_divideComplex, real part

Percentage Accurate: 62.0% → 85.6%

Time: 12.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+304)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.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 ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+304) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_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 (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+304)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999997e304

   1. Initial program 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.5%

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

      4. hypot-def 79.5%

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

      5. fma-def 79.5%

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

      6. hypot-def 95.4%

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

   4. Applied egg-rr 95.4%

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

      1. associate-*l/ 95.6%

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

      2. *-un-lft-identity 95.6%

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

   6. Applied egg-rr 95.6%

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

   if 4.9999999999999997e304 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 8.3%

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

   3. Taylor expanded in y.re around inf 44.5%

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

      1. associate-/l* 52.9%

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

      2. associate-/r/ 52.9%

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

   5. Simplified 52.9%

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

      1. *-un-lft-identity 52.9%

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

      2. pow2 52.9%

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

      3. times-frac 65.0%

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

   7. Applied egg-rr 65.0%

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

      1. associate-*l/ 65.0%

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

      2. *-lft-identity 65.0%

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

   9. Simplified 65.0%

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

       1. associate-*l/ 65.1%

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

   11. Applied egg-rr 65.1%

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

       1. associate-/l* 65.1%

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

   13. Simplified 65.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -2.06 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{t\_0}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im)))
        (t_1 (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))
   (if (<= y.re -2.06e+102)
     t_1
     (if (<= y.re -2.3e-164)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.05e-105)
         (* (/ -1.0 y.im) (- (- x.im) (* x.re (/ y.re y.im))))
         (if (<= y.re 6.6e+78) (/ t_0 (pow (hypot y.re y.im) 2.0)) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -2.06e+102) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-164) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.05e-105) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 6.6e+78) {
		tmp = t_0 / pow(hypot(y_46_re, y_46_im), 2.0);
	} else {
		tmp = t_1;
	}
	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_re * y_46_re) + (x_46_im * y_46_im);
	double t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -2.06e+102) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-164) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.05e-105) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 6.6e+78) {
		tmp = t_0 / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -2.06e+102:
		tmp = t_1
	elif y_46_re <= -2.3e-164:
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.05e-105:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)))
	elif y_46_re <= 6.6e+78:
		tmp = t_0 / math.pow(math.hypot(y_46_re, y_46_im), 2.0)
	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(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.06e+102)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-164)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.05e-105)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(x_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 6.6e+78)
		tmp = Float64(t_0 / (hypot(y_46_re, y_46_im) ^ 2.0));
	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 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.06e+102)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-164)
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.05e-105)
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	elseif (y_46_re <= 6.6e+78)
		tmp = t_0 / (hypot(y_46_re, y_46_im) ^ 2.0);
	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[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.06e+102], t$95$1, If[LessEqual[y$46$re, -2.3e-164], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-105], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+78], N[(t$95$0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.05999999999999999e102 or 6.6e78 < y.re

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around inf 71.0%

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

      1. associate-/l* 75.7%

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

      2. associate-/r/ 77.8%

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

   5. Simplified 77.8%

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

      1. *-un-lft-identity 77.8%

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

      2. pow2 77.8%

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

      3. times-frac 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. associate-*l/ 85.6%

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

      2. *-lft-identity 85.6%

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

   9. Simplified 85.6%

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

       1. associate-*l/ 86.9%

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

   11. Applied egg-rr 86.9%

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

       1. associate-/l* 86.9%

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

   13. Simplified 86.9%

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

   if -2.05999999999999999e102 < y.re < -2.29999999999999985e-164

   1. Initial program 83.0%

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

   if -2.29999999999999985e-164 < y.re < 1.05e-105

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y.im around -inf 52.2%

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

      1. distribute-lft-out 52.2%

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

      2. associate-/l* 52.2%

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

   7. Simplified 52.2%

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

      1. distribute-lft-in 52.2%

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

      2. distribute-lft-in 50.7%

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

      3. mul-1-neg 50.7%

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

      4. div-inv 50.7%

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

      5. clear-num 50.7%

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

      6. associate-*r* 50.7%

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

      7. neg-mul-1 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. distribute-lft-out 52.2%

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

   11. Simplified 52.2%

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

   12. Taylor expanded in y.im around -inf 92.1%

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

   if 1.05e-105 < y.re < 6.6e78

   1. Initial program 81.5%

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

      1. expm1-log1p-u 78.9%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)}} \]

