_divideComplex, real part

Percentage Accurate: 62.3% → 82.2%

Time: 12.7s

Alternatives: 20

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: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + {y.im}^{2} \cdot \left(\frac{x.im}{y.im \cdot y.re} - \frac{x.re}{{y.re}^{2}}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
      INFINITY)
   (/ (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (/
    (+
     (- x.re (* x.im (pow (/ y.im y.re) 3.0)))
     (* (pow y.im 2.0) (- (/ x.im (* y.im y.re)) (/ x.re (pow y.re 2.0)))))
    y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = ((x_46_re - (x_46_im * pow((y_46_im / y_46_re), 3.0))) + (pow(y_46_im, 2.0) * ((x_46_im / (y_46_im * y_46_re)) - (x_46_re / pow(y_46_re, 2.0))))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_re - Float64(x_46_im * (Float64(y_46_im / y_46_re) ^ 3.0))) + Float64((y_46_im ^ 2.0) * Float64(Float64(x_46_im / Float64(y_46_im * y_46_re)) - Float64(x_46_re / (y_46_re ^ 2.0))))) / 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[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re - N[(x$46$im * N[Power[N[(y$46$im / y$46$re), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[y$46$im, 2.0], $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. add-sqr-sqrt 75.0%

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

      3. times-frac 75.0%

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

      4. hypot-define 75.1%

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

      5. fma-define 75.1%

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

      6. hypot-define 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. associate-*l/ 93.7%

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

      2. *-un-lft-identity 93.7%

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

   6. Applied egg-rr 93.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 31.3%

      \[\leadsto \color{blue}{\frac{\left(x.re + \left(-1 \cdot \frac{x.im \cdot {y.im}^{3}}{{y.re}^{3}} + \frac{x.im \cdot y.im}{y.re}\right)\right) - \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}}}{y.re}} \]
   4. Step-by-step derivation

   5. Simplified 35.9%

      \[\leadsto \color{blue}{\frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + \frac{x.im \cdot y.im - {y.im}^{2} \cdot \frac{x.re}{y.re}}{y.re}}{y.re}} \]

   6. Taylor expanded in y.im around inf 35.5%

      \[\leadsto \frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + \color{blue}{{y.im}^{2} \cdot \left(-1 \cdot \frac{x.re}{{y.re}^{2}} + \frac{x.im}{y.im \cdot y.re}\right)}}{y.re} \]
   7. Step-by-step derivation

      1. mul-1-neg 35.5%

         \[\leadsto \frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + {y.im}^{2} \cdot \left(\color{blue}{\left(-\frac{x.re}{{y.re}^{2}}\right)} + \frac{x.im}{y.im \cdot y.re}\right)}{y.re} \]

      2. +-commutative 35.5%

         \[\leadsto \frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + {y.im}^{2} \cdot \color{blue}{\left(\frac{x.im}{y.im \cdot y.re} + \left(-\frac{x.re}{{y.re}^{2}}\right)\right)}}{y.re} \]

      3. sub-neg 35.5%

         \[\leadsto \frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + {y.im}^{2} \cdot \color{blue}{\left(\frac{x.im}{y.im \cdot y.re} - \frac{x.re}{{y.re}^{2}}\right)}}{y.re} \]

   8. Simplified 35.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + {y.im}^{2} \cdot \left(\frac{x.im}{y.im \cdot y.re} - \frac{x.re}{{y.re}^{2}}\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cbrt (fma x.re y.re (* y.im x.im)))))
   (if (<= y.im -8.5e+98)
     (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
     (* (/ (pow t_0 2.0) (hypot y.re y.im)) (/ t_0 (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cbrt(fma(x_46_re, y_46_re, (y_46_im * x_46_im)));
	double tmp;
	if (y_46_im <= -8.5e+98) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else {
		tmp = (pow(t_0, 2.0) / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cbrt(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)))
	tmp = 0.0
	if (y_46_im <= -8.5e+98)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	else
		tmp = Float64(Float64((t_0 ^ 2.0) / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[y$46$im, -8.5e+98], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.4999999999999996e98

