_divideComplex, real part

Percentage Accurate: 61.2% → 83.6%

Time: 10.8s

Alternatives: 10

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: 61.2% 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: 83.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. add-sqr-sqrt 73.7%

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

      3. times-frac 73.6%

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

      4. hypot-define 73.6%

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

      5. fma-define 73.6%

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

      6. hypot-define 93.6%

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

   4. Applied egg-rr 93.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 57.3%

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

4. Final simplification 86.7%

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

Alternative 2: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -4.6 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -4.7e+104)
     (+ (/ x.re y.re) (* x.im (/ y.im (pow y.re 2.0))))
     (if (<= y.re -4.6e-107)
       t_0
       (if (<= y.re 5.8e-152)
         (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ (* x.re y.re) y.im)))
         (if (<= y.re 2.2e+142)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.re (* x.im (/ y.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -4.7e+104) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_re <= -4.6e-107) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-152) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 2.2e+142) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -4.7e+104) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_re <= -4.6e-107) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-152) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 2.2e+142) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -4.7e+104:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_re <= -4.6e-107:
		tmp = t_0
	elif y_46_re <= 5.8e-152:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 2.2e+142:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -4.7e+104)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_re <= -4.6e-107)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-152)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 2.2e+142)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -4.7e+104)
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_re <= -4.6e-107)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-152)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 2.2e+142)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.7e+104], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.6e-107], t$95$0, If[LessEqual[y$46$re, 5.8e-152], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+142], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.70000000000000017e104

   1. Initial program 25.5%

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

   3. Taylor expanded in y.re around inf 76.6%

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

      1. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

   if -4.70000000000000017e104 < y.re < -4.60000000000000007e-107 or 5.8000000000000003e-152 < y.re < 2.19999999999999987e142

   1. Initial program 79.0%

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

   if -4.60000000000000007e-107 < y.re < 5.8000000000000003e-152

   1. Initial program 68.7%

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

   3. Taylor expanded in y.re around 0 89.4%

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

      1. *-commutative 89.4%

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

   5. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. pow2 89.4%

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

      3. times-frac 94.7%

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

      4. *-commutative 94.7%

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

   7. Applied egg-rr 94.7%

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

   if 2.19999999999999987e142 < y.re

   1. Initial program 23.4%

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

      1. *-un-lft-identity 23.4%

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

      2. add-sqr-sqrt 23.4%

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

      3. times-frac 23.4%

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

      4. hypot-define 23.4%

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

      5. fma-define 23.4%

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

      6. hypot-define 48.3%

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in y.re around inf 84.3%

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

      1. associate-/l* 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -4.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -2.02 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (* x.im (/ y.im (pow y.re 2.0))))))
   (if (<= y.re -4.5e+103)
     t_1
     (if (<= y.re -2.02e-106)
       t_0
       (if (<= y.re 1.45e-151)
         (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ (* x.re y.re) y.im)))
         (if (<= y.re 1.8e+112) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (x_46_im * (y_46_im / pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -4.5e+103) {
		tmp = t_1;
	} else if (y_46_re <= -2.02e-106) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.8e+112) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re ** 2.0d0)))
    if (y_46re <= (-4.5d+103)) then
        tmp = t_1
    else if (y_46re <= (-2.02d-106)) then
        tmp = t_0
    else if (y_46re <= 1.45d-151) then
        tmp = (x_46im / y_46im) + ((1.0d0 / y_46im) * ((x_46re * y_46re) / y_46im))
    else if (y_46re <= 1.8d+112) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (x_46_im * (y_46_im / Math.pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -4.5e+103) {
		tmp = t_1;
	} else if (y_46_re <= -2.02e-106) {
		tmp = t_0;
	} else if (y_46_re <= 1.45e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.8e+112) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + (x_46_im * (y_46_im / math.pow(y_46_re, 2.0)))
	tmp = 0
	if y_46_re <= -4.5e+103:
		tmp = t_1
	elif y_46_re <= -2.02e-106:
		tmp = t_0
	elif y_46_re <= 1.45e-151:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.8e+112:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / (y_46_re ^ 2.0))))
	tmp = 0.0
	if (y_46_re <= -4.5e+103)
		tmp = t_1;
	elseif (y_46_re <= -2.02e-106)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-151)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.8e+112)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -4.5e+103)
		tmp = t_1;
	elseif (y_46_re <= -2.02e-106)
		tmp = t_0;
	elseif (y_46_re <= 1.45e-151)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.8e+112)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.5e+103], t$95$1, If[LessEqual[y$46$re, -2.02e-106], t$95$0, If[LessEqual[y$46$re, 1.45e-151], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e+112], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.50000000000000001e103 or 1.8e112 < y.re