      2. expm1-udef 50.8%

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

      3. add-sqr-sqrt 50.8%

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

      4. pow2 50.8%

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

      5. hypot-def 50.8%

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

   4. Applied egg-rr 50.8%

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

      1. expm1-def 78.9%

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

      2. expm1-log1p 81.5%

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

   6. Simplified 81.5%

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))
   (if (<= y.re -2.2e+102)
     t_1
     (if (<= y.re -2.3e-164)
       t_0
       (if (<= y.re 3.9e-106)
         (* (/ -1.0 y.im) (- (- x.im) (* x.re (/ y.re y.im))))
         (if (<= y.re 4.2e+78) 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -2.2e+102) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-164) {
		tmp = t_0;
	} else if (y_46_re <= 3.9e-106) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 4.2e+78) {
		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 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    if (y_46re <= (-2.2d+102)) then
        tmp = t_1
    else if (y_46re <= (-2.3d-164)) then
        tmp = t_0
    else if (y_46re <= 3.9d-106) then
        tmp = ((-1.0d0) / y_46im) * (-x_46im - (x_46re * (y_46re / y_46im)))
    else if (y_46re <= 4.2d+78) 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -2.2e+102) {
		tmp = t_1;
	} else if (y_46_re <= -2.3e-164) {
		tmp = t_0;
	} else if (y_46_re <= 3.9e-106) {
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 4.2e+78) {
		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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -2.2e+102:
		tmp = t_1
	elif y_46_re <= -2.3e-164:
		tmp = t_0
	elif y_46_re <= 3.9e-106:
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)))
	elif y_46_re <= 4.2e+78:
		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(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.2e+102)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-164)
		tmp = t_0;
	elseif (y_46_re <= 3.9e-106)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(x_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 4.2e+78)
		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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.2e+102)
		tmp = t_1;
	elseif (y_46_re <= -2.3e-164)
		tmp = t_0;
	elseif (y_46_re <= 3.9e-106)
		tmp = (-1.0 / y_46_im) * (-x_46_im - (x_46_re * (y_46_re / y_46_im)));
	elseif (y_46_re <= 4.2e+78)
		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[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+102], t$95$1, If[LessEqual[y$46$re, -2.3e-164], t$95$0, If[LessEqual[y$46$re, 3.9e-106], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+78], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.20000000000000007e102 or 4.2000000000000002e78 < y.re

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around inf 71.0%

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

      1. associate-/l* 75.7%

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

      2. associate-/r/ 77.8%

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

   5. Simplified 77.8%

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

      1. *-un-lft-identity 77.8%

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

      2. pow2 77.8%

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

      3. times-frac 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. associate-*l/ 85.6%

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

      2. *-lft-identity 85.6%

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

   9. Simplified 85.6%

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

       1. associate-*l/ 86.9%

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

   11. Applied egg-rr 86.9%

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

       1. associate-/l* 86.9%

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

   13. Simplified 86.9%

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

   if -2.20000000000000007e102 < y.re < -2.29999999999999985e-164 or 3.9000000000000001e-106 < y.re < 4.2000000000000002e78

   1. Initial program 82.5%

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

   if -2.29999999999999985e-164 < y.re < 3.9000000000000001e-106

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in y.im around -inf 52.2%

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

      1. distribute-lft-out 52.2%

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

      2. associate-/l* 52.2%

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

   7. Simplified 52.2%

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

      1. distribute-lft-in 52.2%

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

      2. distribute-lft-in 50.7%

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

      3. mul-1-neg 50.7%

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

      4. div-inv 50.7%

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

      5. clear-num 50.7%

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

      6. associate-*r* 50.7%

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

      7. neg-mul-1 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. distribute-lft-out 52.2%

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

   11. Simplified 52.2%

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

   12. Taylor expanded in y.im around -inf 92.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ -1.0 y.im) (- (- x.im) (/ x.re (/ y.im y.re))))))
   (if (<= y.im -1.12e-35)
     t_0
     (if (<= y.im 1e-12)
       (+ (/ x.re y.re) (/ (/ (* x.im y.im) y.re) y.re))
       (if (<= y.im 1.85e+67)
         (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (-1.0 / y_46_im) * (-x_46_im - (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -1.12e-35) {
		tmp = t_0;
	} else if (y_46_im <= 1e-12) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.85e+67) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / y_46im) * (-x_46im - (x_46re / (y_46im / y_46re)))
    if (y_46im <= (-1.12d-35)) then
        tmp = t_0
    else if (y_46im <= 1d-12) then
        tmp = (x_46re / y_46re) + (((x_46im * y_46im) / y_46re) / y_46re)
    else if (y_46im <= 1.85d+67) then
        tmp = (x_46re / y_46re) + (y_46im * ((x_46im / y_46re) / y_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (-1.0 / y_46_im) * (-x_46_im - (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -1.12e-35) {
		tmp = t_0;
	} else if (y_46_im <= 1e-12) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.85e+67) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (-1.0 / y_46_im) * (-x_46_im - (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_im <= -1.12e-35:
		tmp = t_0
	elif y_46_im <= 1e-12:
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re)
	elif y_46_im <= 1.85e+67:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-1.0 / y_46_im) * Float64(Float64(-x_46_im) - Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_im <= -1.12e-35)
		tmp = t_0;
	elseif (y_46_im <= 1e-12)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(x_46_im * y_46_im) / y_46_re) / y_46_re));
	elseif (y_46_im <= 1.85e+67)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (-1.0 / y_46_im) * (-x_46_im - (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_im <= -1.12e-35)
		tmp = t_0;
	elseif (y_46_im <= 1e-12)
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	elseif (y_46_im <= 1.85e+67)
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[((-x$46$im) - N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.12e-35], t$95$0, If[LessEqual[y$46$im, 1e-12], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.85e+67], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.12e-35 or 1.8499999999999999e67 < y.im

