_divideComplex, real part

Percentage Accurate: 61.5% → 84.8%

Time: 13.3s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 76.7%

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

      1. +-commutative 76.7%

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

      2. fma-udef 76.8%

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

      3. *-un-lft-identity 76.8%

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

      4. associate-*r/ 76.8%

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

      5. add-sqr-sqrt 76.7%

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

      6. times-frac 76.7%

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

      7. fma-udef 76.7%

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

      8. +-commutative 76.7%

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

      9. hypot-def 76.7%

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

      10. fma-def 76.7%

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

      11. fma-udef 76.7%

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

      12. +-commutative 76.7%

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

      13. hypot-def 93.2%

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

   4. Applied egg-rr 93.2%

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

      1. associate-*l/ 93.4%

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

      2. *-un-lft-identity 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. fma-def 93.4%

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

   8. Applied egg-rr 93.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x.re around 0 1.2%

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

      1. *-commutative 1.2%

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

      2. add-sqr-sqrt 1.2%

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

      3. hypot-udef 1.2%

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

      4. hypot-udef 1.2%

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

      5. times-frac 65.2%

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

   5. Applied egg-rr 65.2%

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

4. Final simplification 88.3%

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

Alternative 2: 79.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.2e+133)
   (/ (- (/ (- x.re) (/ y.im y.re)) x.im) (hypot y.re y.im))
   (if (<= y.im -8.5e-109)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 1.2e+23)
       (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
       (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+133) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -8.5e-109) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.2e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+133) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -8.5e-109) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.2e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (y_46_im / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.2e+133:
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -8.5e-109:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.2e+23:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (y_46_im / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+133)
		tmp = Float64(Float64(Float64(Float64(-x_46_re) / Float64(y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -8.5e-109)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.2e+23)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.2e+133)
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -8.5e-109)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.2e+23)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	else
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.2e+133], N[(N[(N[((-x$46$re) / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -8.5e-109], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+23], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.1999999999999999e133

   1. Initial program 36.6%

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

      1. +-commutative 36.6%

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

      2. fma-udef 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. associate-*r/ 36.6%

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

      5. add-sqr-sqrt 36.6%

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

      6. times-frac 36.7%

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

      7. fma-udef 36.7%

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

      8. +-commutative 36.7%

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

      9. hypot-def 36.7%

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

      10. fma-def 36.7%

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

      11. fma-udef 36.7%

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

      12. +-commutative 36.7%

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

      13. hypot-def 57.6%

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

   4. Applied egg-rr 57.6%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in y.im around -inf 87.6%

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

      1. neg-mul-1 87.6%

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

      2. +-commutative 87.6%

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

      3. unsub-neg 87.6%

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

      4. mul-1-neg 87.6%

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

      5. associate-/l* 87.7%

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

      6. distribute-neg-frac 87.7%

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

   9. Simplified 87.7%

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

   if -1.1999999999999999e133 < y.im < -8.50000000000000005e-109

   1. Initial program 82.6%

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

   if -8.50000000000000005e-109 < y.im < 1.2e23

   1. Initial program 70.7%

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

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

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

      1. associate-/l* 77.9%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. pow2 76.9%

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

      2. associate-*l/ 79.7%

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

      3. associate-/r* 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 1.2e23 < y.im

   1. Initial program 46.3%

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

   3. Taylor expanded in x.re around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. add-sqr-sqrt 41.7%

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

      3. hypot-udef 41.7%

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

      4. hypot-udef 41.7%

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

      5. times-frac 81.5%

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

   5. Applied egg-rr 81.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.4e+130)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im -7.5e-108)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 1.4e+24)
       (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
       (/ (+ x.im (/ x.re (/ y.im y.re))) (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.4e+130) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -7.5e-108) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.4e+24) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.4e+130) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -7.5e-108) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.4e+24) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.4e+130:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= -7.5e-108:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.4e+24:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.4e+130)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -7.5e-108)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.4e+24)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.4e+130)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= -7.5e-108)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.4e+24)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.4e+130], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.5e-108], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e+24], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.4000000000000001e130