   1. Initial program 28.9%

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

   3. Taylor expanded in y.re around inf 74.5%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -4.50000000000000001e103 < y.re < -2.02000000000000011e-106 or 1.45000000000000006e-151 < y.re < 1.8e112

   1. Initial program 78.8%

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

   if -2.02000000000000011e-106 < y.re < 1.45000000000000006e-151

   1. Initial program 68.7%

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

   3. Taylor expanded in y.re around 0 89.4%

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

      1. *-commutative 89.4%

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

   5. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. pow2 89.4%

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

      3. times-frac 94.7%

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

      4. *-commutative 94.7%

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -2.02 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re + x.im \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_0 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ x.re (* x.im (/ y.im y.re)))))
   (if (<= y.re -2e-15)
     (* t_0 (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re 3.8e-151)
       (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ (* x.re y.re) y.im)))
       (if (<= y.re 1.16e+139)
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (* (/ 1.0 (hypot y.re y.im)) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re + (x_46_im * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -2e-15) {
		tmp = t_0 * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 3.8e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.16e+139) {
		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 {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_0;
	}
	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 + (x_46_im * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -2e-15) {
		tmp = t_0 * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 3.8e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 1.16e+139) {
		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 {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re + (x_46_im * (y_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -2e-15:
		tmp = t_0 * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= 3.8e-151:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 1.16e+139:
		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:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2e-15)
		tmp = Float64(t_0 * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 3.8e-151)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 1.16e+139)
		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)));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re + (x_46_im * (y_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -2e-15)
		tmp = t_0 * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 3.8e-151)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 1.16e+139)
		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
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * 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[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2e-15], N[(t$95$0 * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.8e-151], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.16e+139], 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], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.0000000000000002e-15

   1. Initial program 45.4%

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

      1. *-un-lft-identity 45.4%

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

      2. add-sqr-sqrt 45.4%

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

      3. times-frac 45.4%

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

      4. hypot-define 45.4%

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

      5. fma-define 45.4%

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

      6. hypot-define 64.1%

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

   4. Applied egg-rr 64.1%

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

   5. Taylor expanded in y.re around -inf 81.5%

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

      1. distribute-lft-out 81.5%

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

      2. associate-/l* 87.3%

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

   7. Simplified 87.3%

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

   if -2.0000000000000002e-15 < y.re < 3.7999999999999997e-151

   1. Initial program 71.9%

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

   3. Taylor expanded in y.re around 0 81.7%

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

      1. *-commutative 81.7%

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

   5. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. pow2 81.7%

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

      3. times-frac 87.7%

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

      4. *-commutative 87.7%

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

   7. Applied egg-rr 87.7%

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

   if 3.7999999999999997e-151 < y.re < 1.16000000000000004e139

   1. Initial program 75.1%

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

   if 1.16000000000000004e139 < y.re

   1. Initial program 23.4%

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

      1. *-un-lft-identity 23.4%

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

      2. add-sqr-sqrt 23.4%

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

      3. times-frac 23.4%

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

      4. hypot-define 23.4%

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

      5. fma-define 23.4%

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

      6. hypot-define 48.3%

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in y.re around inf 84.3%

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

      1. associate-/l* 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -7.5e+114)
     (/ x.re y.re)
     (if (<= y.re -4.2e-107)
       t_0
       (if (<= y.re 1.7e-151)
         (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ (* x.re y.re) y.im)))
         (if (<= y.re 3.1e+140) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -7.5e+114) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4.2e-107) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 3.1e+140) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-7.5d+114)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-4.2d-107)) then
        tmp = t_0
    else if (y_46re <= 1.7d-151) then
        tmp = (x_46im / y_46im) + ((1.0d0 / y_46im) * ((x_46re * y_46re) / y_46im))
    else if (y_46re <= 3.1d+140) then
        tmp = t_0
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -7.5e+114) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4.2e-107) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e-151) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_re <= 3.1e+140) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -7.5e+114:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -4.2e-107:
		tmp = t_0
	elif y_46_re <= 1.7e-151:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im))
	elif y_46_re <= 3.1e+140:
		tmp = t_0
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -7.5e+114)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -4.2e-107)
		tmp = t_0;
	elseif (y_46_re <= 1.7e-151)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_re <= 3.1e+140)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -7.5e+114)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -4.2e-107)
		tmp = t_0;
	elseif (y_46_re <= 1.7e-151)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_re <= 3.1e+140)
		tmp = t_0;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+114], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.2e-107], t$95$0, If[LessEqual[y$46$re, 1.7e-151], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.1e+140], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.5000000000000001e114 or 3.1e140 < y.re

