_divideComplex, imaginary part

Percentage Accurate: 62.0% → 91.7%

Time: 10.7s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}\\ t_2 := t_0 \cdot \frac{y.re}{t_1} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ t_3 := \frac{t_0}{\frac{t_1}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+141}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-169}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (/ (hypot y.re y.im) x.im))
        (t_2
         (-
          (* t_0 (/ y.re t_1))
          (* y.im (/ x.re (pow (hypot y.re y.im) 2.0)))))
        (t_3 (- (/ t_0 (/ t_1 y.re)) (/ x.re y.im))))
   (if (<= y.im -8e+141)
     t_3
     (if (<= y.im -2e-134)
       t_2
       (if (<= y.im 1.7e-169)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 7.5e+144) t_2 t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = hypot(y_46_re, y_46_im) / x_46_im;
	double t_2 = (t_0 * (y_46_re / t_1)) - (y_46_im * (x_46_re / pow(hypot(y_46_re, y_46_im), 2.0)));
	double t_3 = (t_0 / (t_1 / y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -8e+141) {
		tmp = t_3;
	} else if (y_46_im <= -2e-134) {
		tmp = t_2;
	} else if (y_46_im <= 1.7e-169) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.5e+144) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = Math.hypot(y_46_re, y_46_im) / x_46_im;
	double t_2 = (t_0 * (y_46_re / t_1)) - (y_46_im * (x_46_re / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0)));
	double t_3 = (t_0 / (t_1 / y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -8e+141) {
		tmp = t_3;
	} else if (y_46_im <= -2e-134) {
		tmp = t_2;
	} else if (y_46_im <= 1.7e-169) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.5e+144) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = math.hypot(y_46_re, y_46_im) / x_46_im
	t_2 = (t_0 * (y_46_re / t_1)) - (y_46_im * (x_46_re / math.pow(math.hypot(y_46_re, y_46_im), 2.0)))
	t_3 = (t_0 / (t_1 / y_46_re)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -8e+141:
		tmp = t_3
	elif y_46_im <= -2e-134:
		tmp = t_2
	elif y_46_im <= 1.7e-169:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 7.5e+144:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(hypot(y_46_re, y_46_im) / x_46_im)
	t_2 = Float64(Float64(t_0 * Float64(y_46_re / t_1)) - Float64(y_46_im * Float64(x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0))))
	t_3 = Float64(Float64(t_0 / Float64(t_1 / y_46_re)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -8e+141)
		tmp = t_3;
	elseif (y_46_im <= -2e-134)
		tmp = t_2;
	elseif (y_46_im <= 1.7e-169)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 7.5e+144)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = hypot(y_46_re, y_46_im) / x_46_im;
	t_2 = (t_0 * (y_46_re / t_1)) - (y_46_im * (x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0)));
	t_3 = (t_0 / (t_1 / y_46_re)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -8e+141)
		tmp = t_3;
	elseif (y_46_im <= -2e-134)
		tmp = t_2;
	elseif (y_46_im <= 1.7e-169)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 7.5e+144)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(y$46$re / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 / N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8e+141], t$95$3, If[LessEqual[y$46$im, -2e-134], t$95$2, If[LessEqual[y$46$im, 1.7e-169], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+144], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.00000000000000014e141 or 7.5000000000000006e144 < y.im

   1. Initial program 24.5%

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

      1. div-sub 24.5%

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

      2. *-un-lft-identity 24.5%

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

      3. add-sqr-sqrt 24.5%

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

      4. times-frac 24.5%

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

      5. fma-neg 24.5%

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

      6. hypot-def 24.5%

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

      7. hypot-def 33.1%

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

      8. associate-/l* 37.3%

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

      9. add-sqr-sqrt 37.3%

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

      10. pow2 37.3%

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

      11. hypot-def 37.3%

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

   3. Applied egg-rr 37.3%

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

      1. fma-neg 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-/l* 50.6%

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

      4. associate-/r/ 47.7%

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

      5. *-commutative 47.7%

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

   5. Simplified 47.7%

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

      1. clear-num 47.7%

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

      2. un-div-inv 47.8%

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

   7. Applied egg-rr 47.8%

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

   8. Taylor expanded in y.im around inf 94.1%

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

   if -8.00000000000000014e141 < y.im < -2.00000000000000008e-134 or 1.7e-169 < y.im < 7.5000000000000006e144