   1. Initial program 51.1%

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

      1. *-un-lft-identity 51.1%

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

      2. add-sqr-sqrt 51.1%

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

      3. times-frac 51.1%

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

      4. hypot-def 51.1%

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

      5. fma-def 51.1%

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

      6. hypot-def 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in y.im around -inf 54.3%

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

      1. distribute-lft-out 54.3%

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

      2. associate-/l* 57.0%

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

   7. Simplified 57.0%

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

   8. Taylor expanded in y.im around -inf 79.9%

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

   if -1.12e-35 < y.im < 9.9999999999999998e-13

   1. Initial program 73.3%

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

   3. Taylor expanded in y.re around inf 78.9%

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

      1. associate-/l* 77.4%

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

      2. associate-/r/ 76.8%

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

   5. Simplified 76.8%

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

      1. pow2 76.8%

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

      2. associate-*l/ 78.9%

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

      3. associate-/r* 84.7%

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

   7. Applied egg-rr 84.7%

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

   if 9.9999999999999998e-13 < y.im < 1.8499999999999999e67

   1. Initial program 55.1%

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

   3. Taylor expanded in y.re around inf 46.3%

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

      1. associate-/l* 56.7%

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

      2. associate-/r/ 56.7%

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

   5. Simplified 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. pow2 56.7%

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

      3. times-frac 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*l/ 75.9%

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

      2. *-lft-identity 75.9%

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

   9. Simplified 75.9%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.05e-35)
   (/ x.im y.im)
   (if (<= y.im 2e-9)
     (+ (/ x.re y.re) (/ (/ (* x.im y.im) y.re) y.re))
     (if (<= y.im 1.7e+70)
       (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.05e-35) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-9) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.7e+70) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.05d-35)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2d-9) then
        tmp = (x_46re / y_46re) + (((x_46im * y_46im) / y_46re) / y_46re)
    else if (y_46im <= 1.7d+70) then
        tmp = (x_46re / y_46re) + (y_46im * ((x_46im / y_46re) / y_46re))
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.05e-35) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-9) {
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.7e+70) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.05e-35:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2e-9:
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re)
	elif y_46_im <= 1.7e+70:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.05e-35)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2e-9)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(x_46_im * y_46_im) / y_46_re) / y_46_re));
	elseif (y_46_im <= 1.7e+70)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.05e-35)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2e-9)
		tmp = (x_46_re / y_46_re) + (((x_46_im * y_46_im) / y_46_re) / y_46_re);
	elseif (y_46_im <= 1.7e+70)
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.05e-35], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2e-9], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+70], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.05000000000000013e-35 or 1.7e70 < y.im