   1. Initial program 24.3%

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

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

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

   if -7.5000000000000001e114 < y.re < -4.1999999999999998e-107 or 1.7000000000000001e-151 < y.re < 3.1e140

   1. Initial program 77.8%

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

   if -4.1999999999999998e-107 < y.re < 1.7000000000000001e-151

   1. Initial program 68.7%

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

   3. Taylor expanded in y.re around 0 89.4%

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

      1. *-commutative 89.4%

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

   5. Simplified 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. pow2 89.4%

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

      3. times-frac 94.7%

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

      4. *-commutative 94.7%

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-15} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.5e-15) (not (<= y.re 1.5e+92)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ (* x.re y.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.5e-15) || !(y_46_re <= 1.5e+92)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.5d-15)) .or. (.not. (y_46re <= 1.5d+92))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((1.0d0 / y_46im) * ((x_46re * y_46re) / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.5e-15) || !(y_46_re <= 1.5e+92)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.5e-15) or not (y_46_re <= 1.5e+92):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.5e-15) || !(y_46_re <= 1.5e+92))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.5e-15) || ~((y_46_re <= 1.5e+92)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((x_46_re * y_46_re) / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.5e-15], N[Not[LessEqual[y$46$re, 1.5e+92]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.5e-15 or 1.50000000000000007e92 < y.re

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around inf 72.2%

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

   if -1.5e-15 < y.re < 1.50000000000000007e92

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around 0 71.0%

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

      1. *-commutative 71.0%

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

   5. Simplified 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. pow2 71.0%

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

      3. times-frac 76.3%

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

      4. *-commutative 76.3%

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

   7. Applied egg-rr 76.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-15} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re \cdot y.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{-23} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.9e-23) (not (<= y.im 1.8e+67)))
   (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.9e-23) || !(y_46_im <= 1.8e+67)) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.9d-23)) .or. (.not. (y_46im <= 1.8d+67))) then
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.9e-23) || !(y_46_im <= 1.8e+67)) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.9e-23) or not (y_46_im <= 1.8e+67):
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.9e-23) || !(y_46_im <= 1.8e+67))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.9e-23) || ~((y_46_im <= 1.8e+67)))
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.9e-23], N[Not[LessEqual[y$46$im, 1.8e+67]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.9000000000000002e-23 or 1.7999999999999999e67 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 71.6%

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

      1. *-commutative 71.6%

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

   5. Simplified 71.6%

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

      1. pow2 71.6%

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

      2. times-frac 81.3%

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

   7. Applied egg-rr 81.3%

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

   if -2.9000000000000002e-23 < y.im < 1.7999999999999999e67

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 67.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{-23} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-22} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.2e-22) (not (<= y.im 1.8e+67)))
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e-22) || !(y_46_im <= 1.8e+67)) {
		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;
	}
	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 <= (-8.2d-22)) .or. (.not. (y_46im <= 1.8d+67))) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e-22) || !(y_46_im <= 1.8e+67)) {
		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;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.2e-22) or not (y_46_im <= 1.8e+67):
		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
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.2e-22) || !(y_46_im <= 1.8e+67))
		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(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -8.2e-22) || ~((y_46_im <= 1.8e+67)))
		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;
	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, -8.2e-22], N[Not[LessEqual[y$46$im, 1.8e+67]], $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[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.1999999999999999e-22 or 1.7999999999999999e67 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 71.6%

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

      1. *-commutative 71.6%

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

   5. Simplified 71.6%

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

      1. pow2 71.6%

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

      2. times-frac 81.3%

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

   7. Applied egg-rr 81.3%

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

      1. associate-*l/ 82.1%

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

   9. Applied egg-rr 82.1%

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

   if -8.1999999999999999e-22 < y.im < 1.7999999999999999e67

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 67.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-22} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-21} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.2e-21) (not (<= y.im 7.6e+67)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e-21) || !(y_46_im <= 7.6e+67)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.2d-21)) .or. (.not. (y_46im <= 7.6d+67))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e-21) || !(y_46_im <= 7.6e+67)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.2e-21) or not (y_46_im <= 7.6e+67):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.2e-21) || !(y_46_im <= 7.6e+67))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.2e-21) || ~((y_46_im <= 7.6e+67)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.2e-21], N[Not[LessEqual[y$46$im, 7.6e+67]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2e-21 or 7.60000000000000041e67 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 65.8%

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

   if -1.2e-21 < y.im < 7.60000000000000041e67

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 67.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-21} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in y.re around 0 39.8%

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

4. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2024039 
(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))))