_divideComplex, real part

Percentage Accurate: 62.0% → 85.5%

Time: 10.1s

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.0× speedup

Math

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

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)))
      INFINITY)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (* (/ y.re (hypot y.re y.im)) (/ x.re (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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))) <= ((double) INFINITY)) {
		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 = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], 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[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. add-sqr-sqrt 79.7%

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

      3. times-frac 79.6%

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

      4. hypot-def 79.7%

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

      5. fma-def 79.7%

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

      6. hypot-def 96.4%

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

   3. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   if +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 x.re around inf 1.3%

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

      1. *-commutative 1.3%

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

   4. Simplified 1.3%

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

      1. add-sqr-sqrt 1.3%

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

      2. sqrt-div 0.9%

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

      3. hypot-udef 0.9%

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

      4. sqrt-div 0.9%

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

      5. hypot-udef 1.9%

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

      6. times-frac 0.9%

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

      7. add-sqr-sqrt 1.3%

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

      8. times-frac 61.8%

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

   6. Applied egg-rr 61.8%

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

4. Final simplification 90.1%

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

Alternative 2: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.25e+126)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (if (<= y.re -8.2e-141)
       t_0
       (if (<= y.re 1.05e-119)
         (+ (/ x.im y.im) (/ 1.0 (/ y.im (* y.re (/ x.re y.im)))))
         (if (<= y.re 2.2e+130)
           t_0
           (/ (+ x.re (/ y.im (/ y.re x.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -1.25e+126) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= -8.2e-141) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	} else if (y_46_re <= 2.2e+130) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -1.25e+126) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= -8.2e-141) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-119) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	} else if (y_46_re <= 2.2e+130) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.25e+126:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_re <= -8.2e-141:
		tmp = t_0
	elif y_46_re <= 1.05e-119:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))))
	elif y_46_re <= 2.2e+130:
		tmp = t_0
	else:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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)))
	tmp = 0.0
	if (y_46_re <= -1.25e+126)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= -8.2e-141)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(y_46_re * Float64(x_46_re / y_46_im)))));
	elseif (y_46_re <= 2.2e+130)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.25e+126)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_re <= -8.2e-141)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-119)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	elseif (y_46_re <= 2.2e+130)
		tmp = t_0;
	else
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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]}, If[LessEqual[y$46$re, -1.25e+126], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.2e-141], t$95$0, If[LessEqual[y$46$re, 1.05e-119], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+130], t$95$0, N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.24999999999999994e126

   1. Initial program 47.8%

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

   2. Taylor expanded in y.re around inf 77.1%

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

      1. unpow2 77.1%

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

      2. times-frac 88.6%

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

   4. Simplified 88.6%

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

   if -1.24999999999999994e126 < y.re < -8.20000000000000005e-141 or 1.05e-119 < y.re < 2.19999999999999993e130

   1. Initial program 82.0%

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

   if -8.20000000000000005e-141 < y.re < 1.05e-119

   1. Initial program 64.5%

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

   2. Taylor expanded in y.re around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. *-commutative 80.0%

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

      3. unpow2 80.0%

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

      4. associate-/l* 76.5%

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

   4. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

      3. associate-/l* 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. unpow-1 80.3%

