_divideComplex, real part

Percentage Accurate: 61.8% → 84.7%

Time: 11.2s

Alternatives: 13

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% 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: 84.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq 10^{+251}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 1e+251)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (if (<= t_1 INFINITY)
       (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
       (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= 1e+251) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= 1e+251) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if t_1 <= 1e+251:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	elif t_1 <= math.inf:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_1 <= 1e+251)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (t_1 <= 1e+251)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	elseif (t_1 <= Inf)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+251], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < 1e251

   1. Initial program 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. add-sqr-sqrt 75.7%

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

      3. times-frac 75.7%

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

      4. hypot-def 75.7%

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

      5. fma-def 75.7%

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

      6. hypot-def 94.1%

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

   3. Applied egg-rr 94.1%

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

      1. fma-def 94.1%

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

   5. Applied egg-rr 94.1%

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

   if 1e251 < (/.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 33.2%

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

   2. Taylor expanded in y.re around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. times-frac 77.6%

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

   4. Simplified 77.6%

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

      1. associate-*r/ 83.6%

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

   6. Applied egg-rr 83.6%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y.re around inf 50.0%

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

      1. unpow2 50.0%

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

      2. times-frac 66.3%

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

   4. Simplified 66.3%

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

      1. clear-num 66.3%

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

      2. un-div-inv 66.4%

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

   6. Applied egg-rr 66.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+251}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 2: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.12 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.95e+124)
     (+ (/ x.re y.re) (* (/ 1.0 y.re) (/ (/ x.im y.re) (/ 1.0 y.im))))
     (if (<= y.re -5.2e-26)
       t_0
       (if (<= y.re 2.12e-134)
         (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
         (if (<= y.re 5.8e+26)
           t_0
           (if (<= y.re 1.35e+90)
             (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
             (/ (+ x.re (* y.im (/ x.im y.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 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 <= -1.95e+124) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.2e-26) {
		tmp = t_0;
	} else if (y_46_re <= 2.12e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 5.8e+26) {
		tmp = t_0;
	} else if (y_46_re <= 1.35e+90) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -1.95e+124) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.2e-26) {
		tmp = t_0;
	} else if (y_46_re <= 2.12e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 5.8e+26) {
		tmp = t_0;
	} else if (y_46_re <= 1.35e+90) {
		tmp = (y_46_im / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -1.95e+124:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)))
	elif y_46_re <= -5.2e-26:
		tmp = t_0
	elif y_46_re <= 2.12e-134:
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im)
	elif y_46_re <= 5.8e+26:
		tmp = t_0
	elif y_46_re <= 1.35e+90:
		tmp = (y_46_im / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -1.95e+124)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(Float64(x_46_im / y_46_re) / Float64(1.0 / y_46_im))));
	elseif (y_46_re <= -5.2e-26)
		tmp = t_0;
	elseif (y_46_re <= 2.12e-134)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	elseif (y_46_re <= 5.8e+26)
		tmp = t_0;
	elseif (y_46_re <= 1.35e+90)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / 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 <= -1.95e+124)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	elseif (y_46_re <= -5.2e-26)
		tmp = t_0;
	elseif (y_46_re <= 2.12e-134)
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	elseif (y_46_re <= 5.8e+26)
		tmp = t_0;
	elseif (y_46_re <= 1.35e+90)
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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, -1.95e+124], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-26], t$95$0, If[LessEqual[y$46$re, 2.12e-134], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+26], t$95$0, If[LessEqual[y$46$re, 1.35e+90], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.95e124

   1. Initial program 35.2%

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

   2. Taylor expanded in y.re around inf 85.8%

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

      1. unpow2 85.8%

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

      2. times-frac 93.1%

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

   4. Simplified 93.1%

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

      1. clear-num 93.1%

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

      2. un-div-inv 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-un-lft-identity 93.1%

