_divideComplex, real part

Percentage Accurate: 62.2% → 82.6%

Time: 9.8s

Alternatives: 9

Speedup: 0.7×

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: 62.2% 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: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := y.im \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, t_1, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{t_1}{y.re}\\ \end{array} \end{array} \]

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 (* y.im (/ x.im y.re))))
   (if (<= y.re -2.1e+79)
     (fma (/ 1.0 y.re) t_1 (/ x.re y.re))
     (if (<= y.re -2.6e-79)
       t_0
       (if (<= y.re 1.7e-153)
         (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
         (if (<= y.re 1.4e+74) t_0 (+ (/ x.re y.re) (/ t_1 y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_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 = y_46_im * (x_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.1e+79) {
		tmp = fma((1.0 / y_46_re), t_1, (x_46_re / y_46_re));
	} else if (y_46_re <= -2.6e-79) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e-153) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.4e+74) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (t_1 / y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_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(y_46_im * Float64(x_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.1e+79)
		tmp = fma(Float64(1.0 / y_46_re), t_1, Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -2.6e-79)
		tmp = t_0;
	elseif (y_46_re <= 1.7e-153)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 1.4e+74)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(t_1 / y_46_re));
	end
	return 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[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e+79], N[(N[(1.0 / y$46$re), $MachinePrecision] * t$95$1 + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.6e-79], t$95$0, If[LessEqual[y$46$re, 1.7e-153], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+74], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.10000000000000008e79

   1. Initial program 34.2%

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

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

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

      1. associate-/l* 75.4%

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

      2. associate-/r/ 78.7%

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

   5. Simplified 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. pow2 78.7%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. +-commutative 82.1%

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

      2. associate-*l* 84.4%

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

      3. fma-def 84.4%

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

      4. *-commutative 84.4%

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

   9. Applied egg-rr 84.4%

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

   if -2.10000000000000008e79 < y.re < -2.59999999999999994e-79 or 1.6999999999999999e-153 < y.re < 1.40000000000000001e74

   1. Initial program 77.7%

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

   if -2.59999999999999994e-79 < y.re < 1.6999999999999999e-153

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around 0 86.4%

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

      1. associate-/l* 86.0%

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

   5. Simplified 86.0%

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

      1. pow2 86.0%

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

      2. *-un-lft-identity 86.0%

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

      3. times-frac 92.8%

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

   7. Applied egg-rr 92.8%

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

   if 1.40000000000000001e74 < y.re

   1. Initial program 34.3%

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

   3. Taylor expanded in y.re around inf 83.9%

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

      1. associate-/l* 84.2%

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

      2. associate-/r/ 84.2%

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

   5. Simplified 84.2%

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

      1. *-un-lft-identity 84.2%

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

      2. pow2 84.2%

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

      3. times-frac 88.6%

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

   7. Applied egg-rr 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l/ 88.7%

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

      3. *-un-lft-identity 88.7%

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

      4. associate-*r/ 90.9%

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, y.im \cdot \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

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+290)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((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+290) {
		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_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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+290)
		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_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$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+290], 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$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $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))) < 4.9999999999999998e290

   1. Initial program 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.9%

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

      4. hypot-def 78.9%

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

      5. fma-def 78.9%

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

      6. hypot-def 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. associate-*l/ 95.3%

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

      2. *-un-lft-identity 95.3%

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

   6. Applied egg-rr 95.3%

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

   if 4.9999999999999998e290 < (/.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 6.3%