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

      2. associate-/l/ 92.3%

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

   8. Simplified 92.3%

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

   if 2.19999999999999993e130 < y.re

   1. Initial program 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. add-sqr-sqrt 37.5%

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

      3. times-frac 37.5%

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

      4. hypot-def 37.5%

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

      5. fma-def 37.5%

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

      6. hypot-def 55.8%

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

   3. Applied egg-rr 55.8%

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

      1. associate-*l/ 55.8%

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

      2. *-un-lft-identity 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 88.9%

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

   8. Simplified 88.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.05 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* 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 -4.4e+109)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -2.05e-141)
       t_0
       (if (<= y.re 8e-122)
         (+ (/ x.im y.im) (/ 1.0 (/ y.im (* y.re (/ x.re y.im)))))
         (if (<= y.re 1.1e+131) t_0 (/ (+ x.re t_1) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 <= -4.4e+109) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -2.05e-141) {
		tmp = t_0;
	} else if (y_46_re <= 8e-122) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	} else if (y_46_re <= 1.1e+131) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 <= -4.4e+109) {
		tmp = (-x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -2.05e-141) {
		tmp = t_0;
	} else if (y_46_re <= 8e-122) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	} else if (y_46_re <= 1.1e+131) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((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 <= -4.4e+109:
		tmp = (-x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -2.05e-141:
		tmp = t_0
	elif y_46_re <= 8e-122:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))))
	elif y_46_re <= 1.1e+131:
		tmp = t_0
	else:
		tmp = (x_46_re + t_1) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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 <= -4.4e+109)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -2.05e-141)
		tmp = t_0;
	elseif (y_46_re <= 8e-122)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(y_46_re * Float64(x_46_re / y_46_im)))));
	elseif (y_46_re <= 1.1e+131)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + t_1) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((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 <= -4.4e+109)
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -2.05e-141)
		tmp = t_0;
	elseif (y_46_re <= 8e-122)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	elseif (y_46_re <= 1.1e+131)
		tmp = t_0;
	else
		tmp = (x_46_re + t_1) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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, -4.4e+109], 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, -2.05e-141], t$95$0, If[LessEqual[y$46$re, 8e-122], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.1e+131], t$95$0, N[(N[(x$46$re + t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.3999999999999998e109

   1. Initial program 52.0%

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

      1. *-un-lft-identity 52.0%

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

      2. add-sqr-sqrt 52.0%

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

      3. times-frac 52.0%

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

      4. hypot-def 52.0%

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

      5. fma-def 52.0%

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

      6. hypot-def 64.7%

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

   3. Applied egg-rr 64.7%

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

      1. associate-*l/ 64.9%

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

      2. *-un-lft-identity 64.9%

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

   5. Applied egg-rr 64.9%

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

   6. Taylor expanded in y.re around -inf 85.3%

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

      1. +-commutative 85.3%

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

      2. mul-1-neg 85.3%

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

      3. unsub-neg 85.3%

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

      4. neg-mul-1 85.3%

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

      5. associate-/l* 88.0%

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

   8. Simplified 88.0%

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

   if -4.3999999999999998e109 < y.re < -2.05000000000000001e-141 or 8.00000000000000047e-122 < y.re < 1.0999999999999999e131

   1. Initial program 82.1%

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

   if -2.05000000000000001e-141 < y.re < 8.00000000000000047e-122

   1. Initial program 64.5%

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

   2. Taylor expanded in y.re around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. *-commutative 80.0%

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

      3. unpow2 80.0%

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

      4. associate-/l* 76.5%

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

   4. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

      3. associate-/l* 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. unpow-1 80.3%

