <= -2.5e+65:
		tmp = x_46_im / (y_46_im * (y_46_im / y_46_re))
	elif (y_46_re <= -3.5e-78) or not (y_46_re <= 0.12):
		tmp = t_0
	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)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re * Float64(y_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.35e+86)
		tmp = t_0;
	elseif (y_46_re <= -2.5e+65)
		tmp = Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re)));
	elseif ((y_46_re <= -3.5e-78) || !(y_46_re <= 0.12))
		tmp = t_0;
	else
		tmp = Float64(-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 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.35e+86)
		tmp = t_0;
	elseif (y_46_re <= -2.5e+65)
		tmp = x_46_im / (y_46_im * (y_46_im / y_46_re));
	elseif ((y_46_re <= -3.5e-78) || ~((y_46_re <= 0.12)))
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.35e+86], t$95$0, If[LessEqual[y$46$re, -2.5e+65], N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -3.5e-78], N[Not[LessEqual[y$46$re, 0.12]], $MachinePrecision]], t$95$0, (-N[(x$46$re / y$46$im), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.3500000000000001e86 or -2.49999999999999986e65 < y.re < -3.4999999999999999e-78 or 0.12 < y.re

   1. Initial program 49.3%

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

   3. Taylor expanded in y.re around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. mul-1-neg 69.7%

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

      3. unsub-neg 69.7%

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

      4. associate-/l* 70.8%

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

      5. associate-/r/ 71.4%

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

   5. Simplified 71.4%

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

      1. associate-*l/ 69.7%

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

      2. unpow2 69.7%

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

      3. associate-/r* 75.0%

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

      4. associate-*l/ 79.3%

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

      5. associate-/r/ 79.3%

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

      6. div-inv 79.3%

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

      7. clear-num 79.3%

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

   7. Applied egg-rr 79.3%

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

   if -2.3500000000000001e86 < y.re < -2.49999999999999986e65

   1. Initial program 66.5%

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

   3. Taylor expanded in x.im around inf 66.5%

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

      1. associate-/l* 93.6%

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

      2. unpow2 93.6%

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

      3. fma-udef 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in y.im around inf 80.3%

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

      1. div-inv 80.3%

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

      2. unpow2 80.3%

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

      3. associate-*l* 80.3%

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

      4. div-inv 80.3%

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

   8. Applied egg-rr 80.3%

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

   if -3.4999999999999999e-78 < y.re < 0.12

   1. Initial program 80.0%

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

   3. Taylor expanded in y.re around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-78} \lor \neg \left(y.re \leq 0.12\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-78} \lor \neg \left(y.re \leq 35\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (/ (* x.re (/ y.im y.re)) y.re))))
   (if (<= y.re -2.35e+86)
     t_0
     (if (<= y.re -1.3e+51)
       (* (/ -1.0 y.im) (- (- x.re) (/ x.im (/ y.im y.re))))
       (if (or (<= y.re -1.3e-78) (not (<= y.re 35.0)))
         t_0
         (- (/ 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 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.35e+86) {
		tmp = t_0;
	} else if (y_46_re <= -1.3e+51) {
		tmp = (-1.0 / y_46_im) * (-x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if ((y_46_re <= -1.3e-78) || !(y_46_re <= 35.0)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - ((x_46re * (y_46im / y_46re)) / y_46re)
    if (y_46re <= (-2.35d+86)) then
        tmp = t_0
    else if (y_46re <= (-1.3d+51)) then
        tmp = ((-1.0d0) / y_46im) * (-x_46re - (x_46im / (y_46im / y_46re)))
    else if ((y_46re <= (-1.3d-78)) .or. (.not. (y_46re <= 35.0d0))) then
        tmp = t_0
    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 t_0 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.35e+86) {
		tmp = t_0;
	} else if (y_46_re <= -1.3e+51) {
		tmp = (-1.0 / y_46_im) * (-x_46_re - (x_46_im / (y_46_im / y_46_re)));
	} else if ((y_46_re <= -1.3e-78) || !(y_46_re <= 35.0)) {
		tmp = t_0;
	} 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):
	t_0 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -2.35e+86:
		tmp = t_0
	elif y_46_re <= -1.3e+51:
		tmp = (-1.0 / y_46_im) * (-x_46_re - (x_46_im / (y_46_im / y_46_re)))
	elif (y_46_re <= -1.3e-78) or not (y_46_re <= 35.0):
		tmp = t_0
	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)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re * Float64(y_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.35e+86)
		tmp = t_0;
	elseif (y_46_re <= -1.3e+51)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_re) - Float64(x_46_im / Float64(y_46_im / y_46_re))));
	elseif ((y_46_re <= -1.3e-78) || !(y_46_re <= 35.0))
		tmp = t_0;
	else
		tmp = Float64(-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 = (x_46_im / y_46_re) - ((x_46_re * (y_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.35e+86)
		tmp = t_0;
	elseif (y_46_re <= -1.3e+51)
		tmp = (-1.0 / y_46_im) * (-x_46_re - (x_46_im / (y_46_im / y_46_re)));
	elseif ((y_46_re <= -1.3e-78) || ~((y_46_re <= 35.0)))
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.35e+86], t$95$0, If[LessEqual[y$46$re, -1.3e+51], 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[Or[LessEqual[y$46$re, -1.3e-78], N[Not[LessEqual[y$46$re, 35.0]], $MachinePrecision]], t$95$0, (-N[(x$46$re / y$46$im), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.3500000000000001e86 or -1.3000000000000001e51 < y.re < -1.3000000000000001e-78 or 35 < y.re