   1. Initial program 41.1%

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

   3. Taylor expanded in y.re around 0 81.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 82.0%

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

   5. Simplified 82.0%

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

      1. *-un-lft-identity 82.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{1 \cdot x.re}}{{y.im}^{2}} \cdot y.re \]

      2. pow2 82.0%

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

      3. times-frac 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. associate-*l/ 84.8%

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

      2. *-lft-identity 84.8%

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

   9. Simplified 84.8%

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

       1. associate-*l/ 88.5%

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

   11. Applied egg-rr 88.5%

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

   if -3.4000000000000001e130 < y.im < -7.4999999999999993e-108

   1. Initial program 81.6%

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

   if -7.4999999999999993e-108 < y.im < 1.4000000000000001e24

   1. Initial program 70.7%

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

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

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

      1. associate-/l* 77.9%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. pow2 76.9%

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

      2. associate-*l/ 79.7%

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

      3. associate-/r* 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 1.4000000000000001e24 < y.im

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. fma-udef 46.3%

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

      3. *-un-lft-identity 46.3%

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

      4. associate-*r/ 46.3%

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

      5. add-sqr-sqrt 46.3%

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

      6. times-frac 46.3%

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

      7. fma-udef 46.3%

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

      8. +-commutative 46.3%

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

      9. hypot-def 46.3%

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

      10. fma-def 46.3%

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

      11. fma-udef 46.3%

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

      12. +-commutative 46.3%

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

      13. hypot-def 58.9%

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

   4. Applied egg-rr 58.9%

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

      1. associate-*l/ 59.1%

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

      2. *-un-lft-identity 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in y.re around 0 75.1%

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

      1. associate-/l* 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.2e+133)
   (/ (- (/ (- x.re) (/ y.im y.re)) x.im) (hypot y.re y.im))
   (if (<= y.im -1.05e-105)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 5.8e+23)
       (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
       (/ (+ x.im (/ x.re (/ y.im y.re))) (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+133) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.05e-105) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 5.8e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+133) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.05e-105) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 5.8e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.2e+133:
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -1.05e-105:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 5.8e+23:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+133)
		tmp = Float64(Float64(Float64(Float64(-x_46_re) / Float64(y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.05e-105)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 5.8e+23)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.2e+133)
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -1.05e-105)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 5.8e+23)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.2e+133], N[(N[(N[((-x$46$re) / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.05e-105], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.8e+23], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.1999999999999999e133

   1. Initial program 36.6%

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

      1. +-commutative 36.6%

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

      2. fma-udef 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. associate-*r/ 36.6%

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

      5. add-sqr-sqrt 36.6%

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

      6. times-frac 36.7%

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

      7. fma-udef 36.7%

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

      8. +-commutative 36.7%

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

      9. hypot-def 36.7%

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

      10. fma-def 36.7%

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

      11. fma-udef 36.7%

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

      12. +-commutative 36.7%

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

      13. hypot-def 57.6%

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

   4. Applied egg-rr 57.6%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in y.im around -inf 87.6%

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

      1. neg-mul-1 87.6%

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

      2. +-commutative 87.6%

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

      3. unsub-neg 87.6%

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

      4. mul-1-neg 87.6%

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

      5. associate-/l* 87.7%

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

      6. distribute-neg-frac 87.7%

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

   9. Simplified 87.7%

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

   if -1.1999999999999999e133 < y.im < -1.05e-105

   1. Initial program 82.6%

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

   if -1.05e-105 < y.im < 5.80000000000000025e23

   1. Initial program 70.7%

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

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

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

      1. associate-/l* 77.9%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. pow2 76.9%

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

      2. associate-*l/ 79.7%

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

      3. associate-/r* 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 5.80000000000000025e23 < y.im