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

      2. associate-/l/ 92.3%

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

   8. Simplified 92.3%

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

   if 1.0999999999999999e131 < y.re

   1. Initial program 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. add-sqr-sqrt 37.5%

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

      3. times-frac 37.5%

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

      4. hypot-def 37.5%

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

      5. fma-def 37.5%

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

      6. hypot-def 55.8%

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

   3. Applied egg-rr 55.8%

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

      1. associate-*l/ 55.8%

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

      2. *-un-lft-identity 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 88.9%

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

   8. Simplified 88.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.05 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ t_1 := \frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im)))))
        (t_1 (+ (/ x.im y.im) (/ 1.0 (/ y.im (* y.re (/ x.re y.im)))))))
   (if (<= y.im -2e+120)
     t_1
     (if (<= y.im -1.5e+100)
       t_0
       (if (<= y.im -1.55e-23)
         (* x.im (/ y.im (+ (* y.re y.re) (* y.im y.im))))
         (if (<= y.im 1.55e+28) 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_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	double tmp;
	if (y_46_im <= -2e+120) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e+100) {
		tmp = t_0;
	} else if (y_46_im <= -1.55e-23) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1.55e+28) {
		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_46re * (y_46re / y_46im)))
    t_1 = (x_46im / y_46im) + (1.0d0 / (y_46im / (y_46re * (x_46re / y_46im))))
    if (y_46im <= (-2d+120)) then
        tmp = t_1
    else if (y_46im <= (-1.5d+100)) then
        tmp = t_0
    else if (y_46im <= (-1.55d-23)) then
        tmp = x_46im * (y_46im / ((y_46re * y_46re) + (y_46im * y_46im)))
    else if (y_46im <= 1.55d+28) 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_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	double tmp;
	if (y_46_im <= -2e+120) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e+100) {
		tmp = t_0;
	} else if (y_46_im <= -1.55e-23) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1.55e+28) {
		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_re * (y_46_re / y_46_im)))
	t_1 = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))))
	tmp = 0
	if y_46_im <= -2e+120:
		tmp = t_1
	elif y_46_im <= -1.5e+100:
		tmp = t_0
	elif y_46_im <= -1.55e-23:
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	elif y_46_im <= 1.55e+28:
		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_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(y_46_re * Float64(x_46_re / y_46_im)))))
	tmp = 0.0
	if (y_46_im <= -2e+120)
		tmp = t_1;
	elseif (y_46_im <= -1.5e+100)
		tmp = t_0;
	elseif (y_46_im <= -1.55e-23)
		tmp = Float64(x_46_im * Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 1.55e+28)
		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_re * (y_46_re / y_46_im)));
	t_1 = (x_46_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	tmp = 0.0;
	if (y_46_im <= -2e+120)
		tmp = t_1;
	elseif (y_46_im <= -1.5e+100)
		tmp = t_0;
	elseif (y_46_im <= -1.55e-23)
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	elseif (y_46_im <= 1.55e+28)
		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$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2e+120], t$95$1, If[LessEqual[y$46$im, -1.5e+100], t$95$0, If[LessEqual[y$46$im, -1.55e-23], N[(x$46$im * N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.55e+28], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2e120 or 1.55e28 < y.im

   1. Initial program 50.7%

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

   2. Taylor expanded in y.re around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. *-commutative 75.8%

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

      3. unpow2 75.8%

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

      4. associate-/l* 78.5%

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

   4. Simplified 78.5%

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

      1. clear-num 78.5%

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

      2. inv-pow 78.5%

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

      3. associate-/l* 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. unpow-1 83.4%

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

      2. associate-/l/ 89.6%

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

   8. Simplified 89.6%

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

   if -2e120 < y.im < -1.49999999999999993e100 or -1.5499999999999999e-23 < y.im < 1.55e28