   1. Initial program 49.2%

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

   3. Taylor expanded in y.re around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. mul-1-neg 70.0%

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

      3. unsub-neg 70.0%

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

      4. associate-/l* 71.1%

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

      5. associate-/r/ 71.7%

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

   5. Simplified 71.7%

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

      1. associate-*l/ 70.0%

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

      2. unpow2 70.0%

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

      3. associate-/r* 75.4%

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

      4. associate-*l/ 79.7%

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

      5. associate-/r/ 79.7%

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

      6. div-inv 79.7%

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

      7. clear-num 79.7%

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

   7. Applied egg-rr 79.7%

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

   if -2.3500000000000001e86 < y.re < -1.3000000000000001e51

   1. Initial program 62.8%

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

      1. fma-neg 62.8%

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

      2. *-commutative 62.8%

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

      3. distribute-rgt-neg-out 62.8%

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

      4. *-un-lft-identity 62.8%

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

      5. fma-def 62.8%

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

      6. add-sqr-sqrt 62.8%

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

      7. times-frac 62.8%

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

      8. fma-def 62.8%

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

      9. hypot-def 62.8%

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

      10. *-commutative 62.8%

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

      11. add-sqr-sqrt 29.0%

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

      12. sqrt-unprod 62.3%

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

      13. sqr-neg 62.3%

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

      14. sqrt-unprod 33.8%

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

      15. add-sqr-sqrt 62.8%

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

      16. fma-def 62.8%

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

      17. hypot-def 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in y.im around -inf 51.7%

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

      1. mul-1-neg 51.7%

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

      2. unsub-neg 51.7%

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

      3. mul-1-neg 51.7%

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

      4. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in y.im around -inf 78.7%

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

   if -1.3000000000000001e-78 < y.re < 35

   1. Initial program 80.0%

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

   3. Taylor expanded in y.re around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-78} \lor \neg \left(y.re \leq 35\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.3 \cdot 10^{-65} \lor \neg \left(y.re \leq 50\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.3e-65) (not (<= y.re 50.0)))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
   (- (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5.3e-65) || !(y_46_re <= 50.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-5.3d-65)) .or. (.not. (y_46re <= 50.0d0))) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else
        tmp = -(x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5.3e-65) || !(y_46_re <= 50.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -5.3e-65) or not (y_46_re <= 50.0):
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	else:
		tmp = -(x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -5.3e-65) || !(y_46_re <= 50.0))
		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(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_re <= -5.3e-65) || ~((y_46_re <= 50.0)))
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	else
		tmp = -(x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -5.3e-65], N[Not[LessEqual[y$46$re, 50.0]], $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[(x$46$re / y$46$im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.30000000000000037e-65 or 50 < y.re

   1. Initial program 49.0%

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

   3. Taylor expanded in y.re around inf 67.2%

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

      1. +-commutative 67.2%

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

      2. mul-1-neg 67.2%

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

      3. unsub-neg 67.2%

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

      4. associate-/l* 68.3%

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

      5. associate-/r/ 70.9%

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

   5. Simplified 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. unpow2 70.9%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*l/ 74.5%

