_divideComplex, real part

Percentage Accurate: 61.9% → 79.6%

Time: 14.4s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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: 79.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= y.re -9.2e+50)
     (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
     (if (<= y.re -6.8e-103)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 1.7e-160)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 1.12e-11)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -9.2e+50) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= -6.8e-103) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 1.7e-160) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.12e-11) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -9.2e+50)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= -6.8e-103)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.7e-160)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 1.12e-11)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.2e+50], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.8e-103], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.7e-160], 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], If[LessEqual[y$46$re, 1.12e-11], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -9.19999999999999987e50

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 79.9%

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

      1. unpow2 79.9%

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

      2. times-frac 84.0%

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

   4. Simplified 84.0%

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

      1. *-commutative 84.0%

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

      2. clear-num 84.0%

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

      3. un-div-inv 84.0%

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

   6. Applied egg-rr 84.0%

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

   if -9.19999999999999987e50 < y.re < -6.80000000000000006e-103

   1. Initial program 87.5%

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

      1. fma-def 87.6%

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

   3. Applied egg-rr 87.6%

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

   if -6.80000000000000006e-103 < y.re < 1.70000000000000011e-160

   1. Initial program 73.3%

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

   2. Taylor expanded in y.re around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. times-frac 91.7%

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

   4. Simplified 91.7%

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

   if 1.70000000000000011e-160 < y.re < 1.1200000000000001e-11

   1. Initial program 91.7%

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

   if 1.1200000000000001e-11 < y.re

   1. Initial program 46.5%

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

   2. Taylor expanded in y.re around inf 78.5%

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

      1. unpow2 78.5%

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

      2. times-frac 86.3%

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

   4. Simplified 86.3%

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

      1. associate-*r/ 87.7%

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

   6. Applied egg-rr 87.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      1e+305)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+305) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+305)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.9999999999999994e304

   1. Initial program 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. add-sqr-sqrt 79.6%

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

      3. times-frac 79.6%

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

      4. hypot-def 79.6%

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

      5. fma-def 79.6%

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

      6. hypot-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 9.9999999999999994e304 < (/.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 17.6%

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

   2. Taylor expanded in y.re around inf 53.9%

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

      1. unpow2 53.9%

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

      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%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 3: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.95 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.95e+50)
     (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
     (if (<= y.re -1.45e-108)
       t_0
       (if (<= y.re 1.15e-160)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 1.12e-11)
           t_0
           (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -2.95e+50) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= -1.45e-108) {
		tmp = t_0;
	} else if (y_46_re <= 1.15e-160) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.12e-11) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: 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 <= (-2.95d+50)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else if (y_46re <= (-1.45d-108)) then
        tmp = t_0
    else if (y_46re <= 1.15d-160) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 1.12d-11) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -2.95e+50) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= -1.45e-108) {
		tmp = t_0;
	} else if (y_46_re <= 1.15e-160) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.12e-11) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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 <= -2.95e+50:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	elif y_46_re <= -1.45e-108:
		tmp = t_0
	elif y_46_re <= 1.15e-160:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 1.12e-11:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -2.95e+50)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= -1.45e-108)
		tmp = t_0;
	elseif (y_46_re <= 1.15e-160)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 1.12e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -2.95e+50)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	elseif (y_46_re <= -1.45e-108)
		tmp = t_0;
	elseif (y_46_re <= 1.15e-160)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 1.12e-11)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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, -2.95e+50], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.45e-108], t$95$0, If[LessEqual[y$46$re, 1.15e-160], 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], If[LessEqual[y$46$re, 1.12e-11], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.9499999999999999e50

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 79.9%

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

      1. unpow2 79.9%

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

      2. times-frac 84.0%

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

   4. Simplified 84.0%

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

      1. *-commutative 84.0%

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

      2. clear-num 84.0%

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

      3. un-div-inv 84.0%

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

   6. Applied egg-rr 84.0%

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

   if -2.9499999999999999e50 < y.re < -1.45e-108 or 1.14999999999999992e-160 < y.re < 1.1200000000000001e-11