   1. Initial program 70.2%

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

   2. Taylor expanded in y.re around inf 76.3%

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

      1. *-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. associate-/l* 74.3%

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

   4. Simplified 74.3%

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

   5. Taylor expanded in y.re around 0 74.3%

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

      1. unpow2 74.3%

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

      2. associate-*r/ 81.2%

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

   7. Simplified 81.2%

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

   if -1.49999999999999993e100 < y.im < -1.5499999999999999e-23

   1. Initial program 89.8%

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

   2. Taylor expanded in x.re around 0 85.4%

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

      1. associate-/l* 89.7%

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

      2. associate-/r/ 89.9%

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

      3. unpow2 89.9%

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

      4. unpow2 89.9%

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

      5. fma-udef 89.9%

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

   4. Simplified 89.9%

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

      1. fma-def 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+120}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ t_1 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-23}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re))))
        (t_1 (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))))
   (if (<= y.im -6e+141)
     t_1
     (if (<= y.im -1.5e+100)
       t_0
       (if (<= y.im -1.45e-23)
         (* x.im (/ y.im (+ (* y.re y.re) (* y.im y.im))))
         (if (<= y.im 4.7e+30) 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) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6e+141) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e+100) {
		tmp = t_0;
	} else if (y_46_im <= -1.45e-23) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 4.7e+30) {
		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) + ((y_46im / y_46re) * (x_46im / y_46re))
    t_1 = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    if (y_46im <= (-6d+141)) then
        tmp = t_1
    else if (y_46im <= (-1.5d+100)) then
        tmp = t_0
    else if (y_46im <= (-1.45d-23)) then
        tmp = x_46im * (y_46im / ((y_46re * y_46re) + (y_46im * y_46im)))
    else if (y_46im <= 4.7d+30) 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) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6e+141) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e+100) {
		tmp = t_0;
	} else if (y_46_im <= -1.45e-23) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 4.7e+30) {
		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) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -6e+141:
		tmp = t_1
	elif y_46_im <= -1.5e+100:
		tmp = t_0
	elif y_46_im <= -1.45e-23:
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	elif y_46_im <= 4.7e+30:
		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_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -6e+141)
		tmp = t_1;
	elseif (y_46_im <= -1.5e+100)
		tmp = t_0;
	elseif (y_46_im <= -1.45e-23)
		tmp = Float64(x_46_im * Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 4.7e+30)
		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) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -6e+141)
		tmp = t_1;
	elseif (y_46_im <= -1.5e+100)
		tmp = t_0;
	elseif (y_46_im <= -1.45e-23)
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	elseif (y_46_im <= 4.7e+30)
		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$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+141], t$95$1, If[LessEqual[y$46$im, -1.5e+100], t$95$0, If[LessEqual[y$46$im, -1.45e-23], N[(x$46$im * N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.7e+30], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.9999999999999998e141 or 4.6999999999999999e30 < y.im

   1. Initial program 50.7%

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

   2. Taylor expanded in y.re around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. *-commutative 76.9%

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

      3. unpow2 76.9%

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

      4. times-frac 90.1%

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

   4. Simplified 90.1%

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

   if -5.9999999999999998e141 < y.im < -1.49999999999999993e100 or -1.4500000000000001e-23 < y.im < 4.6999999999999999e30

   1. Initial program 69.7%

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

   2. Taylor expanded in y.re around inf 75.0%

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

      1. unpow2 75.0%

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

      2. times-frac 79.2%

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

   4. Simplified 79.2%

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

   if -1.49999999999999993e100 < y.im < -1.4500000000000001e-23

   1. Initial program 89.8%

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

   2. Taylor expanded in x.re around 0 85.4%

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

      1. associate-/l* 89.7%

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

      2. associate-/r/ 89.9%

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

      3. unpow2 89.9%

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

      4. unpow2 89.9%

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

      5. fma-udef 89.9%

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

   4. Simplified 89.9%

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

      1. fma-def 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-23}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 6: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ t_1 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-21}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im)))))
        (t_1 (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))))
   (if (<= y.im -6e+141)
     t_1
     (if (<= y.im -6.8e+99)
       t_0
       (if (<= y.im -7e-21)
         (* x.im (/ y.im (+ (* y.re y.re) (* y.im y.im))))
         (if (<= y.im 1.9e+29) 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_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6e+141) {
		tmp = t_1;
	} else if (y_46_im <= -6.8e+99) {
		tmp = t_0;
	} else if (y_46_im <= -7e-21) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1.9e+29) {
		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_46re * (y_46re / y_46im)))
    t_1 = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    if (y_46im <= (-6d+141)) then
        tmp = t_1
    else if (y_46im <= (-6.8d+99)) then
        tmp = t_0
    else if (y_46im <= (-7d-21)) then
        tmp = x_46im * (y_46im / ((y_46re * y_46re) + (y_46im * y_46im)))
    else if (y_46im <= 1.9d+29) 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_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6e+141) {
		tmp = t_1;
	} else if (y_46_im <= -6.8e+99) {
		tmp = t_0;
	} else if (y_46_im <= -7e-21) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1.9e+29) {
		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_re * (y_46_re / y_46_im)))
	t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -6e+141:
		tmp = t_1
	elif y_46_im <= -6.8e+99:
		tmp = t_0
	elif y_46_im <= -7e-21:
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	elif y_46_im <= 1.9e+29:
		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_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -6e+141)
		tmp = t_1;
	elseif (y_46_im <= -6.8e+99)
		tmp = t_0;
	elseif (y_46_im <= -7e-21)
		tmp = Float64(x_46_im * Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 1.9e+29)
		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_re * (y_46_re / y_46_im)));
	t_1 = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -6e+141)
		tmp = t_1;
	elseif (y_46_im <= -6.8e+99)
		tmp = t_0;
	elseif (y_46_im <= -7e-21)
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	elseif (y_46_im <= 1.9e+29)
		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$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+141], t$95$1, If[LessEqual[y$46$im, -6.8e+99], t$95$0, If[LessEqual[y$46$im, -7e-21], N[(x$46$im * N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.9e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.9999999999999998e141 or 1.89999999999999985e29 < y.im