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

      2. div-inv 93.1%

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

      3. times-frac 95.0%

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

   8. Applied egg-rr 95.0%

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

   if -1.95e124 < y.re < -5.2000000000000002e-26 or 2.12e-134 < y.re < 5.8e26

   1. Initial program 80.3%

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

   if -5.2000000000000002e-26 < y.re < 2.12e-134

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 81.3%

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

      1. unpow2 81.3%

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

      2. associate-/r* 89.8%

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

      3. *-commutative 89.8%

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

   4. Simplified 89.8%

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

   if 5.8e26 < y.re < 1.35e90

   1. Initial program 33.4%

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

   2. Taylor expanded in x.re around 0 24.9%

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

      1. *-commutative 24.9%

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

      2. add-sqr-sqrt 24.9%

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

      3. hypot-udef 24.9%

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

      4. hypot-udef 24.9%

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

      5. times-frac 91.5%

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

   4. Applied egg-rr 91.5%

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

   if 1.35e90 < y.re

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. add-sqr-sqrt 37.6%

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

      3. times-frac 37.6%

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

      4. hypot-def 37.6%

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

      5. fma-def 37.6%

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

      6. hypot-def 50.3%

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

   3. Applied egg-rr 50.3%

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

   4. Taylor expanded in y.re around inf 71.8%

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

      1. expm1-log1p-u 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)\right)} \]

      2. expm1-udef 26.6%

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

      3. associate-*l/ 26.6%

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

      4. *-un-lft-identity 26.6%

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

      5. div-inv 26.6%

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

      6. *-commutative 26.6%

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

      7. associate-*l* 27.7%

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

      8. div-inv 27.7%

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

   6. Applied egg-rr 27.7%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 81.9%

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

   8. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.12 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.12 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.1e+131)
   (+ (/ x.re y.re) (* (/ 1.0 y.re) (/ (/ x.im y.re) (/ 1.0 y.im))))
   (if (<= y.re -3.1e-21)
     (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
     (if (<= y.re 2.12e-134)
       (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
       (if (<= y.re 1.8e+26)
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 1.35e+90)
           (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
           (/ (+ x.re (* y.im (/ x.im y.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 (y_46_re <= -4.1e+131) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -3.1e-21) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 2.12e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 1.8e+26) {
		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_re <= 1.35e+90) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_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 (y_46_re <= -4.1e+131)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(Float64(x_46_im / y_46_re) / Float64(1.0 / y_46_im))));
	elseif (y_46_re <= -3.1e-21)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.12e-134)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	elseif (y_46_re <= 1.8e+26)
		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_re <= 1.35e+90)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_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[y$46$re, -4.1e+131], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-21], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.12e-134], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e+26], 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, 1.35e+90], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.re < -4.10000000000000007e131

   1. Initial program 30.9%

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

   2. Taylor expanded in y.re around inf 84.8%

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

      1. unpow2 84.8%

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

      2. times-frac 94.7%

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

   4. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. un-div-inv 94.7%

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

   6. Applied egg-rr 94.7%

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

      1. *-un-lft-identity 94.7%

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

      2. div-inv 94.7%

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

      3. times-frac 94.7%

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

   8. Applied egg-rr 94.7%

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

   if -4.10000000000000007e131 < y.re < -3.0999999999999998e-21

   1. Initial program 83.6%

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

      1. fma-def 83.7%

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

      2. fma-def 83.7%

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

   3. Simplified 83.7%

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

   if -3.0999999999999998e-21 < y.re < 2.12e-134

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 81.3%

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

      1. unpow2 81.3%

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

      2. associate-/r* 89.8%

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

      3. *-commutative 89.8%

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

   4. Simplified 89.8%

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

   if 2.12e-134 < y.re < 1.80000000000000012e26

   1. Initial program 78.4%

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

   if 1.80000000000000012e26 < y.re < 1.35e90

   1. Initial program 33.4%

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

   2. Taylor expanded in x.re around 0 24.9%

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

      1. *-commutative 24.9%

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

      2. add-sqr-sqrt 24.9%

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

      3. hypot-udef 24.9%

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

      4. hypot-udef 24.9%

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

      5. times-frac 91.5%

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

   4. Applied egg-rr 91.5%

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

   if 1.35e90 < y.re

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. add-sqr-sqrt 37.6%

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

      3. times-frac 37.6%

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

      4. hypot-def 37.6%

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

      5. fma-def 37.6%

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

      6. hypot-def 50.3%

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

   3. Applied egg-rr 50.3%

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

   4. Taylor expanded in y.re around inf 71.8%

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

      1. expm1-log1p-u 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)\right)} \]