   1. Initial program 89.9%

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

   if -1.45e-108 < y.re < 1.14999999999999992e-160

   1. Initial program 73.3%

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

   2. Taylor expanded in y.re around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. times-frac 91.7%

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

   4. Simplified 91.7%

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

   if 1.1200000000000001e-11 < y.re

   1. Initial program 46.5%

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

   2. Taylor expanded in y.re around inf 78.5%

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

      1. unpow2 78.5%

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

      2. times-frac 86.3%

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

   4. Simplified 86.3%

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

      1. associate-*r/ 87.7%

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

   6. Applied egg-rr 87.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.95 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.95 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{-68}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.38 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 -2.95e-32)
     t_0
     (if (<= y.re 6.8e-105)
       (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
       (if (<= y.re 1e-68)
         (+ (/ x.re y.re) (/ x.im (/ (* y.re y.re) y.im)))
         (if (<= y.re 1.38e-11)
           (* (/ 1.0 y.im) (+ x.im (* y.re (/ 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 <= -2.95e-32) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-105) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1e-68) {
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.38e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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 <= (-2.95d-32)) then
        tmp = t_0
    else if (y_46re <= 6.8d-105) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 1d-68) then
        tmp = (x_46re / y_46re) + (x_46im / ((y_46re * y_46re) / y_46im))
    else if (y_46re <= 1.38d-11) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (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 <= -2.95e-32) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-105) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1e-68) {
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.38e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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 <= -2.95e-32:
		tmp = t_0
	elif y_46_re <= 6.8e-105:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 1e-68:
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.38e-11:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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 <= -2.95e-32)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-105)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 1e-68)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(Float64(y_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.38e-11)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * 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 <= -2.95e-32)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-105)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 1e-68)
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.38e-11)
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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, -2.95e-32], t$95$0, If[LessEqual[y$46$re, 6.8e-105], 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], If[LessEqual[y$46$re, 1e-68], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.38e-11], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.95000000000000027e-32 or 1.38e-11 < y.re

   1. Initial program 48.8%

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

   2. Taylor expanded in y.re around inf 79.3%

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

      1. unpow2 79.3%

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

      2. times-frac 84.2%

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

   4. Simplified 84.2%

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

   if -2.95000000000000027e-32 < y.re < 6.79999999999999984e-105

   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. Taylor expanded in y.re around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. times-frac 85.9%

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

   4. Simplified 85.9%

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

   if 6.79999999999999984e-105 < y.re < 1.00000000000000007e-68

   1. Initial program 92.3%

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

   2. Taylor expanded in y.re around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/l* 70.5%

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

   4. Simplified 70.5%

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

   if 1.00000000000000007e-68 < y.re < 1.38e-11

   1. Initial program 88.9%

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

   2. Taylor expanded in y.re around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x.im around 0 73.5%

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

      1. unpow2 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 75.9%

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

      4. associate-*l/ 75.8%

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

      5. *-rgt-identity 75.8%

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

      6. associate-*r/ 75.7%

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

      7. *-rgt-identity 75.7%

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

      8. associate-*r/ 75.8%

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

      9. distribute-rgt-out 76.1%

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

      10. +-commutative 76.1%

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

      11. fma-def 76.1%

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

   7. Simplified 76.1%

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

      1. fma-udef 76.1%

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

   9. Applied egg-rr 76.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.95 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{-68}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.38 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 5: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.85 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.85e-29)
   (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))
   (if (<= y.re 8.2e-113)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.re 6.8e-69)
       (+ (/ x.re y.re) (/ x.im (/ (* y.re y.re) y.im)))
       (if (<= y.re 1.6e-11)
         (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))
         (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.85e-29) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 8.2e-113) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 6.8e-69) {
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.6e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.85d-29)) then
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    else if (y_46re <= 8.2d-113) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 6.8d-69) then
        tmp = (x_46re / y_46re) + (x_46im / ((y_46re * y_46re) / y_46im))
    else if (y_46re <= 1.6d-11) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.85e-29) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_re <= 8.2e-113) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 6.8e-69) {
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.6e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.85e-29:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	elif y_46_re <= 8.2e-113:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 6.8e-69:
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.6e-11:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.85e-29)
		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 <= 8.2e-113)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 6.8e-69)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(Float64(y_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.6e-11)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.85e-29)
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	elseif (y_46_re <= 8.2e-113)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 6.8e-69)
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.6e-11)
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.85e-29], 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, 8.2e-113], 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], If[LessEqual[y$46$re, 6.8e-69], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-11], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.8500000000000001e-29