   1. Initial program 74.3%

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

      1. div-sub 74.3%

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

      2. *-un-lft-identity 74.3%

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

      3. add-sqr-sqrt 74.3%

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

      4. times-frac 74.3%

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

      5. fma-neg 74.3%

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

      6. hypot-def 74.3%

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

      7. hypot-def 78.9%

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

      8. associate-/l* 83.0%

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

      9. add-sqr-sqrt 83.0%

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

      10. pow2 83.0%

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

      11. hypot-def 83.0%

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

   3. Applied egg-rr 83.0%

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

      1. fma-neg 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-/l* 93.4%

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

      4. associate-/r/ 91.9%

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

      5. *-commutative 91.9%

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

   5. Simplified 91.9%

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

   if -2.00000000000000008e-134 < y.im < 1.7e-169

   1. Initial program 68.2%

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

   2. Taylor expanded in y.re around inf 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. unpow2 90.1%

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

      4. times-frac 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in x.im around 0 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unpow2 90.1%

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

      3. associate-/l* 90.2%

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

      4. associate-*l/ 97.0%

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

      5. sub-neg 97.0%

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

      6. associate-/r* 96.9%

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

      7. associate-/l* 97.0%

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

      8. associate-*r/ 97.0%

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

      9. div-sub 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-169}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 92.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 5e+302)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (-
      (* t_0 (/ y.re (/ (hypot y.re y.im) x.im)))
      (* y.im (/ (/ x.re (hypot y.re y.im)) (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * ((x_46_re / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (t_0 * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * ((x_46_re / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+302)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(t_0 * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(y_46_im * Float64(Float64(x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5e302

   1. Initial program 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

      4. hypot-def 78.7%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 5e302 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 10.4%

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

      1. div-sub 9.0%

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

      2. *-un-lft-identity 9.0%

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

      3. add-sqr-sqrt 9.0%

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

      4. times-frac 9.0%

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

      5. fma-neg 9.0%

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

      6. hypot-def 9.0%

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

      7. hypot-def 9.9%

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

      8. associate-/l* 16.7%

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

      9. add-sqr-sqrt 16.7%

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

      10. pow2 16.7%

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

      11. hypot-def 16.7%

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

   3. Applied egg-rr 16.7%

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

      1. fma-neg 16.7%

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

      2. *-commutative 16.7%

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

      3. associate-/l* 59.0%

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

      4. associate-/r/ 59.0%

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

      5. *-commutative 59.0%

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

   5. Simplified 59.0%

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

      1. *-un-lft-identity 59.0%

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

      2. unpow2 59.0%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. associate-*l/ 88.1%

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

      2. *-lft-identity 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 85.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+302)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+302)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5e302

   1. Initial program 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

      4. hypot-def 78.7%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 5e302 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 10.4%

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

   2. Taylor expanded in y.re around 0 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. unpow2 46.3%

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

      5. times-frac 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 89.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 5e+302)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (- (* t_0 (/ y.re (/ (hypot y.re y.im) x.im))) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (t_0 * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+302)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(t_0 * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+302)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5e302

   1. Initial program 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

      4. hypot-def 78.7%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 5e302 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 10.4%

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

      1. div-sub 9.0%

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

      2. *-un-lft-identity 9.0%

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

      3. add-sqr-sqrt 9.0%

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

      4. times-frac 9.0%

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

      5. fma-neg 9.0%

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

      6. hypot-def 9.0%

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

      7. hypot-def 9.9%

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

      8. associate-/l* 16.7%

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

      9. add-sqr-sqrt 16.7%

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

      10. pow2 16.7%

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

      11. hypot-def 16.7%

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

   3. Applied egg-rr 16.7%

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

      1. fma-neg 16.7%

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

      2. *-commutative 16.7%

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

      3. associate-/l* 59.0%

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

      4. associate-/r/ 59.0%

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

      5. *-commutative 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in y.im around inf 69.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 78.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+18)
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
   (if (<= y.im 7.2e+26)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (* (/ 1.0 (hypot y.re y.im)) (- (/ y.re (/ y.im x.im)) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9e+18) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 7.2e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (y_46_im / x_46_im)) - x_46_re);
	}
	return tmp;
}

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 <= -9e+18) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 7.2e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re / (y_46_im / x_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -9e+18:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_im <= 7.2e+26:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re / (y_46_im / x_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e+18)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 7.2e+26)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -9e+18)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_im <= 7.2e+26)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (y_46_im / x_46_im)) - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9e+18], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+26], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9e18

   1. Initial program 55.9%

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

   2. Taylor expanded in y.re around 0 78.3%

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

      1. +-commutative 78.3%

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

      2. mul-1-neg 78.3%

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

      3. unsub-neg 78.3%

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

      4. unpow2 78.3%

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

      5. times-frac 86.4%

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

   4. Simplified 86.4%

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

   if -9e18 < y.im < 7.20000000000000048e26

   1. Initial program 70.3%

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

   2. Taylor expanded in y.re around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unsub-neg 82.9%

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

      3. unpow2 82.9%

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

      4. times-frac 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in x.im around 0 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unpow2 82.9%