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. div-inv 78.2%

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

   5. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r* 86.7%

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

      3. div-inv 86.7%

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

      4. clear-num 86.7%

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

      5. un-div-inv 86.7%

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

   7. Applied egg-rr 86.7%

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

   if -8.4999999999999996e98 < y.im

   1. Initial program 63.8%

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

      1. add-cube-cbrt 63.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}\right) \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt 63.1%

         \[\leadsto \frac{\left(\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}\right) \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. times-frac 63.1%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. pow2 63.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}\right)}^{2}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. fma-define 63.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}\right)}^{2}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      6. hypot-define 63.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      7. fma-define 63.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      8. hypot-define 79.3%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Applied egg-rr 79.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot y.re y.im) -0.5)))
   (if (<= y.im -1.2e+98)
     (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
     (* t_0 (* (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(y_46_re, y_46_im), -0.5);
	double tmp;
	if (y_46_im <= -1.2e+98) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else {
		tmp = t_0 * ((fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) * t_0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(y_46_re, y_46_im) ^ -0.5
	tmp = 0.0
	if (y_46_im <= -1.2e+98)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	else
		tmp = Float64(t_0 * Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) * t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e+98], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(t$95$0 * N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.1999999999999999e98

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. div-inv 78.2%

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

   5. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r* 86.7%

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

      3. div-inv 86.7%

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

      4. clear-num 86.7%

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

      5. un-div-inv 86.7%

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

   7. Applied egg-rr 86.7%

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

   if -1.1999999999999999e98 < y.im

   1. Initial program 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. times-frac 63.8%

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

      4. hypot-define 63.8%

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

      5. fma-define 63.8%

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

      6. hypot-define 80.3%

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

   4. Applied egg-rr 80.3%

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

      1. associate-*l/ 80.4%

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

      2. *-un-lft-identity 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. div-inv 80.3%

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

      2. add-sqr-sqrt 80.0%

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

      3. associate-*r* 80.0%

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

      4. inv-pow 80.0%

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

      5. sqrt-pow1 80.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      6. metadata-eval 80.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{\color{blue}{-0.5}}\right) \cdot \sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      7. inv-pow 80.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\right) \cdot \sqrt{\color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-1}}} \]

      8. sqrt-pow1 80.0%

         \[\leadsto \left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\right) \cdot \color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{\left(\frac{-1}{2}\right)}} \]

      9. metadata-eval 80.0%

         \[\leadsto \left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{\color{blue}{-0.5}} \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5} \cdot \left(\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.2e+97)
   (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
   (*
    (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
    (pow (sqrt (hypot y.re y.im)) -2.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -7.2e+97) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) * pow(sqrt(hypot(y_46_re, y_46_im)), -2.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.2e+97)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) * (sqrt(hypot(y_46_re, y_46_im)) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -7.2e+97], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.19999999999999932e97

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. div-inv 78.2%

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

   5. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r* 86.7%

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

      3. div-inv 86.7%

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

      4. clear-num 86.7%

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

      5. un-div-inv 86.7%

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

   7. Applied egg-rr 86.7%

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

   if -7.19999999999999932e97 < y.im

   1. Initial program 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. times-frac 63.8%

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

      4. hypot-define 63.8%

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

      5. fma-define 63.8%

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

      6. hypot-define 80.3%

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

   4. Applied egg-rr 80.3%

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

      1. inv-pow 80.3%

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

      2. add-sqr-sqrt 80.0%

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

      3. unpow-prod-down 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. pow-sqr 80.1%

         \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{\left(2 \cdot -1\right)}} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. metadata-eval 80.1%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{\color{blue}{-2}} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   8. Simplified 80.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot {\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 1.45 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\log \left(\mathsf{hypot}\left(y.re, y.im\right)\right)} \cdot \mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 1.45e+168)
   (/ (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (* (exp (- (log (hypot y.re y.im)))) (fma y.re (/ x.re y.im) x.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 1.45e+168) {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = exp(-log(hypot(y_46_re, y_46_im))) * fma(y_46_re, (x_46_re / y_46_im), x_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 1.45e+168)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(exp(Float64(-log(hypot(y_46_re, y_46_im)))) * fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 1.45e+168], N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-N[Log[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.45e168