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. fma-udef 46.3%

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

      3. *-un-lft-identity 46.3%

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

      4. associate-*r/ 46.3%

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

      5. add-sqr-sqrt 46.3%

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

      6. times-frac 46.3%

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

      7. fma-udef 46.3%

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

      8. +-commutative 46.3%

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

      9. hypot-def 46.3%

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

      10. fma-def 46.3%

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

      11. fma-udef 46.3%

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

      12. +-commutative 46.3%

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

      13. hypot-def 58.9%

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

   4. Applied egg-rr 58.9%

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

      1. associate-*l/ 59.1%

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

      2. *-un-lft-identity 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in y.re around 0 75.1%

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

      1. associate-/l* 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))))
   (if (<= y.im -4e+130)
     t_0
     (if (<= y.im -5.8e-107)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 1.5e+23)
         (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -4e+130) {
		tmp = t_0;
	} else if (y_46_im <= -5.8e-107) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.5e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    if (y_46im <= (-4d+130)) then
        tmp = t_0
    else if (y_46im <= (-5.8d-107)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 1.5d+23) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -4e+130) {
		tmp = t_0;
	} else if (y_46_im <= -5.8e-107) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.5e+23) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	tmp = 0
	if y_46_im <= -4e+130:
		tmp = t_0
	elif y_46_im <= -5.8e-107:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.5e+23:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4e+130)
		tmp = t_0;
	elseif (y_46_im <= -5.8e-107)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.5e+23)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4e+130)
		tmp = t_0;
	elseif (y_46_im <= -5.8e-107)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.5e+23)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+130], t$95$0, If[LessEqual[y$46$im, -5.8e-107], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.5e+23], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.0000000000000002e130 or 1.5e23 < y.im

   1. Initial program 44.1%

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

   3. Taylor expanded in y.re around 0 76.5%

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

      1. associate-/l* 76.0%

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

      2. associate-/r/ 78.9%

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

   5. Simplified 78.9%

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

      1. *-un-lft-identity 78.9%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{1 \cdot x.re}}{{y.im}^{2}} \cdot y.re \]

      2. pow2 78.9%

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

      3. times-frac 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ 80.2%

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

      2. *-lft-identity 80.2%

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

   9. Simplified 80.2%

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

       1. associate-*l/ 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -4.0000000000000002e130 < y.im < -5.7999999999999996e-107

   1. Initial program 81.6%

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

   if -5.7999999999999996e-107 < y.im < 1.5e23

   1. Initial program 70.7%

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

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

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

      1. associate-/l* 77.9%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. pow2 76.9%

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

      2. associate-*l/ 79.7%

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

      3. associate-/r* 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+59} \lor \neg \left(y.re \leq 1.9 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.9e+59) (not (<= y.re 1.9e+19)))
   (/ x.re y.re)
   (+ (/ x.im y.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 <= -3.9e+59) || !(y_46_re <= 1.9e+19)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= (-3.9d+59)) .or. (.not. (y_46re <= 1.9d+19))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_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 <= -3.9e+59) || !(y_46_re <= 1.9e+19)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= -3.9e+59) or not (y_46_re <= 1.9e+19):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_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 <= -3.9e+59) || !(y_46_re <= 1.9e+19))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(Float64(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 <= -3.9e+59) || ~((y_46_re <= 1.9e+19)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_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[Or[LessEqual[y$46$re, -3.9e+59], N[Not[LessEqual[y$46$re, 1.9e+19]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.90000000000000021e59 or 1.9e19 < y.re

   1. Initial program 43.6%

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

   3. Taylor expanded in y.re around inf 73.4%

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

   if -3.90000000000000021e59 < y.re < 1.9e19

   1. Initial program 75.9%

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

   3. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.1%

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

      2. associate-/r/ 74.0%

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

   5. Simplified 74.0%

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

      1. *-un-lft-identity 74.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{1 \cdot x.re}}{{y.im}^{2}} \cdot y.re \]