   1. Initial program 50.3%

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

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

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

      1. unpow2 78.9%

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

      2. times-frac 80.9%

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

   4. Simplified 80.9%

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

   if -4.8500000000000001e-29 < y.re < 8.1999999999999999e-113

   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. Taylor expanded in y.re around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. times-frac 85.9%

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

   4. Simplified 85.9%

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

   if 8.1999999999999999e-113 < y.re < 6.80000000000000016e-69

   1. Initial program 92.3%

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

   2. Taylor expanded in y.re around inf 70.4%

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

      1. unpow2 70.4%

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

      2. associate-/l* 70.5%

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

   4. Simplified 70.5%

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

   if 6.80000000000000016e-69 < y.re < 1.59999999999999997e-11

   1. Initial program 88.9%

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

   2. Taylor expanded in y.re around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x.im around 0 73.5%

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

      1. unpow2 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 75.9%

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

      4. associate-*l/ 75.8%

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

      5. *-rgt-identity 75.8%

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

      6. associate-*r/ 75.7%

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

      7. *-rgt-identity 75.7%

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

      8. associate-*r/ 75.8%

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

      9. distribute-rgt-out 76.1%

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

      10. +-commutative 76.1%

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

      11. fma-def 76.1%

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

   7. Simplified 76.1%

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

      1. fma-udef 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 1.59999999999999997e-11 < y.re

   1. Initial program 47.2%

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

   2. Taylor expanded in y.re around inf 79.7%

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

      1. unpow2 79.7%

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

      2. times-frac 87.6%

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

   4. Simplified 87.6%

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

      1. associate-*r/ 89.0%

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

   6. Applied egg-rr 89.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.85 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ (/ x.im (/ y.re y.im)) y.re))))
   (if (<= y.re -8.4e-27)
     t_0
     (if (<= y.re 5.4e-105)
       (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
       (if (<= y.re 5.5e-68)
         t_0
         (if (<= y.re 1.2e-11)
           (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))
           (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -8.4e-27) {
		tmp = t_0;
	} else if (y_46_re <= 5.4e-105) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 5.5e-68) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + ((x_46im / (y_46re / y_46im)) / y_46re)
    if (y_46re <= (-8.4d-27)) then
        tmp = t_0
    else if (y_46re <= 5.4d-105) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 5.5d-68) then
        tmp = t_0
    else if (y_46re <= 1.2d-11) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -8.4e-27) {
		tmp = t_0;
	} else if (y_46_re <= 5.4e-105) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 5.5e-68) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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 <= -8.4e-27:
		tmp = t_0
	elif y_46_re <= 5.4e-105:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 5.5e-68:
		tmp = t_0
	elif y_46_re <= 1.2e-11:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / Float64(y_46_re / y_46_im)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -8.4e-27)
		tmp = t_0;
	elseif (y_46_re <= 5.4e-105)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 5.5e-68)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-11)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -8.4e-27)
		tmp = t_0;
	elseif (y_46_re <= 5.4e-105)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 5.5e-68)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-11)
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8.4e-27], t$95$0, If[LessEqual[y$46$re, 5.4e-105], 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], If[LessEqual[y$46$re, 5.5e-68], t$95$0, If[LessEqual[y$46$re, 1.2e-11], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.40000000000000061e-27 or 5.39999999999999985e-105 < y.re < 5.5000000000000003e-68

   1. Initial program 57.5%

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

   2. Taylor expanded in y.re around inf 77.4%

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

      1. unpow2 77.4%

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

      2. times-frac 76.6%

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

   4. Simplified 76.6%

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

      1. associate-*l/ 80.3%

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

      2. clear-num 80.3%

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

      3. un-div-inv 80.3%

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

   6. Applied egg-rr 80.3%

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

   if -8.40000000000000061e-27 < y.re < 5.39999999999999985e-105

   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. Taylor expanded in y.re around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. times-frac 85.9%

