_divideComplex, real part

Percentage Accurate: 62.4% → 85.7%

Time: 9.3s

Alternatives: 9

Speedup: 2.1×

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: 62.4% 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: 85.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.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 t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_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):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_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)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(t_0 / 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(1.0 / Float64(y_46_im / Float64(x_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)
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Inf)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_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_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. add-sqr-sqrt 77.6%

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

      3. times-frac 77.5%

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

      4. hypot-def 77.5%

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

      5. fma-def 77.5%

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

      6. hypot-def 94.0%

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

   3. Applied egg-rr 94.0%

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

      1. associate-*l/ 94.3%

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

      2. *-un-lft-identity 94.3%

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

   5. Applied egg-rr 94.3%

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

      1. fma-def 94.3%

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

   7. Applied egg-rr 94.3%

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

   if +inf.0 < (/.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 0.0%

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

   2. Taylor expanded in y.re around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. associate-/l* 48.0%

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

   4. Simplified 48.0%

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

      1. clear-num 48.0%

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

      2. inv-pow 48.0%

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

      3. associate-/l* 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow-1 51.7%

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

      2. associate-/l/ 55.3%

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

   8. Simplified 55.3%

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

4. Final simplification 86.3%

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

Alternative 2: 83.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -1.46 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + t_1\right)\\ \end{array} \end{array} \]

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 (/ y.re x.im))))
   (if (<= y.re -1.46e+85)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -9e-46)
       t_0
       (if (<= y.re 2.6e-159)
         (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
         (if (<= y.re 2.45e+36)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.re 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 = y_46_im / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -1.46e+85) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9e-46) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-159) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 2.45e+36) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + 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)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = y_46_im / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -1.46e+85) {
		tmp = (-x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9e-46) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-159) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 2.45e+36) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re + 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 = y_46_im / (y_46_re / x_46_im)
	tmp = 0
	if y_46_re <= -1.46e+85:
		tmp = (-x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -9e-46:
		tmp = t_0
	elif y_46_re <= 2.6e-159:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 2.45e+36:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re + 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(y_46_im / Float64(y_46_re / x_46_im))
	tmp = 0.0
	if (y_46_re <= -1.46e+85)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -9e-46)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-159)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 2.45e+36)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re + 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 = y_46_im / (y_46_re / x_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.46e+85)
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -9e-46)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-159)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 2.45e+36)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + 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[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.46e+85], N[(N[((-x$46$re) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e-46], t$95$0, If[LessEqual[y$46$re, 2.6e-159], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.45e+36], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.46e85

   1. Initial program 41.5%

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

      1. *-un-lft-identity 41.5%

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

      2. add-sqr-sqrt 41.5%

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

      3. times-frac 41.5%

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

      4. hypot-def 41.5%

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

      5. fma-def 41.5%

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

      6. hypot-def 57.7%

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

   3. Applied egg-rr 57.7%

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

      1. associate-*l/ 57.9%

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

      2. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

   6. Taylor expanded in y.re around -inf 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

      4. mul-1-neg 83.6%

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

      5. associate-/l* 87.7%

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

   8. Simplified 87.7%

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

   if -1.46e85 < y.re < -9.00000000000000001e-46 or 2.5999999999999998e-159 < y.re < 2.4499999999999999e36

   1. Initial program 90.0%

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

   if -9.00000000000000001e-46 < y.re < 2.5999999999999998e-159

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. associate-/l* 77.1%

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

   4. Simplified 77.1%

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

      1. clear-num 77.1%

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

      2. inv-pow 77.1%

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

      3. associate-/l* 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. unpow-1 82.2%