      2. expm1-udef 26.6%

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

      3. associate-*l/ 26.6%

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

      4. *-un-lft-identity 26.6%

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

      5. div-inv 26.6%

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

      6. *-commutative 26.6%

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

      7. associate-*l* 27.7%

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

      8. div-inv 27.7%

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

   6. Applied egg-rr 27.7%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 81.9%

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

   8. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.1 \cdot 10^{+131}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.12 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -8.5e+123)
     (+ (/ x.re y.re) (* (/ 1.0 y.re) (/ (/ x.im y.re) (/ 1.0 y.im))))
     (if (<= y.re -5.5e-26)
       t_0
       (if (<= y.re 2.7e-134)
         (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
         (if (<= y.re 4.1e+26)
           t_0
           (if (<= y.re 1.7e+90)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (/ (+ x.re (* y.im (/ x.im y.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 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 <= -8.5e+123) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.5e-26) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 4.1e+26) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e+90) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -8.5e+123) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.5e-26) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 4.1e+26) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e+90) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -8.5e+123:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)))
	elif y_46_re <= -5.5e-26:
		tmp = t_0
	elif y_46_re <= 2.7e-134:
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im)
	elif y_46_re <= 4.1e+26:
		tmp = t_0
	elif y_46_re <= 1.7e+90:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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 <= -8.5e+123)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(Float64(x_46_im / y_46_re) / Float64(1.0 / y_46_im))));
	elseif (y_46_re <= -5.5e-26)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-134)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	elseif (y_46_re <= 4.1e+26)
		tmp = t_0;
	elseif (y_46_re <= 1.7e+90)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / 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 <= -8.5e+123)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	elseif (y_46_re <= -5.5e-26)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-134)
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	elseif (y_46_re <= 4.1e+26)
		tmp = t_0;
	elseif (y_46_re <= 1.7e+90)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / 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, -8.5e+123], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.5e-26], t$95$0, If[LessEqual[y$46$re, 2.7e-134], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.1e+26], t$95$0, If[LessEqual[y$46$re, 1.7e+90], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -8.5e123

   1. Initial program 35.2%

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

   2. Taylor expanded in y.re around inf 85.8%

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

      1. unpow2 85.8%

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

      2. times-frac 93.1%

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

   4. Simplified 93.1%

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

      1. clear-num 93.1%

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

      2. un-div-inv 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-un-lft-identity 93.1%

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

      2. div-inv 93.1%

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

      3. times-frac 95.0%

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

   8. Applied egg-rr 95.0%

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

   if -8.5e123 < y.re < -5.5000000000000005e-26 or 2.6999999999999998e-134 < y.re < 4.09999999999999983e26

   1. Initial program 80.3%

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

   if -5.5000000000000005e-26 < y.re < 2.6999999999999998e-134

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 81.3%

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

      1. unpow2 81.3%

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

      2. associate-/r* 89.8%

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

      3. *-commutative 89.8%

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

   4. Simplified 89.8%

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

   if 4.09999999999999983e26 < y.re < 1.70000000000000009e90

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

   if 1.70000000000000009e90 < y.re

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. add-sqr-sqrt 37.6%

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

      3. times-frac 37.6%

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

      4. hypot-def 37.6%

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

      5. fma-def 37.6%

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

      6. hypot-def 50.3%

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

   3. Applied egg-rr 50.3%

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

   4. Taylor expanded in y.re around inf 71.8%

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

      1. expm1-log1p-u 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)\right)\right)} \]