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

   4. Simplified 85.9%

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

   if 5.5000000000000003e-68 < y.re < 1.2000000000000001e-11

   1. Initial program 88.9%

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

   2. Taylor expanded in y.re around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x.im around 0 73.5%

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

      1. unpow2 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 75.9%

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

      4. associate-*l/ 75.8%

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

      5. *-rgt-identity 75.8%

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

      6. associate-*r/ 75.7%

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

      7. *-rgt-identity 75.7%

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

      8. associate-*r/ 75.8%

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

      9. distribute-rgt-out 76.1%

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

      10. +-commutative 76.1%

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

      11. fma-def 76.1%

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

   7. Simplified 76.1%

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

      1. fma-udef 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 1.2000000000000001e-11 < y.re

   1. Initial program 47.2%

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

   2. Taylor expanded in y.re around inf 79.7%

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

      1. unpow2 79.7%

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

      2. times-frac 87.6%

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

   4. Simplified 87.6%

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

      1. associate-*r/ 89.0%

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

   6. Applied egg-rr 89.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.8e-32)
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (if (<= y.re 4e-106)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.re 4e-69)
       (+ (/ x.re y.re) (/ (/ x.im (/ y.re y.im)) y.re))
       (if (<= y.re 1.15e-11)
         (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.re y.im))))
         (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e-32) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 4e-106) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 4e-69) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_re <= 1.15e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.8d-32)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else if (y_46re <= 4d-106) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 4d-69) then
        tmp = (x_46re / y_46re) + ((x_46im / (y_46re / y_46im)) / y_46re)
    else if (y_46re <= 1.15d-11) then
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (x_46re / y_46im)))
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e-32) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 4e-106) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 4e-69) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_re <= 1.15e-11) {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.8e-32:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	elif y_46_re <= 4e-106:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 4e-69:
		tmp = (x_46_re / y_46_re) + ((x_46_im / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_re <= 1.15e-11:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.8e-32)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 4e-106)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 4e-69)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_re <= 1.15e-11)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.8e-32)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	elseif (y_46_re <= 4e-106)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 4e-69)
		tmp = (x_46_re / y_46_re) + ((x_46_im / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_re <= 1.15e-11)
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.8e-32], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4e-106], 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], If[LessEqual[y$46$re, 4e-69], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.15e-11], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.7999999999999999e-32

   1. Initial program 50.3%

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

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

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

      1. unpow2 78.9%

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

      2. times-frac 80.9%

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

   4. Simplified 80.9%

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

      1. *-commutative 80.9%

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

      2. clear-num 82.3%

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

      3. un-div-inv 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -2.7999999999999999e-32 < y.re < 3.99999999999999976e-106

   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. Taylor expanded in y.re around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. times-frac 85.9%

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

   4. Simplified 85.9%

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

   if 3.99999999999999976e-106 < y.re < 3.9999999999999999e-69

   1. Initial program 92.3%

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

   2. Taylor expanded in y.re around inf 70.4%

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

      1. unpow2 70.4%

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

      2. times-frac 56.2%

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

   4. Simplified 56.2%

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

      1. associate-*l/ 70.5%

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

      2. clear-num 70.5%

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

      3. un-div-inv 70.5%

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

   6. Applied egg-rr 70.5%

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

   if 3.9999999999999999e-69 < y.re < 1.15000000000000007e-11

   1. Initial program 88.9%

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

   2. Taylor expanded in y.re around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x.im around 0 73.5%

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

      1. unpow2 73.5%

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

      2. *-commutative 73.5%

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

      3. times-frac 75.9%

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

      4. associate-*l/ 75.8%

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

      5. *-rgt-identity 75.8%

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

      6. associate-*r/ 75.7%

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

      7. *-rgt-identity 75.7%

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

      8. associate-*r/ 75.8%

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

      9. distribute-rgt-out 76.1%

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

      10. +-commutative 76.1%

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

      11. fma-def 76.1%

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

   7. Simplified 76.1%

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

      1. fma-udef 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 1.15000000000000007e-11 < y.re