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

      2. associate-/l/ 85.4%

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

   8. Simplified 85.4%

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

   if 2.4499999999999999e36 < y.re

   1. Initial program 43.4%

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

      1. *-un-lft-identity 43.4%

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

      2. add-sqr-sqrt 43.4%

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

      3. times-frac 43.4%

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

      4. hypot-def 43.4%

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

      5. fma-def 43.4%

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

      6. hypot-def 62.3%

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

   3. Applied egg-rr 62.3%

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

   4. Taylor expanded in y.re around inf 78.6%

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

      1. associate-/l* 80.5%

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

   6. Simplified 80.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.46 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 3: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + t_1\right) \cdot \frac{1}{y.re}\\ \end{array} \end{array} \]

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 (/ y.re x.im))))
   (if (<= y.re -2.15e+85)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -1.15e-45)
       t_0
       (if (<= y.re 4.5e-159)
         (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
         (if (<= y.re 1.2e+34) t_0 (* (+ x.re t_1) (/ 1.0 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 / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -2.15e+85) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.15e-45) {
		tmp = t_0;
	} else if (y_46_re <= 4.5e-159) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 1.2e+34) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) * (1.0 / y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -2.15e+85) {
		tmp = (-x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.15e-45) {
		tmp = t_0;
	} else if (y_46_re <= 4.5e-159) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 1.2e+34) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) * (1.0 / y_46_re);
	}
	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 = y_46_im / (y_46_re / x_46_im)
	tmp = 0
	if y_46_re <= -2.15e+85:
		tmp = (-x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -1.15e-45:
		tmp = t_0
	elif y_46_re <= 4.5e-159:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 1.2e+34:
		tmp = t_0
	else:
		tmp = (x_46_re + t_1) * (1.0 / 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(y_46_re / x_46_im))
	tmp = 0.0
	if (y_46_re <= -2.15e+85)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.15e-45)
		tmp = t_0;
	elseif (y_46_re <= 4.5e-159)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 1.2e+34)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + t_1) * Float64(1.0 / y_46_re));
	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 = y_46_im / (y_46_re / x_46_im);
	tmp = 0.0;
	if (y_46_re <= -2.15e+85)
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -1.15e-45)
		tmp = t_0;
	elseif (y_46_re <= 4.5e-159)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 1.2e+34)
		tmp = t_0;
	else
		tmp = (x_46_re + t_1) * (1.0 / y_46_re);
	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[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.15e+85], N[(N[((-x$46$re) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.15e-45], t$95$0, If[LessEqual[y$46$re, 4.5e-159], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.2e+34], t$95$0, N[(N[(x$46$re + t$95$1), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.15e85

   1. Initial program 41.5%

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

      1. *-un-lft-identity 41.5%

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

      2. add-sqr-sqrt 41.5%

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

      3. times-frac 41.5%

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

      4. hypot-def 41.5%

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

      5. fma-def 41.5%

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

      6. hypot-def 57.7%

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

   3. Applied egg-rr 57.7%

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

      1. associate-*l/ 57.9%

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

      2. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

   6. Taylor expanded in y.re around -inf 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

      4. mul-1-neg 83.6%

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

      5. associate-/l* 87.7%

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

   8. Simplified 87.7%

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

   if -2.15e85 < y.re < -1.14999999999999996e-45 or 4.49999999999999989e-159 < y.re < 1.19999999999999993e34

   1. Initial program 90.0%

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

   if -1.14999999999999996e-45 < y.re < 4.49999999999999989e-159

   1. Initial program 63.0%

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

   2. Taylor expanded in y.re around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. associate-/l* 77.1%

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

   4. Simplified 77.1%

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

      1. clear-num 77.1%

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

      2. inv-pow 77.1%

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

      3. associate-/l* 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. unpow-1 82.2%