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

   3. Taylor expanded in y.re around 0 48.3%

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

      1. associate-/l* 50.2%

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

   5. Simplified 50.2%

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

      1. pow2 50.2%

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

      2. *-un-lft-identity 50.2%

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

      3. times-frac 56.0%

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

   7. Applied egg-rr 56.0%

      \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{1} \cdot \frac{y.im}{y.re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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) (/ (* y.im (/ x.im y.re)) y.re))))
   (if (<= y.re -1.75e+27)
     t_1
     (if (<= y.re -2.6e-79)
       t_0
       (if (<= y.re 2.1e-155)
         (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
         (if (<= y.re 1.25e+69) 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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -1.75e+27) {
		tmp = t_1;
	} else if (y_46_re <= -2.6e-79) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e-155) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.25e+69) {
		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) + ((y_46im * (x_46im / y_46re)) / y_46re)
    if (y_46re <= (-1.75d+27)) then
        tmp = t_1
    else if (y_46re <= (-2.6d-79)) then
        tmp = t_0
    else if (y_46re <= 2.1d-155) then
        tmp = (x_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    else if (y_46re <= 1.25d+69) 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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -1.75e+27) {
		tmp = t_1;
	} else if (y_46_re <= -2.6e-79) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e-155) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.25e+69) {
		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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -1.75e+27:
		tmp = t_1
	elif y_46_re <= -2.6e-79:
		tmp = t_0
	elif y_46_re <= 2.1e-155:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	elif y_46_re <= 1.25e+69:
		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(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.75e+27)
		tmp = t_1;
	elseif (y_46_re <= -2.6e-79)
		tmp = t_0;
	elseif (y_46_re <= 2.1e-155)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 1.25e+69)
		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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.75e+27)
		tmp = t_1;
	elseif (y_46_re <= -2.6e-79)
		tmp = t_0;
	elseif (y_46_re <= 2.1e-155)
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	elseif (y_46_re <= 1.25e+69)
		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[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e+27], t$95$1, If[LessEqual[y$46$re, -2.6e-79], t$95$0, If[LessEqual[y$46$re, 2.1e-155], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+69], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.7500000000000001e27 or 1.25000000000000009e69 < y.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.re around inf 74.4%

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

      1. associate-/l* 76.4%

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

      2. associate-/r/ 78.9%

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

   5. Simplified 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. pow2 78.9%

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

      3. times-frac 82.3%

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

   7. Applied egg-rr 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*l/ 82.3%

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

      3. *-un-lft-identity 82.3%

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

      4. associate-*r/ 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -1.7500000000000001e27 < y.re < -2.59999999999999994e-79 or 2.1000000000000002e-155 < y.re < 1.25000000000000009e69

   1. Initial program 81.1%

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

   if -2.59999999999999994e-79 < y.re < 2.1000000000000002e-155

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around 0 86.4%

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

      1. associate-/l* 86.0%

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

   5. Simplified 86.0%

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

      1. pow2 86.0%

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

      2. *-un-lft-identity 86.0%

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

      3. times-frac 92.8%

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

   7. Applied egg-rr 92.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1050000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re)))))
        (t_1 (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))
   (if (<= y.re -2.3e-7)
     t_1
     (if (<= y.re -2.9e-36)
       t_0
       (if (<= y.re -3.4e-78)
         (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 1050000000.0) 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_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.3e-7) {
		tmp = t_1;
	} else if (y_46_re <= -2.9e-36) {
		tmp = t_0;
	} else if (y_46_re <= -3.4e-78) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1050000000.0) {
		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_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    t_1 = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    if (y_46re <= (-2.3d-7)) then
        tmp = t_1
    else if (y_46re <= (-2.9d-36)) then
        tmp = t_0
    else if (y_46re <= (-3.4d-78)) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1050000000.0d0) 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_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.3e-7) {
		tmp = t_1;
	} else if (y_46_re <= -2.9e-36) {
		tmp = t_0;
	} else if (y_46_re <= -3.4e-78) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1050000000.0) {
		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_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -2.3e-7:
		tmp = t_1
	elif y_46_re <= -2.9e-36:
		tmp = t_0
	elif y_46_re <= -3.4e-78:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1050000000.0:
		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(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.3e-7)
		tmp = t_1;
	elseif (y_46_re <= -2.9e-36)
		tmp = t_0;
	elseif (y_46_re <= -3.4e-78)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1050000000.0)
		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_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.3e-7)
		tmp = t_1;
	elseif (y_46_re <= -2.9e-36)
		tmp = t_0;
	elseif (y_46_re <= -3.4e-78)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1050000000.0)
		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[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e-7], t$95$1, If[LessEqual[y$46$re, -2.9e-36], t$95$0, If[LessEqual[y$46$re, -3.4e-78], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1050000000.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.29999999999999995e-7 or 1.05e9 < y.re