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

      2. *-lft-identity 74.5%

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

   9. Simplified 74.5%

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

   if -5.30000000000000037e-65 < y.re < 50

   1. Initial program 80.5%

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

   3. Taylor expanded in y.re around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.3 \cdot 10^{-65} \lor \neg \left(y.re \leq 50\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.00017 \lor \neg \left(y.re \leq 90\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -0.00017) (not (<= y.re 90.0)))
   (/ x.im y.re)
   (- (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.00017) || !(y_46_re <= 90.0)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-0.00017d0)) .or. (.not. (y_46re <= 90.0d0))) then
        tmp = x_46im / y_46re
    else
        tmp = -(x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.00017) || !(y_46_re <= 90.0)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -0.00017) or not (y_46_re <= 90.0):
		tmp = x_46_im / y_46_re
	else:
		tmp = -(x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -0.00017) || !(y_46_re <= 90.0))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(-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_re <= -0.00017) || ~((y_46_re <= 90.0)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -(x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -0.00017], N[Not[LessEqual[y$46$re, 90.0]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], (-N[(x$46$re / y$46$im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.7e-4 or 90 < y.re

   1. Initial program 45.5%

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

   3. Taylor expanded in y.re around inf 66.6%

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

   if -1.7e-4 < y.re < 90

   1. Initial program 80.9%

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

   3. Taylor expanded in y.re around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   5. Simplified 67.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00017 \lor \neg \left(y.re \leq 90\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 6.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 6.5e+200) (/ 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 <= 6.5e+200) {
		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 <= 6.5d+200) 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 <= 6.5e+200) {
		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 <= 6.5e+200:
		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 <= 6.5e+200)
		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 <= 6.5e+200)
		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, 6.5e+200], 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 < 6.49999999999999963e200

   1. Initial program 65.4%

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

   3. Taylor expanded in y.re around inf 48.7%

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

   if 6.49999999999999963e200 < 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. fma-neg 39.8%

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

      2. *-commutative 39.8%

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

      3. distribute-rgt-neg-out 39.8%

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

      4. *-un-lft-identity 39.8%

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

      5. fma-def 39.8%

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

      6. add-sqr-sqrt 39.8%

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

      7. times-frac 39.8%

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

      8. fma-def 39.8%

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

      9. hypot-def 39.8%

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

      10. *-commutative 39.8%

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

      11. add-sqr-sqrt 26.3%

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

      12. sqrt-unprod 39.8%

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

      13. sqr-neg 39.8%

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

      14. sqrt-unprod 13.5%

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

      15. add-sqr-sqrt 39.8%

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

      16. fma-def 39.8%

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

      17. hypot-def 48.9%

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

   4. Applied egg-rr 48.9%

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

   5. Taylor expanded in y.re around 0 41.0%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 6.5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.8% 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 62.9%

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

   1. fma-neg 62.9%

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

   2. *-commutative 62.9%

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

   3. distribute-rgt-neg-out 62.9%

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

   4. *-un-lft-identity 62.9%

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

   5. fma-def 62.9%

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

   6. add-sqr-sqrt 62.9%

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

   7. times-frac 63.0%

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

   8. fma-def 63.0%

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

   9. hypot-def 63.0%

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

   10. *-commutative 63.0%

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

   11. add-sqr-sqrt 29.8%

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

   12. sqrt-unprod 41.8%

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

   13. sqr-neg 41.8%

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

   14. sqrt-unprod 18.2%

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

   15. add-sqr-sqrt 37.0%

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

   16. fma-def 37.0%

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

   17. hypot-def 44.6%

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

4. Applied egg-rr 44.6%

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

5. Taylor expanded in y.im around -inf 32.6%

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

   1. mul-1-neg 32.6%

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

   2. unsub-neg 32.6%

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

   3. mul-1-neg 32.6%

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

   4. associate-/l* 33.8%

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

7. Simplified 33.8%

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

8. Taylor expanded in y.re around -inf 10.2%

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

9. Final simplification 10.2%

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

Alternative 12: 42.5% 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 62.9%

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

3. Taylor expanded in y.re around inf 44.6%

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

4. Final simplification 44.6%

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

Reproduce

herbie shell --seed 2024021 
(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))))