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

      2. associate-/l/ 85.4%

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

   8. Simplified 85.4%

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

   if 1.19999999999999993e34 < y.re

   1. Initial program 43.4%

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

      1. *-un-lft-identity 43.4%

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

      2. add-sqr-sqrt 43.4%

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

      3. times-frac 43.4%

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

      4. hypot-def 43.4%

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

      5. fma-def 43.4%

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

      6. hypot-def 62.3%

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

   3. Applied egg-rr 62.3%

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

   4. Taylor expanded in y.re around inf 78.6%

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

      1. associate-/l* 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in y.re around inf 80.2%

      \[\leadsto \color{blue}{\frac{1}{y.re}} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 4: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+18} \lor \neg \left(y.im \leq 7 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.1e+18) (not (<= y.im 7e+26)))
   (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.1e+18) || !(y_46_im <= 7e+26)) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.1d+18)) .or. (.not. (y_46im <= 7d+26))) then
        tmp = (x_46im / y_46im) + (1.0d0 / (y_46im / (x_46re * (y_46re / y_46im))))
    else
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.1e+18) || !(y_46_im <= 7e+26)) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.1e+18) or not (y_46_im <= 7e+26):
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	else:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.1e+18) || !(y_46_im <= 7e+26))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.1e+18) || ~((y_46_im <= 7e+26)))
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	else
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.1e+18], N[Not[LessEqual[y$46$im, 7e+26]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1e18 or 6.9999999999999998e26 < y.im

   1. Initial program 48.9%

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

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

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

      1. +-commutative 74.4%

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

      2. unpow2 74.4%

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

      3. associate-/l* 70.8%

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

   4. Simplified 70.8%

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

      1. clear-num 70.8%

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

      2. inv-pow 70.8%

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

      3. associate-/l* 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. unpow-1 73.2%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

   if -4.1e18 < y.im < 6.9999999999999998e26

   1. Initial program 73.4%

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

      1. *-un-lft-identity 73.4%

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

      2. add-sqr-sqrt 73.4%

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

      3. times-frac 73.4%

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

      4. hypot-def 73.4%

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

      5. fma-def 73.4%

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

      6. hypot-def 85.4%

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

   3. Applied egg-rr 85.4%

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

   4. Taylor expanded in y.re around inf 54.2%

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

      1. associate-/l* 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in y.re around inf 86.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+18} \lor \neg \left(y.im \leq 7 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 5: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+38} \lor \neg \left(y.im \leq 4 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2e+38) (not (<= y.im 4e+37)))
   (/ x.im y.im)
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 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 <= -2e+38) || !(y_46_im <= 4e+37)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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 <= (-2d+38)) .or. (.not. (y_46im <= 4d+37))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / 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 <= -2e+38) || !(y_46_im <= 4e+37)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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 <= -2e+38) or not (y_46_im <= 4e+37):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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 <= -2e+38) || !(y_46_im <= 4e+37))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / 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 <= -2e+38) || ~((y_46_im <= 4e+37)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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, -2e+38], N[Not[LessEqual[y$46$im, 4e+37]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.99999999999999995e38 or 3.99999999999999982e37 < y.im

   1. Initial program 47.1%

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

   2. Taylor expanded in y.re around 0 67.1%

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

   if -1.99999999999999995e38 < y.im < 3.99999999999999982e37

   1. Initial program 73.9%

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

      1. *-un-lft-identity 73.9%

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

      2. add-sqr-sqrt 73.8%

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

      3. times-frac 73.8%

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

      4. hypot-def 73.8%

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

      5. fma-def 73.9%

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

      6. hypot-def 85.3%

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

   3. Applied egg-rr 85.3%

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

   4. Taylor expanded in y.re around inf 52.7%

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

      1. associate-/l* 52.1%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in y.re around inf 84.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+38} \lor \neg \left(y.im \leq 4 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+18} \lor \neg \left(y.im \leq 8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.8e+18) (not (<= y.im 8e+26)))
   (+ (/ x.im y.im) (* y.re (/ x.re (* y.im y.im))))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.8e+18) || !(y_46_im <= 8e+26)) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.8d+18)) .or. (.not. (y_46im <= 8d+26))) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im * y_46im)))
    else
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.8e+18) || !(y_46_im <= 8e+26)) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.8e+18) or not (y_46_im <= 8e+26):
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)))
	else:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.8e+18) || !(y_46_im <= 8e+26))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.8e+18) || ~((y_46_im <= 8e+26)))
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	else
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.8e+18], N[Not[LessEqual[y$46$im, 8e+26]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.8e18 or 8.00000000000000038e26 < y.im