   1. Initial program 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. add-sqr-sqrt 65.6%

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

      3. times-frac 65.6%

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

      4. hypot-define 65.6%

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

      5. fma-define 65.6%

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

      6. hypot-define 79.9%

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

   4. Applied egg-rr 79.9%

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

      1. associate-*l/ 80.0%

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

      2. *-un-lft-identity 80.0%

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

   6. Applied egg-rr 80.0%

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

   if 1.45e168 < y.im

   1. Initial program 18.0%

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

      1. *-un-lft-identity 18.0%

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

      2. add-sqr-sqrt 18.0%

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

      3. times-frac 18.0%

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

      4. hypot-define 18.0%

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

      5. fma-define 18.0%

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

      6. hypot-define 45.3%

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

   4. Applied egg-rr 45.3%

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

   5. Taylor expanded in y.re around 0 85.4%

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

      1. associate-*r/ 89.6%

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

      2. +-commutative 89.6%

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

      3. associate-*r/ 85.4%

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

      4. *-commutative 85.4%

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

      5. associate-*r/ 89.6%

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

      6. fma-define 89.6%

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

   7. Simplified 89.6%

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

      1. add-exp-log 82.7%

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

      2. log-rec 82.7%

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

   9. Applied egg-rr 82.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 1.45 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\log \left(\mathsf{hypot}\left(y.re, y.im\right)\right)} \cdot \mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.6e+39)
   (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
   (* (fma x.re y.re (* y.im x.im)) (/ 1.0 (pow (hypot y.re y.im) 2.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.6e+39) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.6e+39)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	else
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.6e+39], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.60000000000000003e39

   1. Initial program 50.4%

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

   3. Taylor expanded in y.im around inf 77.2%

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

      1. div-inv 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*r* 83.6%

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

      3. div-inv 83.6%

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

      4. clear-num 83.7%

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

      5. un-div-inv 83.7%

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

   7. Applied egg-rr 83.7%

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

   if -5.60000000000000003e39 < y.im

   1. Initial program 63.2%

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

      1. div-inv 63.0%

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

      2. fma-define 63.0%

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

      3. add-sqr-sqrt 63.0%

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

      4. pow2 63.0%

         \[\leadsto \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}} \]

      5. hypot-define 63.0%

         \[\leadsto \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}} \]

   4. Applied egg-rr 63.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + {\left(\sqrt[3]{x.re \cdot \frac{y.re}{y.im}}\right)}^{3}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 3.5e+81)
   (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
   (/ (+ x.im (pow (cbrt (* x.re (/ y.re y.im))) 3.0)) (hypot y.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 3.5e+81) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + pow(cbrt((x_46_re * (y_46_re / y_46_im))), 3.0)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 3.5e+81) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + Math.pow(Math.cbrt((x_46_re * (y_46_re / y_46_im))), 3.0)) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 3.5e+81)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + (cbrt(Float64(x_46_re * Float64(y_46_re / y_46_im))) ^ 3.0)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 3.5e+81], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[Power[N[Power[N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 3.5e81

   1. Initial program 67.0%

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

   if 3.5e81 < y.im

   1. Initial program 29.1%

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

      1. *-un-lft-identity 29.1%

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

      2. add-sqr-sqrt 29.1%

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

      3. times-frac 29.3%

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

      4. hypot-define 29.3%

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

      5. fma-define 29.3%

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

      6. hypot-define 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. associate-*l/ 59.5%

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

      2. *-un-lft-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in y.re around 0 75.3%

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

      1. add-cube-cbrt 75.2%

         \[\leadsto \frac{x.im + \color{blue}{\left(\sqrt[3]{\frac{x.re \cdot y.re}{y.im}} \cdot \sqrt[3]{\frac{x.re \cdot y.re}{y.im}}\right) \cdot \sqrt[3]{\frac{x.re \cdot y.re}{y.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. pow3 75.2%

         \[\leadsto \frac{x.im + \color{blue}{{\left(\sqrt[3]{\frac{x.re \cdot y.re}{y.im}}\right)}^{3}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      3. associate-/l* 77.9%

         \[\leadsto \frac{x.im + {\left(\sqrt[3]{\color{blue}{x.re \cdot \frac{y.re}{y.im}}}\right)}^{3}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   9. Applied egg-rr 77.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + {\left(\sqrt[3]{x.re \cdot \frac{y.re}{y.im}}\right)}^{3}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + e^{\log \left(x.re \cdot \frac{y.re}{y.im}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 8e+149)
   (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
   (/ (+ x.im (exp (log (* x.re (/ y.re y.im))))) (hypot y.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 8e+149) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + exp(log((x_46_re * (y_46_re / y_46_im))))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 8e+149) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + Math.exp(Math.log((x_46_re * (y_46_re / y_46_im))))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 8e+149:
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im + math.exp(math.log((x_46_re * (y_46_re / y_46_im))))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 8e+149)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + exp(log(Float64(x_46_re * Float64(y_46_re / y_46_im))))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 8e+149)
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im + exp(log((x_46_re * (y_46_re / y_46_im))))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 8e+149], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[Exp[N[Log[N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 8.00000000000000039e149

   1. Initial program 66.2%

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

   if 8.00000000000000039e149 < y.im

   1. Initial program 19.4%

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

      1. *-un-lft-identity 19.4%

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

      2. add-sqr-sqrt 19.4%

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

      3. times-frac 19.4%

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

      4. hypot-define 19.4%

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

      5. fma-define 19.4%

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

      6. hypot-define 49.4%

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

   4. Applied egg-rr 49.4%

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

      1. associate-*l/ 49.5%

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

      2. *-un-lft-identity 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in y.re around 0 84.3%