   1. Initial program 50.7%

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

   2. Taylor expanded in y.re around 0 76.9%

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

      1. +-commutative 76.9%

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

      2. *-commutative 76.9%

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

      3. unpow2 76.9%

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

      4. times-frac 90.1%

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

   4. Simplified 90.1%

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

   if -5.9999999999999998e141 < y.im < -6.79999999999999968e99 or -7.0000000000000007e-21 < y.im < 1.89999999999999985e29

   1. Initial program 69.7%

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

   2. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. unpow2 75.0%

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

      3. associate-/l* 73.0%

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

   4. Simplified 73.0%

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

   5. Taylor expanded in y.re around 0 73.0%

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

      1. unpow2 73.0%

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

      2. associate-*r/ 80.4%

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

   7. Simplified 80.4%

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

   if -6.79999999999999968e99 < y.im < -7.0000000000000007e-21

   1. Initial program 89.8%

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

   2. Taylor expanded in x.re around 0 85.4%

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

      1. associate-/l* 89.7%

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

      2. associate-/r/ 89.9%

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

      3. unpow2 89.9%

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

      4. unpow2 89.9%

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

      5. fma-udef 89.9%

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

   4. Simplified 89.9%

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

      1. fma-def 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-21}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 7: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ 1.0 (/ y.im (* y.re (/ x.re y.im)))))))
   (if (<= y.im -1.5e+120)
     t_0
     (if (<= y.im -8.5e-117)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 2.3e+28)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re 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_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	double tmp;
	if (y_46_im <= -1.5e+120) {
		tmp = t_0;
	} else if (y_46_im <= -8.5e-117) {
		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));
	} else if (y_46_im <= 2.3e+28) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / 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_46im / y_46im) + (1.0d0 / (y_46im / (y_46re * (x_46re / y_46im))))
    if (y_46im <= (-1.5d+120)) then
        tmp = t_0
    else if (y_46im <= (-8.5d-117)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 2.3d+28) then
        tmp = (x_46re / y_46re) + (x_46im / (y_46re * (y_46re / 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_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	double tmp;
	if (y_46_im <= -1.5e+120) {
		tmp = t_0;
	} else if (y_46_im <= -8.5e-117) {
		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));
	} else if (y_46_im <= 2.3e+28) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / 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_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))))
	tmp = 0
	if y_46_im <= -1.5e+120:
		tmp = t_0
	elif y_46_im <= -8.5e-117:
		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))
	elif y_46_im <= 2.3e+28:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / 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_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(y_46_re * Float64(x_46_re / y_46_im)))))
	tmp = 0.0
	if (y_46_im <= -1.5e+120)
		tmp = t_0;
	elseif (y_46_im <= -8.5e-117)
		tmp = 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)));
	elseif (y_46_im <= 2.3e+28)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / 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_im / y_46_im) + (1.0 / (y_46_im / (y_46_re * (x_46_re / y_46_im))));
	tmp = 0.0;
	if (y_46_im <= -1.5e+120)
		tmp = t_0;
	elseif (y_46_im <= -8.5e-117)
		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));
	elseif (y_46_im <= 2.3e+28)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / 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$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.5e+120], t$95$0, If[LessEqual[y$46$im, -8.5e-117], 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], If[LessEqual[y$46$im, 2.3e+28], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.5e120 or 2.29999999999999984e28 < y.im

   1. Initial program 50.2%

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

   2. Taylor expanded in y.re around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. *-commutative 75.0%

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

      3. unpow2 75.0%

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

      4. associate-/l* 77.7%

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

   4. Simplified 77.7%

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

      1. clear-num 77.7%

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

      2. inv-pow 77.7%

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

      3. associate-/l* 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. unpow-1 82.6%