   1. Initial program 48.9%

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

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

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

      1. +-commutative 74.4%

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

      2. unpow2 74.4%

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

      3. associate-/l* 70.8%

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

      4. associate-/r/ 76.3%

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

   4. Simplified 76.3%

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

   if -4.8e18 < y.im < 8.00000000000000038e26

   1. Initial program 73.4%

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

      1. *-un-lft-identity 73.4%

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

      2. add-sqr-sqrt 73.4%

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

      3. times-frac 73.4%

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

      4. hypot-def 73.4%

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

      5. fma-def 73.4%

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

      6. hypot-def 85.4%

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

   3. Applied egg-rr 85.4%

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

   4. Taylor expanded in y.re around inf 54.2%

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

      1. associate-/l* 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in y.re around inf 86.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+18} \lor \neg \left(y.im \leq 8 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 7: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+38} \lor \neg \left(y.im \leq 3.3 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.9e+38) (not (<= y.im 3.3e+34)))
   (/ x.im y.im)
   (/ x.re (+ y.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 ((y_46_im <= -1.9e+38) || !(y_46_im <= 3.3e+34)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.9d+38)) .or. (.not. (y_46im <= 3.3d+34))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / (y_46re + ((y_46im * y_46im) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.9e+38) || !(y_46_im <= 3.3e+34)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.9e+38) or not (y_46_im <= 3.3e+34):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / (y_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 ((y_46_im <= -1.9e+38) || !(y_46_im <= 3.3e+34))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.9e+38) || ~((y_46_im <= 3.3e+34)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.9e+38], N[Not[LessEqual[y$46$im, 3.3e+34]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8999999999999999e38 or 3.29999999999999988e34 < y.im

   1. Initial program 47.1%

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

   2. Taylor expanded in y.re around 0 67.1%

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

   if -1.8999999999999999e38 < y.im < 3.29999999999999988e34

   1. Initial program 73.9%

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

      1. *-un-lft-identity 73.9%

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

      2. add-sqr-sqrt 73.8%

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

      3. times-frac 73.8%

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

      4. hypot-def 73.8%

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

      5. fma-def 73.9%

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

      6. hypot-def 85.3%

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

   3. Applied egg-rr 85.3%

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

   4. Taylor expanded in x.re around inf 49.1%

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

      1. associate-/l* 53.8%

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

      2. unpow2 53.8%

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

      3. unpow2 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in y.re around 0 69.4%

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

      1. unpow2 69.4%

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

   9. Simplified 69.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+38} \lor \neg \left(y.im \leq 3.3 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \end{array} \]

Alternative 8: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e+38)
   (/ x.im y.im)
   (if (<= y.im 7e+26) (/ 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_im <= -2.1e+38) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7e+26) {
		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_46im <= (-2.1d+38)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 7d+26) 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_im <= -2.1e+38) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7e+26) {
		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_im <= -2.1e+38:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 7e+26:
		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_im <= -2.1e+38)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 7e+26)
		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_im <= -2.1e+38)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 7e+26)
		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[LessEqual[y$46$im, -2.1e+38], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7e+26], 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.im < -2.1e38 or 6.9999999999999998e26 < y.im

   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. Taylor expanded in y.re around 0 66.3%

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

   if -2.1e38 < y.im < 6.9999999999999998e26

   1. Initial program 74.0%

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

   2. Taylor expanded in y.re around inf 65.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 42.0% 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 61.8%

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

2. Taylor expanded in y.re around 0 38.3%

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

3. Final simplification 38.3%

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

Reproduce

herbie shell --seed 2023230 
(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))))