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

      1. add-exp-log 66.7%

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

      2. associate-/l* 73.8%

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

   9. Applied egg-rr 73.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + e^{\log \left(x.re \cdot \frac{y.re}{y.im}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, 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 + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.9%

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

   1. *-un-lft-identity 60.9%

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

   2. add-sqr-sqrt 60.9%

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

   3. times-frac 61.0%

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

   4. hypot-define 61.0%

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

   5. fma-define 61.0%

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

   6. hypot-define 76.5%

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

4. Applied egg-rr 76.5%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

6. Applied egg-rr 76.7%

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

7. Final simplification 76.7%

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

Alternative 10: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 31000000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 31000000.0)
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ (* y.re x.re) (+ (* 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) {
	double tmp;
	if (y_46_re <= 31000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_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 <= 31000000.0d0) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (y_46re * x_46re) / ((y_46re * y_46re) + (y_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 <= 31000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_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 <= 31000000.0:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_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 <= 31000000.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(y_46_re * x_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_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 <= 31000000.0)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 31000000.0], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 3.1e7

   1. Initial program 63.7%

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

   3. Taylor expanded in y.im around inf 64.4%

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

      1. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

   if 3.1e7 < y.re

   1. Initial program 54.2%

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

   3. Taylor expanded in x.re around inf 46.6%

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

      1. *-commutative 46.6%

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

   5. Simplified 46.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 31000000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \left(y.re \cdot x.re\right) \cdot \frac{1}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e-19)
   (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
   (/ (+ x.im (* (* y.re x.re) (/ 1.0 y.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 <= -2.6e-19) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) * (1.0 / y_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 <= (-2.6d-19)) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else
        tmp = (x_46im + ((y_46re * x_46re) * (1.0d0 / y_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 <= -2.6e-19) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) * (1.0 / y_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 <= -2.6e-19:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) * (1.0 / y_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 <= -2.6e-19)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) * Float64(1.0 / y_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 <= -2.6e-19)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) * (1.0 / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.6e-19], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.60000000000000013e-19

   1. Initial program 52.0%

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

   3. Taylor expanded in y.re around inf 77.0%

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

   if -2.60000000000000013e-19 < y.re

   1. Initial program 63.7%

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

   3. Taylor expanded in y.im around inf 58.2%

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

      1. div-inv 58.2%

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

   5. Applied egg-rr 58.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \left(y.re \cdot x.re\right) \cdot \frac{1}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* 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 ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((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 = ((y_46im * x_46im) + (y_46re * x_46re)) / ((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 ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((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 ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((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(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / 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 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((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[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.9%

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

3. Final simplification 60.9%

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

Alternative 13: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.5e+76)
   (/ x.re y.re)
   (/ (+ x.im (* x.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) {
	double tmp;
	if (y_46_re <= -6.5e+76) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= (-6.5d+76)) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_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 <= -6.5e+76) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -6.5e+76:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -6.5e+76)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_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 <= -6.5e+76)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.5e+76], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5000000000000005e76

   1. Initial program 40.5%

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

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

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

   if -6.5000000000000005e76 < y.re

   1. Initial program 64.8%

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

   3. Taylor expanded in y.im around inf 56.7%

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

      1. associate-/l* 57.8%

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

   5. Simplified 57.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.5e+77)
   (/ x.re y.re)
   (/ (+ x.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 <= -1.5e+77) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_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 <= (-1.5d+77)) then
        tmp = x_46re / y_46re
    else
        tmp = (x_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 <= -1.5e+77) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_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 <= -1.5e+77:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_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 <= -1.5e+77)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / 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 <= -1.5e+77)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_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[LessEqual[y$46$re, -1.5e+77], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.4999999999999999e77

   1. Initial program 40.5%

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

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

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

   if -1.4999999999999999e77 < y.re

   1. Initial program 64.8%

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

   3. Taylor expanded in y.im around inf 56.7%

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

      1. div-inv 56.7%

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

   5. Applied egg-rr 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-*r* 56.9%

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

      3. div-inv 56.9%

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

      4. clear-num 56.9%

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

      5. un-div-inv 56.9%

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

   7. Applied egg-rr 56.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.5e+76)
   (/ x.re y.re)
   (/ (+ x.im (/ (* y.re x.re) y.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 <= -6.5e+76) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_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 <= (-6.5d+76)) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + ((y_46re * x_46re) / y_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 <= -6.5e+76) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_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 <= -6.5e+76:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_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 <= -6.5e+76)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_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 <= -6.5e+76)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.5e+76], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5000000000000005e76