   1. Initial program 47.2%

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

   2. Taylor expanded in y.re around inf 79.7%

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

      1. unpow2 79.7%

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

      2. times-frac 87.6%

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

   4. Simplified 87.6%

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

      1. associate-*r/ 89.0%

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

   6. Applied egg-rr 89.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e-47) (not (<= y.re 1.2e-11)))
   (/ x.re (+ y.re (/ (* y.im y.im) y.re)))
   (* (/ 1.0 y.im) (+ x.im (* y.re (/ x.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 <= -6.5e-47) || !(y_46_re <= 1.2e-11)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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) :: tmp
    if ((y_46re <= (-6.5d-47)) .or. (.not. (y_46re <= 1.2d-11))) then
        tmp = x_46re / (y_46re + ((y_46im * y_46im) / y_46re))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (y_46re * (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 tmp;
	if ((y_46_re <= -6.5e-47) || !(y_46_re <= 1.2e-11)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_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 <= -6.5e-47) or not (y_46_re <= 1.2e-11):
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (x_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 <= -6.5e-47) || !(y_46_re <= 1.2e-11))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(y_46_re * 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)
	tmp = 0.0;
	if ((y_46_re <= -6.5e-47) || ~((y_46_re <= 1.2e-11)))
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (y_46_re * (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_] := If[Or[LessEqual[y$46$re, -6.5e-47], N[Not[LessEqual[y$46$re, 1.2e-11]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5000000000000004e-47 or 1.2000000000000001e-11 < y.re

   1. Initial program 49.6%

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

   2. Taylor expanded in x.re around inf 43.5%

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

      1. associate-/l* 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. fma-udef 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in y.im around 0 76.1%

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

      1. unpow2 76.1%

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

   7. Simplified 76.1%

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

   if -6.5000000000000004e-47 < y.re < 1.2000000000000001e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in y.re around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. times-frac 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in x.im around 0 73.6%

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

      1. unpow2 73.6%

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

      2. *-commutative 73.6%

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

      3. times-frac 80.3%

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

      4. associate-*l/ 80.3%

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

      5. *-rgt-identity 80.3%

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

      6. associate-*r/ 80.0%

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

      7. *-rgt-identity 80.0%

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

      8. associate-*r/ 80.0%

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

      9. distribute-rgt-out 80.0%

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

      10. +-commutative 80.0%

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

      11. fma-def 80.0%

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

   7. Simplified 80.0%

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

      1. fma-udef 80.0%

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

   9. Applied egg-rr 80.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 9: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.26 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.5e-47) (not (<= y.re 1.26e-11)))
   (/ x.re (+ y.re (/ (* y.im y.im) y.re)))
   (+ (/ 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 tmp;
	if ((y_46_re <= -5.5e-47) || !(y_46_re <= 1.26e-11)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} 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) :: tmp
    if ((y_46re <= (-5.5d-47)) .or. (.not. (y_46re <= 1.26d-11))) then
        tmp = x_46re / (y_46re + ((y_46im * y_46im) / y_46re))
    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 tmp;
	if ((y_46_re <= -5.5e-47) || !(y_46_re <= 1.26e-11)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} 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):
	tmp = 0
	if (y_46_re <= -5.5e-47) or not (y_46_re <= 1.26e-11):
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))
	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)
	tmp = 0.0
	if ((y_46_re <= -5.5e-47) || !(y_46_re <= 1.26e-11))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	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)
	tmp = 0.0;
	if ((y_46_re <= -5.5e-47) || ~((y_46_re <= 1.26e-11)))
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	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_] := If[Or[LessEqual[y$46$re, -5.5e-47], N[Not[LessEqual[y$46$re, 1.26e-11]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 2 regimes

2. if y.re < -5.5000000000000002e-47 or 1.26e-11 < y.re

   1. Initial program 49.6%

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

   2. Taylor expanded in x.re around inf 43.5%

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

      1. associate-/l* 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. fma-udef 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in y.im around 0 76.1%

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

      1. unpow2 76.1%

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

   7. Simplified 76.1%

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

   if -5.5000000000000002e-47 < y.re < 1.26e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in y.re around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. times-frac 80.3%