   1. Initial program 51.1%

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

   3. Taylor expanded in y.re around 0 67.2%

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

   if -2.05000000000000013e-35 < y.im < 2.00000000000000012e-9

   1. Initial program 73.3%

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

   3. Taylor expanded in y.re around inf 78.9%

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

      1. associate-/l* 77.4%

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

      2. associate-/r/ 76.8%

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

   5. Simplified 76.8%

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

      1. pow2 76.8%

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

      2. associate-*l/ 78.9%

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

      3. associate-/r* 84.7%

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

   7. Applied egg-rr 84.7%

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

   if 2.00000000000000012e-9 < y.im < 1.7e70

   1. Initial program 55.1%

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

   3. Taylor expanded in y.re around inf 46.3%

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

      1. associate-/l* 56.7%

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

      2. associate-/r/ 56.7%

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

   5. Simplified 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. pow2 56.7%

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

      3. times-frac 75.9%

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

   7. Applied egg-rr 75.9%

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

      1. associate-*l/ 75.9%

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

      2. *-lft-identity 75.9%

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

   9. Simplified 75.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35} \lor \neg \left(y.im \leq 1.75 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.05e-35) (not (<= y.im 1.75e+70)))
   (/ x.im y.im)
   (+ (/ x.re y.re) (* y.im (/ (/ x.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 <= -2.05e-35) || !(y_46_im <= 1.75e+70)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_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 <= (-2.05d-35)) .or. (.not. (y_46im <= 1.75d+70))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re / y_46re) + (y_46im * ((x_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 <= -2.05e-35) || !(y_46_im <= 1.75e+70)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_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 <= -2.05e-35) or not (y_46_im <= 1.75e+70):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_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 <= -2.05e-35) || !(y_46_im <= 1.75e+70))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_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 <= -2.05e-35) || ~((y_46_im <= 1.75e+70)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_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, -2.05e-35], N[Not[LessEqual[y$46$im, 1.75e+70]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.05000000000000013e-35 or 1.75000000000000001e70 < y.im

   1. Initial program 51.1%

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

   3. Taylor expanded in y.re around 0 67.2%

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

   if -2.05000000000000013e-35 < y.im < 1.75000000000000001e70

   1. Initial program 70.7%

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

   3. Taylor expanded in y.re around inf 74.2%

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

      1. associate-/l* 74.4%

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

      2. associate-/r/ 73.9%

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

   5. Simplified 73.9%

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

      1. *-un-lft-identity 73.9%

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

      2. pow2 73.9%

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

      3. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. associate-*l/ 79.5%

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

      2. *-lft-identity 79.5%

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

   9. Simplified 79.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35} \lor \neg \left(y.im \leq 1.75 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35} \lor \neg \left(y.im \leq 1.12 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.05e-35) (not (<= y.im 1.12e+70)))
   (/ x.im y.im)
   (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.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 <= -2.05e-35) || !(y_46_im <= 1.12e+70)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_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 <= (-2.05d-35)) .or. (.not. (y_46im <= 1.12d+70))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_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 <= -2.05e-35) || !(y_46_im <= 1.12e+70)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_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 <= -2.05e-35) or not (y_46_im <= 1.12e+70):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_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 <= -2.05e-35) || !(y_46_im <= 1.12e+70))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(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)
	tmp = 0.0;
	if ((y_46_im <= -2.05e-35) || ~((y_46_im <= 1.12e+70)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (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_] := If[Or[LessEqual[y$46$im, -2.05e-35], N[Not[LessEqual[y$46$im, 1.12e+70]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.05000000000000013e-35 or 1.11999999999999993e70 < y.im

   1. Initial program 51.1%

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

   3. Taylor expanded in y.re around 0 67.2%

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

   if -2.05000000000000013e-35 < y.im < 1.11999999999999993e70

   1. Initial program 70.7%

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

   3. Taylor expanded in y.re around inf 74.2%

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

      1. associate-/l* 74.4%

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

      2. associate-/r/ 73.9%

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

   5. Simplified 73.9%

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

      1. *-un-lft-identity 73.9%

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

      2. pow2 73.9%

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

      3. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

      1. associate-*l/ 79.5%

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

      2. *-lft-identity 79.5%

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

   9. Simplified 79.5%

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

       1. associate-*l/ 80.8%

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

   11. Applied egg-rr 80.8%

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

       1. associate-/l* 80.1%

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

   13. Simplified 80.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-35} \lor \neg \left(y.im \leq 1.12 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-32} \lor \neg \left(y.re \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.5e-32) (not (<= y.re 2.05e+77)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5.5e-32) || !(y_46_re <= 2.05e+77)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-5.5d-32)) .or. (.not. (y_46re <= 2.05d+77))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5.5e-32) || !(y_46_re <= 2.05e+77)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -5.5e-32) or not (y_46_re <= 2.05e+77):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -5.5e-32) || !(y_46_re <= 2.05e+77))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -5.5e-32) || ~((y_46_re <= 2.05e+77)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -5.5e-32], N[Not[LessEqual[y$46$re, 2.05e+77]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.50000000000000024e-32 or 2.05e77 < y.re

   1. Initial program 47.8%

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

   3. Taylor expanded in y.re around inf 68.8%

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

   if -5.50000000000000024e-32 < y.re < 2.05e77

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around 0 61.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-32} \lor \neg \left(y.re \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.8% 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 61.7%

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

3. Taylor expanded in y.re around 0 40.0%

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

4. Final simplification 40.0%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))