      2. pow2 74.0%

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

      3. times-frac 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+59} \lor \neg \left(y.re \leq 1.9 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5e+59) (not (<= y.re 2.6e+19)))
   (/ x.re y.re)
   (+ (/ x.im y.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 <= -5e+59) || !(y_46_re <= 2.6e+19)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= (-5d+59)) .or. (.not. (y_46re <= 2.6d+19))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_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 <= -5e+59) || !(y_46_re <= 2.6e+19)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= -5e+59) or not (y_46_re <= 2.6e+19):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_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 <= -5e+59) || !(y_46_re <= 2.6e+19))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(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 <= -5e+59) || ~((y_46_re <= 2.6e+19)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_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[Or[LessEqual[y$46$re, -5e+59], N[Not[LessEqual[y$46$re, 2.6e+19]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.9999999999999997e59 or 2.6e19 < y.re

   1. Initial program 43.6%

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

   3. Taylor expanded in y.re around inf 73.4%

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

   if -4.9999999999999997e59 < y.re < 2.6e19

   1. Initial program 75.9%

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

   3. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.1%

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

      2. associate-/r/ 74.0%

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

   5. Simplified 74.0%

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

      1. *-un-lft-identity 74.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{1 \cdot x.re}}{{y.im}^{2}} \cdot y.re \]

      2. pow2 74.0%

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

      3. times-frac 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

   9. Simplified 74.7%

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

       1. associate-*l/ 77.9%

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

   11. Applied egg-rr 77.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.4 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.9e+59) (not (<= y.re 2.4e+18)))
   (/ x.re y.re)
   (+ (/ x.im y.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 <= -3.9e+59) || !(y_46_re <= 2.4e+18)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= (-3.9d+59)) .or. (.not. (y_46re <= 2.4d+18))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_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 <= -3.9e+59) || !(y_46_re <= 2.4e+18)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_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 <= -3.9e+59) or not (y_46_re <= 2.4e+18):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_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 <= -3.9e+59) || !(y_46_re <= 2.4e+18))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * 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 <= -3.9e+59) || ~((y_46_re <= 2.4e+18)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_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[Or[LessEqual[y$46$re, -3.9e+59], N[Not[LessEqual[y$46$re, 2.4e+18]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.90000000000000021e59 or 2.4e18 < y.re

   1. Initial program 43.6%

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

   3. Taylor expanded in y.re around inf 73.4%

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

   if -3.90000000000000021e59 < y.re < 2.4e18

   1. Initial program 75.9%

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

   3. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.1%

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

      2. associate-/r/ 74.0%

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

   5. Simplified 74.0%

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

      1. pow2 74.0%

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

      2. associate-*l/ 74.0%

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

      3. associate-/r* 78.6%

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

   7. Applied egg-rr 78.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.4 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+53} \lor \neg \left(y.re \leq 30500\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3e+53) (not (<= y.re 30500.0)))
   (+ (/ x.re y.re) (* y.im (/ (/ x.im y.re) y.re)))
   (+ (/ x.im y.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 <= -3e+53) || !(y_46_re <= 30500.0)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_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 <= (-3d+53)) .or. (.not. (y_46re <= 30500.0d0))) then
        tmp = (x_46re / y_46re) + (y_46im * ((x_46im / y_46re) / y_46re))
    else
        tmp = (x_46im / y_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 <= -3e+53) || !(y_46_re <= 30500.0)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_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 <= -3e+53) or not (y_46_re <= 30500.0):
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re))
	else:
		tmp = (x_46_im / y_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 <= -3e+53) || !(y_46_re <= 30500.0))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * 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 <= -3e+53) || ~((y_46_re <= 30500.0)))
		tmp = (x_46_re / y_46_re) + (y_46_im * ((x_46_im / y_46_re) / y_46_re));
	else
		tmp = (x_46_im / y_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[Or[LessEqual[y$46$re, -3e+53], N[Not[LessEqual[y$46$re, 30500.0]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.99999999999999998e53 or 30500 < y.re