   1. Initial program 43.5%

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

   3. Taylor expanded in y.re around inf 72.6%

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

      1. associate-/l* 74.3%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. *-un-lft-identity 76.5%

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

      2. pow2 76.5%

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

      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. *-commutative 79.5%

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

      2. associate-*l/ 79.5%

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

      3. *-un-lft-identity 79.5%

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

      4. associate-*r/ 81.2%

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

   9. Applied egg-rr 81.2%

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

   if -2.29999999999999995e-7 < y.re < -2.90000000000000013e-36 or -3.40000000000000012e-78 < y.re < 1.05e9

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around 0 83.7%

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

      1. associate-/l* 82.5%

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

   5. Simplified 82.5%

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

      1. pow2 82.5%

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

      2. *-un-lft-identity 82.5%

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

      3. times-frac 87.7%

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

   7. Applied egg-rr 87.7%

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

   if -2.90000000000000013e-36 < y.re < -3.40000000000000012e-78

   1. Initial program 99.6%

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

   3. Taylor expanded in x.re around inf 76.5%

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

      1. *-commutative 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1050000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 750000:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))
   (if (<= y.re -2.4e-78)
     t_0
     (if (<= y.re 3e-153)
       (/ x.im y.im)
       (if (<= y.re 750000.0)
         (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.4e-78) {
		tmp = t_0;
	} else if (y_46_re <= 3e-153) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 750000.0) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} 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 = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    if (y_46re <= (-2.4d-78)) then
        tmp = t_0
    else if (y_46re <= 3d-153) then
        tmp = x_46im / y_46im
    else if (y_46re <= 750000.0d0) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    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 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -2.4e-78) {
		tmp = t_0;
	} else if (y_46_re <= 3e-153) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 750000.0) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	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) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -2.4e-78:
		tmp = t_0
	elif y_46_re <= 3e-153:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 750000.0:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	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(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.4e-78)
		tmp = t_0;
	elseif (y_46_re <= 3e-153)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 750000.0)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	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 = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.4e-78)
		tmp = t_0;
	elseif (y_46_re <= 3e-153)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 750000.0)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.4e-78], t$95$0, If[LessEqual[y$46$re, 3e-153], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 750000.0], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.4e-78 or 7.5e5 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in y.re around inf 70.1%

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

      1. associate-/l* 71.6%

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

      2. associate-/r/ 72.9%

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

   5. Simplified 72.9%

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

      1. *-un-lft-identity 72.9%

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

      2. pow2 72.9%

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

      3. times-frac 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*l/ 75.5%

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

      3. *-un-lft-identity 75.5%

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

      4. associate-*r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -2.4e-78 < y.re < 3e-153

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around 0 85.2%

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

   if 3e-153 < y.re < 7.5e5

   1. Initial program 83.2%

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

   3. Taylor expanded in x.re around inf 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 750000:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+99} \lor \neg \left(y.re \leq -2.7 \cdot 10^{-36} \lor \neg \left(y.re \leq -3.5 \cdot 10^{-78}\right) \land y.re \leq 2.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.9e+99)
         (not
          (or (<= y.re -2.7e-36)
              (and (not (<= y.re -3.5e-78)) (<= y.re 2.7e+54)))))
   (/ 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 <= -1.9e+99) || !((y_46_re <= -2.7e-36) || (!(y_46_re <= -3.5e-78) && (y_46_re <= 2.7e+54)))) {
		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 <= (-1.9d+99)) .or. (.not. (y_46re <= (-2.7d-36)) .or. (.not. (y_46re <= (-3.5d-78))) .and. (y_46re <= 2.7d+54))) 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 <= -1.9e+99) || !((y_46_re <= -2.7e-36) || (!(y_46_re <= -3.5e-78) && (y_46_re <= 2.7e+54)))) {
		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 <= -1.9e+99) or not ((y_46_re <= -2.7e-36) or (not (y_46_re <= -3.5e-78) and (y_46_re <= 2.7e+54))):
		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 <= -1.9e+99) || !((y_46_re <= -2.7e-36) || (!(y_46_re <= -3.5e-78) && (y_46_re <= 2.7e+54))))
		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 <= -1.9e+99) || ~(((y_46_re <= -2.7e-36) || (~((y_46_re <= -3.5e-78)) && (y_46_re <= 2.7e+54)))))
		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, -1.9e+99], N[Not[Or[LessEqual[y$46$re, -2.7e-36], And[N[Not[LessEqual[y$46$re, -3.5e-78]], $MachinePrecision], LessEqual[y$46$re, 2.7e+54]]]], $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 < -1.9e99 or -2.70000000000000007e-36 < y.re < -3.4999999999999999e-78 or 2.70000000000000011e54 < y.re