      2. expm1-udef 26.6%

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

      3. associate-*l/ 26.6%

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

      4. *-un-lft-identity 26.6%

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

      5. div-inv 26.6%

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

      6. *-commutative 26.6%

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

      7. associate-*l* 27.7%

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

      8. div-inv 27.7%

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

   6. Applied egg-rr 27.7%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 81.9%

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

   8. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.6e+123)
     (+ (/ x.re y.re) (* (/ 1.0 y.re) (/ (/ x.im y.re) (/ 1.0 y.im))))
     (if (<= y.re -5.6e-26)
       t_0
       (if (<= y.re 1.45e-134)
         (+ (/ x.im y.im) (/ (/ (* x.re y.re) y.im) y.im))
         (if (<= y.re 5e+26)
           t_0
           (if (<= y.re 2.1e+91)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -1.6e+123) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.6e-26) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 5e+26) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1.6d+123)) then
        tmp = (x_46re / y_46re) + ((1.0d0 / y_46re) * ((x_46im / y_46re) / (1.0d0 / y_46im)))
    else if (y_46re <= (-5.6d-26)) then
        tmp = t_0
    else if (y_46re <= 1.45d-134) then
        tmp = (x_46im / y_46im) + (((x_46re * y_46re) / y_46im) / y_46im)
    else if (y_46re <= 5d+26) then
        tmp = t_0
    else if (y_46re <= 2.1d+91) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -1.6e+123) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= -5.6e-26) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-134) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	} else if (y_46_re <= 5e+26) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

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 <= -1.6e+123:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)))
	elif y_46_re <= -5.6e-26:
		tmp = t_0
	elif y_46_re <= 1.45e-134:
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im)
	elif y_46_re <= 5e+26:
		tmp = t_0
	elif y_46_re <= 2.1e+91:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -1.6e+123)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(Float64(x_46_im / y_46_re) / Float64(1.0 / y_46_im))));
	elseif (y_46_re <= -5.6e-26)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-134)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) / y_46_im));
	elseif (y_46_re <= 5e+26)
		tmp = t_0;
	elseif (y_46_re <= 2.1e+91)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

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 <= -1.6e+123)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	elseif (y_46_re <= -5.6e-26)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-134)
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) / y_46_im);
	elseif (y_46_re <= 5e+26)
		tmp = t_0;
	elseif (y_46_re <= 2.1e+91)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	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, -1.6e+123], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.6e-26], t$95$0, If[LessEqual[y$46$re, 1.45e-134], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+26], t$95$0, If[LessEqual[y$46$re, 2.1e+91], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.60000000000000002e123

   1. Initial program 35.2%

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

   2. Taylor expanded in y.re around inf 85.8%

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

      1. unpow2 85.8%

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

      2. times-frac 93.1%

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

   4. Simplified 93.1%

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

      1. clear-num 93.1%

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

      2. un-div-inv 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-un-lft-identity 93.1%

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

      2. div-inv 93.1%

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

      3. times-frac 95.0%

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

   8. Applied egg-rr 95.0%

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

   if -1.60000000000000002e123 < y.re < -5.6000000000000002e-26 or 1.44999999999999997e-134 < y.re < 5.0000000000000001e26

   1. Initial program 80.3%

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

   if -5.6000000000000002e-26 < y.re < 1.44999999999999997e-134

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 81.3%

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

      1. unpow2 81.3%

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

      2. associate-/r* 89.8%

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

      3. *-commutative 89.8%

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

   4. Simplified 89.8%

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

   if 5.0000000000000001e26 < y.re < 2.10000000000000008e91

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

   if 2.10000000000000008e91 < y.re

   1. Initial program 37.6%

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

   2. Taylor expanded in y.re around inf 66.6%

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

      1. unpow2 66.6%

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

      2. times-frac 79.9%

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

   4. Simplified 79.9%

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

      1. clear-num 79.9%

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

      2. un-div-inv 80.0%

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

   6. Applied egg-rr 80.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 6: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3600000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+26} \lor \neg \left(y.re \leq 1.6 \cdot 10^{+90}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))))
   (if (<= y.re -4.6e+52)
     t_0
     (if (<= y.re 3600000000.0)
       (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
       (if (or (<= y.re 1.45e+26) (not (<= y.re 1.6e+90)))
         t_0
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -4.6e+52) {
		tmp = t_0;
	} else if (y_46_re <= 3600000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if ((y_46_re <= 1.45e+26) || !(y_46_re <= 1.6e+90)) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    if (y_46re <= (-4.6d+52)) then
        tmp = t_0
    else if (y_46re <= 3600000000.0d0) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if ((y_46re <= 1.45d+26) .or. (.not. (y_46re <= 1.6d+90))) then
        tmp = t_0
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -4.6e+52) {
		tmp = t_0;
	} else if (y_46_re <= 3600000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if ((y_46_re <= 1.45e+26) || !(y_46_re <= 1.6e+90)) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -4.6e+52:
		tmp = t_0
	elif y_46_re <= 3600000000.0:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif (y_46_re <= 1.45e+26) or not (y_46_re <= 1.6e+90):
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.6e+52)
		tmp = t_0;
	elseif (y_46_re <= 3600000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif ((y_46_re <= 1.45e+26) || !(y_46_re <= 1.6e+90))
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -4.6e+52)
		tmp = t_0;
	elseif (y_46_re <= 3600000000.0)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif ((y_46_re <= 1.45e+26) || ~((y_46_re <= 1.6e+90)))
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+52], t$95$0, If[LessEqual[y$46$re, 3600000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 1.45e+26], N[Not[LessEqual[y$46$re, 1.6e+90]], $MachinePrecision]], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e52 or 3.6e9 < y.re < 1.45e26 or 1.59999999999999999e90 < y.re