   1. Initial program 40.5%

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

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

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

   if -6.5000000000000005e76 < y.re

   1. Initial program 64.8%

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

   3. Taylor expanded in y.im around inf 56.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 2e+52) (/ x.im y.im) (/ (/ (* 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 <= 2e+52) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = ((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 <= 2d+52) then
        tmp = x_46im / y_46im
    else
        tmp = ((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 <= 2e+52) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = ((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 <= 2e+52:
		tmp = x_46_im / y_46_im
	else:
		tmp = ((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 <= 2e+52)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(Float64(y_46_im * 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 <= 2e+52)
		tmp = x_46_im / y_46_im;
	else
		tmp = ((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, 2e+52], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 2e52

   1. Initial program 64.9%

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

   3. Taylor expanded in y.re around 0 49.9%

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

   if 2e52 < y.re

   1. Initial program 49.4%

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

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

      \[\leadsto \color{blue}{\frac{\left(x.re + \left(-1 \cdot \frac{x.im \cdot {y.im}^{3}}{{y.re}^{3}} + \frac{x.im \cdot y.im}{y.re}\right)\right) - \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}}}{y.re}} \]
   4. Step-by-step derivation

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\left(x.re - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}\right) + \frac{x.im \cdot y.im - {y.im}^{2} \cdot \frac{x.re}{y.re}}{y.re}}{y.re}} \]

   6. Taylor expanded in x.re around 0 29.6%

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

      1. associate-/l* 29.7%

         \[\leadsto \frac{\frac{x.im \cdot y.im}{y.re} - \color{blue}{x.im \cdot \frac{{y.im}^{3}}{{y.re}^{3}}}}{y.re} \]

      2. cube-div 37.7%

         \[\leadsto \frac{\frac{x.im \cdot y.im}{y.re} - x.im \cdot \color{blue}{{\left(\frac{y.im}{y.re}\right)}^{3}}}{y.re} \]

      3. associate-/l* 39.3%

         \[\leadsto \frac{\color{blue}{x.im \cdot \frac{y.im}{y.re}} - x.im \cdot {\left(\frac{y.im}{y.re}\right)}^{3}}{y.re} \]

      4. distribute-lft-out-- 39.3%

         \[\leadsto \frac{\color{blue}{x.im \cdot \left(\frac{y.im}{y.re} - {\left(\frac{y.im}{y.re}\right)}^{3}\right)}}{y.re} \]

   8. Simplified 39.3%

      \[\leadsto \frac{\color{blue}{x.im \cdot \left(\frac{y.im}{y.re} - {\left(\frac{y.im}{y.re}\right)}^{3}\right)}}{y.re} \]

   9. Taylor expanded in y.im around 0 37.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 7.2e+135) (/ x.re y.re) (/ (/ (* y.re x.re) y.im) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 7.2e+135) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = ((y_46_re * x_46_re) / y_46_im) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 7.2d+135) then
        tmp = x_46re / y_46re
    else
        tmp = ((y_46re * x_46re) / y_46im) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 7.2e+135) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = ((y_46_re * x_46_re) / y_46_im) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 7.2e+135:
		tmp = x_46_re / y_46_re
	else:
		tmp = ((y_46_re * x_46_re) / y_46_im) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 7.2e+135)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * x_46_re) / y_46_im) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 7.2e+135)
		tmp = x_46_re / y_46_re;
	else
		tmp = ((y_46_re * x_46_re) / y_46_im) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 7.2e+135], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 7.1999999999999996e135

   1. Initial program 66.2%

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

   3. Taylor expanded in y.re around inf 49.7%

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

   if 7.1999999999999996e135 < y.im

   1. Initial program 24.1%

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

   3. Taylor expanded in y.im around inf 82.7%

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

   4. Taylor expanded in x.im around 0 26.3%

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

      1. *-commutative 26.3%

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

   6. Simplified 26.3%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 7.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 + (x_46im * (y_46im / y_46re))) / y_46re
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 + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in y.re around inf 56.4%

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

   1. associate-/l* 56.8%

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

5. Simplified 56.8%

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

6. Final simplification 56.8%

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

Alternative 19: 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 60.9%

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

3. Taylor expanded in y.re around 0 41.5%

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

4. Final simplification 41.5%

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

Alternative 20: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

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;
}

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
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;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in y.re around inf 44.8%

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

4. Final simplification 44.8%

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

Reproduce

herbie shell --seed 2024066 
(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))))