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

   4. Simplified 80.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.26 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 10: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-24} \lor \neg \left(y.re \leq 1.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.6e-24) (not (<= y.re 1.6e-11)))
   (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))
   (+ (/ 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 tmp;
	if ((y_46_re <= -4.6e-24) || !(y_46_re <= 1.6e-11)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} 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) :: tmp
    if ((y_46re <= (-4.6d-24)) .or. (.not. (y_46re <= 1.6d-11))) then
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    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 tmp;
	if ((y_46_re <= -4.6e-24) || !(y_46_re <= 1.6e-11)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} 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):
	tmp = 0
	if (y_46_re <= -4.6e-24) or not (y_46_re <= 1.6e-11):
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	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)
	tmp = 0.0
	if ((y_46_re <= -4.6e-24) || !(y_46_re <= 1.6e-11))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	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)
	tmp = 0.0;
	if ((y_46_re <= -4.6e-24) || ~((y_46_re <= 1.6e-11)))
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	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_] := If[Or[LessEqual[y$46$re, -4.6e-24], N[Not[LessEqual[y$46$re, 1.6e-11]], $MachinePrecision]], 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], 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 2 regimes

2. if y.re < -4.6000000000000002e-24 or 1.59999999999999997e-11 < y.re

   1. Initial program 48.8%

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

   2. Taylor expanded in y.re around inf 79.3%

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

      1. unpow2 79.3%

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

      2. times-frac 84.2%

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

   4. Simplified 84.2%

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

   if -4.6000000000000002e-24 < y.re < 1.59999999999999997e-11

   1. Initial program 78.5%

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

   2. Taylor expanded in y.re around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. times-frac 79.9%

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

   4. Simplified 79.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-24} \lor \neg \left(y.re \leq 1.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 11: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-100} \lor \neg \left(y.re \leq 7.6 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e-100) (not (<= y.re 7.6e-121)))
   (/ x.re (+ y.re (/ (* y.im y.im) y.re)))
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6e-100) || !(y_46_re <= 7.6e-121)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6d-100)) .or. (.not. (y_46re <= 7.6d-121))) then
        tmp = x_46re / (y_46re + ((y_46im * y_46im) / y_46re))
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6e-100) || !(y_46_re <= 7.6e-121)) {
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6e-100) or not (y_46_re <= 7.6e-121):
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re))
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6e-100) || !(y_46_re <= 7.6e-121))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6e-100) || ~((y_46_re <= 7.6e-121)))
		tmp = x_46_re / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6e-100], N[Not[LessEqual[y$46$re, 7.6e-121]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.0000000000000001e-100 or 7.6000000000000002e-121 < y.re

   1. Initial program 58.5%

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

   2. Taylor expanded in x.re around inf 45.8%

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

      1. associate-/l* 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. fma-udef 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in y.im around 0 70.7%

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

      1. unpow2 70.7%

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

   7. Simplified 70.7%

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

   if -6.0000000000000001e-100 < y.re < 7.6000000000000002e-121

   1. Initial program 74.5%

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

   2. Taylor expanded in y.re around 0 79.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-100} \lor \neg \left(y.re \leq 7.6 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 61.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9.8e-8)
   (/ x.re y.re)
   (if (<= y.re 9.5e-121) (/ 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 <= -9.8e-8) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 9.5e-121) {
		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 <= (-9.8d-8)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 9.5d-121) 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 <= -9.8e-8) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 9.5e-121) {
		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 <= -9.8e-8:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 9.5e-121:
		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 <= -9.8e-8)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 9.5e-121)
		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 <= -9.8e-8)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 9.5e-121)
		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, -9.8e-8], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-121], 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 < -9.8000000000000004e-8 or 9.4999999999999994e-121 < y.re

   1. Initial program 55.2%

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

   2. Taylor expanded in y.re around inf 69.2%

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

   if -9.8000000000000004e-8 < y.re < 9.4999999999999994e-121

   1. Initial program 76.1%

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

   2. Taylor expanded in y.re around 0 72.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 13: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

2. Taylor expanded in y.re around 0 41.2%

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

3. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2023279 
(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))))