   1. Initial program 44.0%

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

   2. Taylor expanded in y.re around inf 76.5%

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

      1. unpow2 76.5%

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

      2. times-frac 85.2%

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

   4. Simplified 85.2%

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

   if -4.6e52 < y.re < 3.6e9

   1. Initial program 74.9%

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

   2. Taylor expanded in y.re around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 83.5%

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

   6. Applied egg-rr 83.5%

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

   if 1.45e26 < y.re < 1.59999999999999999e90

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3600000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+26} \lor \neg \left(y.re \leq 1.6 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 280000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))))
   (if (<= y.re -1.15e+53)
     t_0
     (if (<= y.re 280000000.0)
       (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
       (if (<= y.re 1.05e+22)
         (+ (/ x.re y.re) (/ (* x.im y.im) (* y.re y.re)))
         (if (<= y.re 1.4e+90)
           (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.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_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -1.15e+53) {
		tmp = t_0;
	} else if (y_46_re <= 280000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 1.05e+22) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 1.4e+90) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    if (y_46re <= (-1.15d+53)) then
        tmp = t_0
    else if (y_46re <= 280000000.0d0) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46re <= 1.05d+22) then
        tmp = (x_46re / y_46re) + ((x_46im * y_46im) / (y_46re * y_46re))
    else if (y_46re <= 1.4d+90) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_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_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -1.15e+53) {
		tmp = t_0;
	} else if (y_46_re <= 280000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 1.05e+22) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 1.4e+90) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -1.15e+53:
		tmp = t_0
	elif y_46_re <= 280000000.0:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_re <= 1.05e+22:
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re))
	elif y_46_re <= 1.4e+90:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.15e+53)
		tmp = t_0;
	elseif (y_46_re <= 280000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_re <= 1.05e+22)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * y_46_im) / Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 1.4e+90)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_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_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -1.15e+53)
		tmp = t_0;
	elseif (y_46_re <= 280000000.0)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 1.05e+22)
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	elseif (y_46_re <= 1.4e+90)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+53], t$95$0, If[LessEqual[y$46$re, 280000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+22], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+90], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.1500000000000001e53 or 1.4e90 < y.re

   1. Initial program 42.3%

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

   2. Taylor expanded in y.re around inf 75.7%

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

      1. unpow2 75.7%

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

      2. times-frac 84.7%

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

   4. Simplified 84.7%

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

   if -1.1500000000000001e53 < y.re < 2.8e8

   1. Initial program 74.9%

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

   2. Taylor expanded in y.re around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 83.5%

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

   6. Applied egg-rr 83.5%

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

   if 2.8e8 < y.re < 1.0499999999999999e22

   1. Initial program 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. add-sqr-sqrt 81.5%

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

      3. times-frac 81.5%

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

      4. hypot-def 81.8%

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

      5. fma-def 81.8%

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

      6. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.re around inf 95.5%

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

      1. unpow2 95.5%

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

   6. Simplified 95.5%

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

   if 1.0499999999999999e22 < y.re < 1.4e90

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 280000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 8: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3800000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.6e+52)
   (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))
   (if (<= y.re 3800000000.0)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (if (<= y.re 2.3e+26)
       (+ (/ x.re y.re) (/ (* x.im y.im) (* y.re y.re)))
       (if (<= y.re 2.1e+91)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.6e+52) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 3800000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 2.3e+26) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 2.1e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