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

      3. associate-/l* 82.3%

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

      4. associate-*l/ 85.8%

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

      5. sub-neg 85.8%

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

      6. associate-/r* 89.3%

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

      7. associate-/l* 89.3%

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

      8. associate-*r/ 89.3%

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

      9. div-sub 89.3%

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

   7. Simplified 89.3%

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

   if 7.20000000000000048e26 < y.im

   1. Initial program 42.8%

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

      1. *-un-lft-identity 42.8%

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

      2. add-sqr-sqrt 42.8%

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

      3. times-frac 42.8%

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

      4. hypot-def 42.8%

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

      5. hypot-def 56.6%

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

   3. Applied egg-rr 56.6%

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

   4. Taylor expanded in y.re around 0 69.0%

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

      1. neg-mul-1 69.0%

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

      2. unsub-neg 69.0%

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

      3. associate-/l* 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} - x.re\right)\\ \end{array} \]

Alternative 6: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+18} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.1e+18) (not (<= y.im 8.5e+26)))
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.1e+18) || !(y_46_im <= 8.5e+26)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.1d+18)) .or. (.not. (y_46im <= 8.5d+26))) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.1e+18) || !(y_46_im <= 8.5e+26)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.1e+18) or not (y_46_im <= 8.5e+26):
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.1e+18) || !(y_46_im <= 8.5e+26))
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.1e+18) || ~((y_46_im <= 8.5e+26)))
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.1e+18], N[Not[LessEqual[y$46$im, 8.5e+26]], $MachinePrecision]], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.1e18 or 8.5e26 < y.im

   1. Initial program 48.1%

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

   2. Taylor expanded in y.re around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. times-frac 80.9%

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

   4. Simplified 80.9%

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

   if -3.1e18 < y.im < 8.5e26

   1. Initial program 70.3%

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

   2. Taylor expanded in y.re around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unsub-neg 82.9%

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

      3. unpow2 82.9%

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

      4. times-frac 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in x.im around 0 82.9%

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

      1. mul-1-neg 82.9%

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

      2. unpow2 82.9%

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

      3. associate-/l* 82.3%

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

      4. associate-*l/ 85.8%

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

      5. sub-neg 85.8%

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

      6. associate-/r* 89.3%

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

      7. associate-/l* 89.3%

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

      8. associate-*r/ 89.3%

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

      9. div-sub 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+18} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+39} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.5e+39) (not (<= y.im 7.6e+26)))
   (/ (- x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.5e+39) || !(y_46_im <= 7.6e+26)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.5d+39)) .or. (.not. (y_46im <= 7.6d+26))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.5e+39) || !(y_46_im <= 7.6e+26)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.5e+39) or not (y_46_im <= 7.6e+26):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.5e+39) || !(y_46_im <= 7.6e+26))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.5e+39) || ~((y_46_im <= 7.6e+26)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.5e+39], N[Not[LessEqual[y$46$im, 7.6e+26]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.5000000000000001e39 or 7.6000000000000004e26 < y.im

   1. Initial program 46.8%

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

   2. Taylor expanded in y.re around 0 63.5%

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

      1. associate-*r/ 63.5%

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

      2. neg-mul-1 63.5%

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

   4. Simplified 63.5%

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

   if -6.5000000000000001e39 < y.im < 7.6000000000000004e26

   1. Initial program 70.9%

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

   2. Taylor expanded in y.re around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unsub-neg 82.0%

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

      3. unpow2 82.0%

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

      4. times-frac 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in x.im around 0 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unpow2 82.0%

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

      3. associate-/l* 81.3%

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

      4. associate-*l/ 84.7%

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

      5. sub-neg 84.7%

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

      6. associate-/r* 88.1%

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

      7. associate-/l* 88.2%

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

      8. associate-*r/ 88.2%

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

      9. div-sub 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+39} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 63.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+21} \lor \neg \left(y.im \leq 5.5 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.4e+21) (not (<= y.im 5.5e+16)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.4e+21) || !(y_46_im <= 5.5e+16)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.4d+21)) .or. (.not. (y_46im <= 5.5d+16))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.4e+21) || !(y_46_im <= 5.5e+16)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.4e+21) or not (y_46_im <= 5.5e+16):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.4e+21) || !(y_46_im <= 5.5e+16))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.4e+21) || ~((y_46_im <= 5.5e+16)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.4e+21], N[Not[LessEqual[y$46$im, 5.5e+16]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.4e21 or 5.5e16 < y.im

   1. Initial program 48.1%

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

   2. Taylor expanded in y.re around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. neg-mul-1 62.8%

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

   4. Simplified 62.8%

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

   if -4.4e21 < y.im < 5.5e16

   1. Initial program 70.3%

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

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

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+21} \lor \neg \left(y.im \leq 5.5 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9: 43.2% 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 59.8%

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

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

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

3. Final simplification 44.6%

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

Reproduce

herbie shell --seed 2023230 
(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))))