   1. Initial program 41.9%

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

   3. Taylor expanded in y.re around inf 79.4%

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

   if -1.9e99 < y.re < -2.70000000000000007e-36 or -3.4999999999999999e-78 < y.re < 2.70000000000000011e54

   1. Initial program 72.3%

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

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

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+99} \lor \neg \left(y.re \leq -2.7 \cdot 10^{-36} \lor \neg \left(y.re \leq -3.5 \cdot 10^{-78}\right) \land y.re \leq 2.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-78} \lor \neg \left(y.re \leq 6.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.2e-78) (not (<= y.re 6.2e+27)))
   (+ (/ 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_re <= -3.2e-78) || !(y_46_re <= 6.2e+27)) {
		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_46re <= (-3.2d-78)) .or. (.not. (y_46re <= 6.2d+27))) 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_re <= -3.2e-78) || !(y_46_re <= 6.2e+27)) {
		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_re <= -3.2e-78) or not (y_46_re <= 6.2e+27):
		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_re <= -3.2e-78) || !(y_46_re <= 6.2e+27))
		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_re <= -3.2e-78) || ~((y_46_re <= 6.2e+27)))
		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[Or[LessEqual[y$46$re, -3.2e-78], N[Not[LessEqual[y$46$re, 6.2e+27]], $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], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.2e-78 or 6.19999999999999992e27 < y.re

   1. Initial program 47.6%

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

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

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

      1. associate-/l* 72.4%

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

      2. associate-/r/ 73.7%

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

   5. Simplified 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. pow2 73.7%

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

      3. times-frac 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-*l/ 76.4%

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

      2. *-un-lft-identity 76.4%

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

   9. Applied egg-rr 76.4%

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

   if -3.2e-78 < y.re < 6.19999999999999992e27

   1. Initial program 74.8%

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

   3. Taylor expanded in y.re around 0 74.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-78} \lor \neg \left(y.re \leq 6.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-78} \lor \neg \left(y.re \leq 4.15 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.4e-78) (not (<= y.re 4.15e+27)))
   (+ (/ 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_re <= -3.4e-78) || !(y_46_re <= 4.15e+27)) {
		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_46re <= (-3.4d-78)) .or. (.not. (y_46re <= 4.15d+27))) 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_re <= -3.4e-78) || !(y_46_re <= 4.15e+27)) {
		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_re <= -3.4e-78) or not (y_46_re <= 4.15e+27):
		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_re <= -3.4e-78) || !(y_46_re <= 4.15e+27))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * 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_re <= -3.4e-78) || ~((y_46_re <= 4.15e+27)))
		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[Or[LessEqual[y$46$re, -3.4e-78], N[Not[LessEqual[y$46$re, 4.15e+27]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.40000000000000012e-78 or 4.1499999999999998e27 < y.re

   1. Initial program 47.6%

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

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

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

      1. associate-/l* 72.4%

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

      2. associate-/r/ 73.7%

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

   5. Simplified 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. pow2 73.7%

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

      3. times-frac 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l/ 76.4%

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

      3. *-un-lft-identity 76.4%

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

      4. associate-*r/ 77.9%

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

   9. Applied egg-rr 77.9%

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

   if -3.40000000000000012e-78 < y.re < 4.1499999999999998e27

   1. Initial program 74.8%

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

   3. Taylor expanded in y.re around 0 74.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-78} \lor \neg \left(y.re \leq 4.15 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% 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 59.6%

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

3. Taylor expanded in y.re around 0 45.8%

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

4. Final simplification 45.8%

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

Reproduce

herbie shell --seed 2024021 
(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))))