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 <= (-8.6d+52)) then
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    else if (y_46re <= 3800000000.0d0) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46re <= 2.3d+26) then
        tmp = (x_46re / y_46re) + ((x_46im * y_46im) / (y_46re * y_46re))
    else if (y_46re <= 2.1d+91) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.6e+52) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 3800000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 2.3e+26) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 2.1e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8.6e+52:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	elif y_46_re <= 3800000000.0:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_re <= 2.3e+26:
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re))
	elif y_46_re <= 2.1e+91:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.6e+52)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	elseif (y_46_re <= 3800000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_re <= 2.3e+26)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * y_46_im) / Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 2.1e+91)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8.6e+52)
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	elseif (y_46_re <= 3800000000.0)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 2.3e+26)
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	elseif (y_46_re <= 2.1e+91)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.6e+52], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3800000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+26], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e+91], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -8.5999999999999999e52

   1. Initial program 45.9%

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

   2. Taylor expanded in y.re around inf 82.8%

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

      1. unpow2 82.8%

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

      2. times-frac 88.5%

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

   4. Simplified 88.5%

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

   if -8.5999999999999999e52 < y.re < 3.8e9

   1. Initial program 74.9%

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

   2. Taylor expanded in y.re around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 83.5%

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

   6. Applied egg-rr 83.5%

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

   if 3.8e9 < y.re < 2.3000000000000001e26

   1. Initial program 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. add-sqr-sqrt 81.5%

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

      3. times-frac 81.5%

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

      4. hypot-def 81.8%

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

      5. fma-def 81.8%

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

      6. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.re around inf 95.5%

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

      1. unpow2 95.5%

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

   6. Simplified 95.5%

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

   if 2.3000000000000001e26 < y.re < 2.10000000000000008e91

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

   if 2.10000000000000008e91 < y.re

   1. Initial program 37.6%

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

   2. Taylor expanded in y.re around inf 66.6%

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

      1. unpow2 66.6%

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

      2. times-frac 79.9%

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

   4. Simplified 79.9%

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

      1. clear-num 79.9%

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

      2. un-div-inv 80.0%

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

   6. Applied egg-rr 80.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3800000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 9: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 1050000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+55)
   (+ (/ x.re y.re) (* (/ 1.0 y.re) (/ (/ x.im y.re) (/ 1.0 y.im))))
   (if (<= y.re 1050000000.0)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (if (<= y.re 5e+24)
       (+ (/ x.re y.re) (/ (* x.im y.im) (* y.re y.re)))
       (if (<= y.re 3.2e+92)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+55) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= 1050000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 5e+24) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 3.2e+92) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

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 <= (-1.15d+55)) then
        tmp = (x_46re / y_46re) + ((1.0d0 / y_46re) * ((x_46im / y_46re) / (1.0d0 / y_46im)))
    else if (y_46re <= 1050000000.0d0) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46re <= 5d+24) then
        tmp = (x_46re / y_46re) + ((x_46im * y_46im) / (y_46re * y_46re))
    else if (y_46re <= 3.2d+92) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+55) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	} else if (y_46_re <= 1050000000.0) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 5e+24) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 3.2e+92) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.15e+55:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)))
	elif y_46_re <= 1050000000.0:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_re <= 5e+24:
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re))
	elif y_46_re <= 3.2e+92:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e+55)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(Float64(x_46_im / y_46_re) / Float64(1.0 / y_46_im))));
	elseif (y_46_re <= 1050000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_re <= 5e+24)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * y_46_im) / Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 3.2e+92)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.15e+55)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * ((x_46_im / y_46_re) / (1.0 / y_46_im)));
	elseif (y_46_re <= 1050000000.0)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 5e+24)
		tmp = (x_46_re / y_46_re) + ((x_46_im * y_46_im) / (y_46_re * y_46_re));
	elseif (y_46_re <= 3.2e+92)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e+55], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1050000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+24], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.2e+92], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.14999999999999994e55

   1. Initial program 45.9%

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

   2. Taylor expanded in y.re around inf 82.8%

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

      1. unpow2 82.8%

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

      2. times-frac 88.5%

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

   4. Simplified 88.5%

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

      1. clear-num 88.5%

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

      2. un-div-inv 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. *-un-lft-identity 88.5%