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

      2. associate-/l/ 88.7%

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

   8. Simplified 88.7%

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

   if -1.5e120 < y.im < -8.49999999999999981e-117

   1. Initial program 82.3%

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

   if -8.49999999999999981e-117 < y.im < 2.29999999999999984e28

   1. Initial program 69.5%

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

   2. Taylor expanded in y.re around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. associate-/l* 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in y.re around 0 76.7%

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

      1. unpow2 76.7%

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

      2. associate-*r/ 83.6%

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

   7. Simplified 83.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re \cdot \frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 8: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1700:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.8e+120)
   (/ x.im y.im)
   (if (<= y.im -1.35e-127)
     (* x.im (/ y.im (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 1700.0) (/ 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 <= -1.8e+120) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.35e-127) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1700.0) {
		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 <= (-1.8d+120)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-1.35d-127)) then
        tmp = x_46im * (y_46im / ((y_46re * y_46re) + (y_46im * y_46im)))
    else if (y_46im <= 1700.0d0) 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 <= -1.8e+120) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.35e-127) {
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	} else if (y_46_im <= 1700.0) {
		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 <= -1.8e+120:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -1.35e-127:
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	elif y_46_im <= 1700.0:
		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 <= -1.8e+120)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.35e-127)
		tmp = Float64(x_46_im * Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 1700.0)
		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 <= -1.8e+120)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -1.35e-127)
		tmp = x_46_im * (y_46_im / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	elseif (y_46_im <= 1700.0)
		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, -1.8e+120], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.35e-127], N[(x$46$im * N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1700.0], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.80000000000000008e120 or 1700 < y.im

   1. Initial program 52.2%

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

   2. Taylor expanded in y.re around 0 67.6%

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

   if -1.80000000000000008e120 < y.im < -1.35e-127

   1. Initial program 83.1%

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

   2. Taylor expanded in x.re around 0 65.4%

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

      1. associate-/l* 59.4%

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

      2. associate-/r/ 64.8%

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

      3. unpow2 64.8%

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

      4. unpow2 64.8%

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

      5. fma-udef 64.8%

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

   4. Simplified 64.8%

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

      1. fma-def 64.8%

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

   6. Applied egg-rr 64.8%

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

   if -1.35e-127 < y.im < 1700

   1. Initial program 67.9%

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

   2. Taylor expanded in y.re around inf 67.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-127}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1700:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.2e+46)
   (/ x.re y.re)
   (if (<= y.re 1.4e+82)
     (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+46) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.4e+82) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.2d+46)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.4d+82) then
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+46) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.4e+82) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.2e+46:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.4e+82:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.2e+46)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.4e+82)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.2e+46)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.4e+82)
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.2e+46], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+82], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.20000000000000004e46 or 1.4e82 < y.re

   1. Initial program 52.5%

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

   2. Taylor expanded in y.re around inf 72.0%

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

   if -1.20000000000000004e46 < y.re < 1.4e82

   1. Initial program 73.1%

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

   2. Taylor expanded in y.re around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. *-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. times-frac 73.6%

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

   4. Simplified 73.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100} \lor \neg \left(y.im \leq -2.75 \cdot 10^{-108}\right) \land y.im \leq 1.6:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6e+141)
   (/ x.im y.im)
   (if (or (<= y.im -1.5e+100) (and (not (<= y.im -2.75e-108)) (<= y.im 1.6)))
     (/ 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 <= -6e+141) {
		tmp = x_46_im / y_46_im;
	} else if ((y_46_im <= -1.5e+100) || (!(y_46_im <= -2.75e-108) && (y_46_im <= 1.6))) {
		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 <= (-6d+141)) then
        tmp = x_46im / y_46im
    else if ((y_46im <= (-1.5d+100)) .or. (.not. (y_46im <= (-2.75d-108))) .and. (y_46im <= 1.6d0)) 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 <= -6e+141) {
		tmp = x_46_im / y_46_im;
	} else if ((y_46_im <= -1.5e+100) || (!(y_46_im <= -2.75e-108) && (y_46_im <= 1.6))) {
		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 <= -6e+141:
		tmp = x_46_im / y_46_im
	elif (y_46_im <= -1.5e+100) or (not (y_46_im <= -2.75e-108) and (y_46_im <= 1.6)):
		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 <= -6e+141)
		tmp = Float64(x_46_im / y_46_im);
	elseif ((y_46_im <= -1.5e+100) || (!(y_46_im <= -2.75e-108) && (y_46_im <= 1.6)))
		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 <= -6e+141)
		tmp = x_46_im / y_46_im;
	elseif ((y_46_im <= -1.5e+100) || (~((y_46_im <= -2.75e-108)) && (y_46_im <= 1.6)))
		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, -6e+141], N[(x$46$im / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$im, -1.5e+100], And[N[Not[LessEqual[y$46$im, -2.75e-108]], $MachinePrecision], LessEqual[y$46$im, 1.6]]], 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 < -5.9999999999999998e141 or -1.49999999999999993e100 < y.im < -2.75000000000000016e-108 or 1.6000000000000001 < y.im

