_divideComplex, real part

Percentage Accurate: 61.1% → 80.0%

Time: 8.3s

Alternatives: 5

Speedup: 1.2×

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: 61.1% 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: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+125}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (* y.im (/ x.im y.re))) y.re)))
   (if (<= y.re -1.15e-33)
     t_0
     (if (<= y.re 6.8e-162)
       (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
       (if (<= y.re 1e+125)
         (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.15e-33) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-162) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 1e+125) {
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    if (y_46re <= (-1.15d-33)) then
        tmp = t_0
    else if (y_46re <= 6.8d-162) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else if (y_46re <= 1d+125) then
        tmp = ((y_46re * x_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.15e-33) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-162) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 1e+125) {
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -1.15e-33:
		tmp = t_0
	elif y_46_re <= 6.8e-162:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	elif y_46_re <= 1e+125:
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.15e-33)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-162)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_re <= 1e+125)
		tmp = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.15e-33)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-162)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	elseif (y_46_re <= 1e+125)
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e-33], t$95$0, If[LessEqual[y$46$re, 6.8e-162], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e+125], N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.14999999999999993e-33 or 9.9999999999999992e124 < y.re

   1. Initial program 41.9%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.re + \frac{x.im \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{x.im \cdot y.im}{y.re}\right)\right), y.re\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\left(x.im \cdot y.im\right), y.re\right)\right), y.re\right) \]

      4. *-lowering-*.f64 71.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, y.im\right), y.re\right)\right), y.re\right) \]

   5. Simplified 71.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{y.im \cdot x.im}{y.re}\right)\right), y.re\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(y.im \cdot \frac{x.im}{y.re}\right)\right), y.re\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{*.f64}\left(y.im, \left(\frac{x.im}{y.re}\right)\right)\right), y.re\right) \]

      4. /-lowering-/.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(x.im, y.re\right)\right)\right), y.re\right) \]

   7. Applied egg-rr 77.3%

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

   if -1.14999999999999993e-33 < y.re < 6.8e-162

   1. Initial program 73.2%

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

   3. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \frac{x.re \cdot y.re}{y.im}\right), \color{blue}{y.im}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{x.re \cdot y.re}{y.im}\right)\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(x.re \cdot \frac{y.re}{y.im}\right)\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.im\right) \]

      5. /-lowering-/.f64 91.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.im\right) \]

   5. Simplified 91.6%

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

   if 6.8e-162 < y.re < 9.9999999999999992e124

   1. Initial program 81.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+125}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 120:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (* y.im (/ x.im y.re))) y.re)))
   (if (<= y.re -5.6e-35)
     t_0
     (if (<= y.re 120.0) (/ (+ x.im (* x.re (/ y.re y.im))) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -5.6e-35) {
		tmp = t_0;
	} else if (y_46_re <= 120.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    if (y_46re <= (-5.6d-35)) then
        tmp = t_0
    else if (y_46re <= 120.0d0) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -5.6e-35) {
		tmp = t_0;
	} else if (y_46_re <= 120.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -5.6e-35:
		tmp = t_0
	elif y_46_re <= 120.0:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -5.6e-35)
		tmp = t_0;
	elseif (y_46_re <= 120.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -5.6e-35)
		tmp = t_0;
	elseif (y_46_re <= 120.0)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -5.6e-35], t$95$0, If[LessEqual[y$46$re, 120.0], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.5999999999999999e-35 or 120 < y.re

   1. Initial program 51.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.re + \frac{x.im \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{x.im \cdot y.im}{y.re}\right)\right), y.re\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\left(x.im \cdot y.im\right), y.re\right)\right), y.re\right) \]

      4. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, y.im\right), y.re\right)\right), y.re\right) \]

   5. Simplified 70.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{y.im \cdot x.im}{y.re}\right)\right), y.re\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \left(y.im \cdot \frac{x.im}{y.re}\right)\right), y.re\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{*.f64}\left(y.im, \left(\frac{x.im}{y.re}\right)\right)\right), y.re\right) \]

      4. /-lowering-/.f64 75.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(x.im, y.re\right)\right)\right), y.re\right) \]

   7. Applied egg-rr 75.3%

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

   if -5.5999999999999999e-35 < y.re < 120

   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. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \frac{x.re \cdot y.re}{y.im}\right), \color{blue}{y.im}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{x.re \cdot y.re}{y.im}\right)\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(x.re \cdot \frac{y.re}{y.im}\right)\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.im\right) \]

      5. /-lowering-/.f64 86.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.im\right) \]

   5. Simplified 86.4%

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

Alternative 3: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3e+56)
   (/ x.re y.re)
   (if (<= y.re 6.8e+18)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.8e+18) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3d+56)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 6.8d+18) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.8e+18) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3e+56:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 6.8e+18:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3e+56)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 6.8e+18)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3e+56)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 6.8e+18)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3e+56], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.8e+18], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.00000000000000006e56 or 6.8e18 < y.re

   1. Initial program 47.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 68.6%

         \[\leadsto \mathsf{/.f64}\left(x.re, \color{blue}{y.re}\right) \]

   5. Simplified 68.6%

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

   if -3.00000000000000006e56 < y.re < 6.8e18

   1. Initial program 74.7%

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

   3. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \frac{x.re \cdot y.re}{y.im}\right), \color{blue}{y.im}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{x.re \cdot y.re}{y.im}\right)\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \left(x.re \cdot \frac{y.re}{y.im}\right)\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.im\right) \]

      5. /-lowering-/.f64 80.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.im\right) \]

   5. Simplified 80.7%

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

Alternative 4: 62.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.4e-51)
   (/ x.re y.re)
   (if (<= y.re 2.7e-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_re <= -8.4e-51) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e-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_46re <= (-8.4d-51)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.7d-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_re <= -8.4e-51) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e-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_re <= -8.4e-51:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.7e-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_re <= -8.4e-51)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.7e-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_re <= -8.4e-51)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.7e-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[LessEqual[y$46$re, -8.4e-51], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.7e-67], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.40000000000000006e-51 or 2.70000000000000016e-67 < y.re

   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 inf

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

      1. /-lowering-/.f64 62.6%

         \[\leadsto \mathsf{/.f64}\left(x.re, \color{blue}{y.re}\right) \]

   5. Simplified 62.6%

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

   if -8.40000000000000006e-51 < y.re < 2.70000000000000016e-67

   1. Initial program 74.5%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 69.6%

         \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.im}\right) \]

   5. Simplified 69.6%

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

Alternative 5: 43.1% 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 61.7%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 41.0%

      \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.im}\right) \]

5. Simplified 41.0%

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

Reproduce

herbie shell --seed 2024160 
(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))))