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

      2. div-inv 88.5%

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

      3. times-frac 90.0%

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

   8. Applied egg-rr 90.0%

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

   if -1.14999999999999994e55 < y.re < 1.05e9

   1. Initial program 74.9%

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

   2. Taylor expanded in y.re around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 83.5%

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

   6. Applied egg-rr 83.5%

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

   if 1.05e9 < y.re < 5.00000000000000045e24

   1. Initial program 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. add-sqr-sqrt 81.5%

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

      3. times-frac 81.5%

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

      4. hypot-def 81.8%

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

      5. fma-def 81.8%

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

      6. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.re around inf 95.5%

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

      1. unpow2 95.5%

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

   6. Simplified 95.5%

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

   if 5.00000000000000045e24 < y.re < 3.20000000000000025e92

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

   if 3.20000000000000025e92 < y.re

   1. Initial program 37.6%

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

   2. Taylor expanded in y.re around inf 66.6%

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

      1. unpow2 66.6%

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

      2. times-frac 79.9%

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

   4. Simplified 79.9%

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

      1. clear-num 79.9%

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

      2. un-div-inv 80.0%

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

   6. Applied egg-rr 80.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \frac{\frac{x.im}{y.re}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 1050000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 10: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.1e+77)
   (/ x.re y.re)
   (if (<= y.re 2.4e+91)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.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 <= -1.1e+77) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.4e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= (-1.1d+77)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.4d+91) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_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 <= -1.1e+77) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.4e+91) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= -1.1e+77:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.4e+91:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= -1.1e+77)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.4e+91)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_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 <= -1.1e+77)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.4e+91)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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, -1.1e+77], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+91], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$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 < -1.1e77 or 2.39999999999999983e91 < y.re

   1. Initial program 40.5%

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

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

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

   if -1.1e77 < y.re < 2.39999999999999983e91

   1. Initial program 72.0%

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

   2. Taylor expanded in y.re around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. unpow2 72.9%

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

      3. times-frac 77.1%

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

   4. Simplified 77.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.3e+76)
   (/ x.re y.re)
   (if (<= y.re 8.5e+91)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) 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 <= -3.3e+76) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 8.5e+91) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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 <= (-3.3d+76)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 8.5d+91) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / 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 <= -3.3e+76) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 8.5e+91) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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 <= -3.3e+76:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 8.5e+91:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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 <= -3.3e+76)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 8.5e+91)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / 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 <= -3.3e+76)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 8.5e+91)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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, -3.3e+76], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 8.5e+91], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.3000000000000001e76 or 8.4999999999999995e91 < y.re

   1. Initial program 40.5%

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

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

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

   if -3.3000000000000001e76 < y.re < 8.4999999999999995e91

   1. Initial program 72.0%

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

   2. Taylor expanded in y.re around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. unpow2 72.9%

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

      3. times-frac 77.1%

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

   4. Simplified 77.1%

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

      1. associate-*r/ 79.0%

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

   6. Applied egg-rr 79.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12: 63.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.2e+53)
   (/ x.re y.re)
   (if (<= y.re 1.35e+90) (/ x.im 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 <= -5.2e+53) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.35e+90) {
		tmp = x_46_im / 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 <= (-5.2d+53)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.35d+90) then
        tmp = x_46im / 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 <= -5.2e+53) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.35e+90) {
		tmp = x_46_im / 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 <= -5.2e+53:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.35e+90:
		tmp = x_46_im / 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 <= -5.2e+53)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.35e+90)
		tmp = Float64(x_46_im / 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 <= -5.2e+53)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.35e+90)
		tmp = x_46_im / 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, -5.2e+53], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+90], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.19999999999999996e53 or 1.35e90 < y.re

   1. Initial program 42.3%

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

   2. Taylor expanded in y.re around inf 73.5%

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

   if -5.19999999999999996e53 < y.re < 1.35e90

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around 0 64.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 13: 42.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

2. Taylor expanded in y.re around 0 43.0%

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

3. Final simplification 43.0%

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

Reproduce

herbie shell --seed 2023293 
(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))))