   1. Initial program 45.3%

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

   3. Taylor expanded in y.re around inf 73.9%

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

      1. associate-/l* 75.1%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. *-un-lft-identity 76.9%

         \[\leadsto \frac{x.re}{y.re} + \frac{\color{blue}{1 \cdot x.im}}{{y.re}^{2}} \cdot y.im \]

      2. pow2 76.9%

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

      3. times-frac 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. associate-*l/ 81.2%

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

      2. *-lft-identity 81.2%

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

   9. Simplified 81.2%

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

   if -2.99999999999999998e53 < y.re < 30500

   1. Initial program 75.8%

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

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

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

      1. associate-/l* 75.9%

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

      2. associate-/r/ 75.8%

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

   5. Simplified 75.8%

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

      1. pow2 75.8%

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

      2. associate-*l/ 75.8%

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

      3. associate-/r* 79.9%

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

   7. Applied egg-rr 79.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+53} \lor \neg \left(y.re \leq 30500\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -0.0021 \lor \neg \left(y.im \leq 8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -0.0021) (not (<= y.im 8e+22)))
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -0.0021) || !(y_46_im <= 8e+22)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-0.0021d0)) .or. (.not. (y_46im <= 8d+22))) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -0.0021) || !(y_46_im <= 8e+22)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -0.0021) or not (y_46_im <= 8e+22):
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -0.0021) || !(y_46_im <= 8e+22))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -0.0021) || ~((y_46_im <= 8e+22)))
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -0.0021], N[Not[LessEqual[y$46$im, 8e+22]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -0.00209999999999999987 or 8e22 < y.im

   1. Initial program 52.6%

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

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

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

      1. associate-/l* 72.3%

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

      2. associate-/r/ 75.3%

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

   5. Simplified 75.3%

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

      1. *-un-lft-identity 75.3%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{1 \cdot x.re}}{{y.im}^{2}} \cdot y.re \]

      2. pow2 75.3%

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

      3. times-frac 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-*l/ 76.3%

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

      2. *-lft-identity 76.3%

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

   9. Simplified 76.3%

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

       1. associate-*l/ 78.3%

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

   11. Applied egg-rr 78.3%

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

   if -0.00209999999999999987 < y.im < 8e22

   1. Initial program 72.6%

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

   3. Taylor expanded in y.re around inf 76.2%

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

      1. associate-/l* 74.7%

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

      2. associate-/r/ 74.0%

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

   5. Simplified 74.0%

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

      1. pow2 74.0%

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

      2. associate-*l/ 76.2%

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

      3. associate-/r* 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. associate-/l* 81.3%

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

      2. associate-/r/ 81.3%

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

   9. Applied egg-rr 81.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0021 \lor \neg \left(y.im \leq 8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.5e+59) (not (<= y.re 2.5e+18)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.5e+59) || !(y_46_re <= 2.5e+18)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.5d+59)) .or. (.not. (y_46re <= 2.5d+18))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.5e+59) || !(y_46_re <= 2.5e+18)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.5e+59) or not (y_46_re <= 2.5e+18):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.5e+59) || !(y_46_re <= 2.5e+18))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.5e+59) || ~((y_46_re <= 2.5e+18)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.5e+59], N[Not[LessEqual[y$46$re, 2.5e+18]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.49999999999999959e59 or 2.5e18 < y.re

   1. Initial program 43.6%

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

   3. Taylor expanded in y.re around inf 73.4%

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

   if -4.49999999999999959e59 < y.re < 2.5e18

   1. Initial program 75.9%

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

   3. Taylor expanded in y.re around 0 63.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+59} \lor \neg \left(y.re \leq 2.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.6% 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 62.7%

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

3. Taylor expanded in y.re around 0 44.6%

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

4. Final simplification 44.6%

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

Reproduce

herbie shell --seed 2024027 
(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))))