   1. Initial program 61.5%

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

   2. Taylor expanded in y.re around 0 66.5%

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

   if -5.9999999999999998e141 < y.im < -1.49999999999999993e100 or -2.75000000000000016e-108 < y.im < 1.6000000000000001

   1. Initial program 67.8%

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

   2. Taylor expanded in y.re around inf 64.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{+100} \lor \neg \left(y.im \leq -2.75 \cdot 10^{-108}\right) \land y.im \leq 1.6:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq 3.4:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6e+141)
   (/ x.im y.im)
   (if (<= y.im -9.6e+88)
     (/ x.re y.re)
     (if (<= y.im -3.9e-108)
       (/ 1.0 (/ y.im x.im))
       (if (<= y.im 3.4) (/ 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 <= -6e+141) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9.6e+88) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -3.9e-108) {
		tmp = 1.0 / (y_46_im / x_46_im);
	} else if (y_46_im <= 3.4) {
		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 <= (-6d+141)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-9.6d+88)) then
        tmp = x_46re / y_46re
    else if (y_46im <= (-3.9d-108)) then
        tmp = 1.0d0 / (y_46im / x_46im)
    else if (y_46im <= 3.4d0) 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 <= -6e+141) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9.6e+88) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -3.9e-108) {
		tmp = 1.0 / (y_46_im / x_46_im);
	} else if (y_46_im <= 3.4) {
		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 <= -6e+141:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -9.6e+88:
		tmp = x_46_re / y_46_re
	elif y_46_im <= -3.9e-108:
		tmp = 1.0 / (y_46_im / x_46_im)
	elif y_46_im <= 3.4:
		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 <= -6e+141)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -9.6e+88)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= -3.9e-108)
		tmp = Float64(1.0 / Float64(y_46_im / x_46_im));
	elseif (y_46_im <= 3.4)
		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 <= -6e+141)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -9.6e+88)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= -3.9e-108)
		tmp = 1.0 / (y_46_im / x_46_im);
	elseif (y_46_im <= 3.4)
		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, -6e+141], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9.6e+88], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -3.9e-108], N[(1.0 / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.9999999999999998e141 or 3.39999999999999991 < y.im

   1. Initial program 52.8%

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

   2. Taylor expanded in y.re around 0 70.8%

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

   if -5.9999999999999998e141 < y.im < -9.5999999999999996e88 or -3.89999999999999995e-108 < y.im < 3.39999999999999991

   1. Initial program 67.8%

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

   2. Taylor expanded in y.re around inf 64.0%

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

   if -9.5999999999999996e88 < y.im < -3.89999999999999995e-108

   1. Initial program 85.2%

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

   2. Taylor expanded in x.re around 0 74.3%

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

      1. associate-/l* 68.7%

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

      2. associate-/r/ 76.4%

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

      3. unpow2 76.4%

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

      4. unpow2 76.4%

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

      5. fma-udef 76.4%

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

   4. Simplified 76.4%

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

      1. fma-def 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in y.im around inf 54.7%

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

      1. associate-*l/ 54.8%

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

      2. *-un-lft-identity 54.8%

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

      3. clear-num 54.8%

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

   9. Applied egg-rr 54.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq 3.4:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 41.9% 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 64.7%

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

2. Taylor expanded in y.re around 0 41.7%

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

3. Final simplification 41.7%

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

Reproduce

herbie shell --seed 2023194 
(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))))