_divideComplex, real part

Percentage Accurate: 62.0% → 83.0%

Time: 8.3s

Alternatives: 7

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, \frac{x.re \cdot y.re}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -0.00012:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma x.im (/ y.im t_0) (/ (* x.re y.re) t_0)))
        (t_2 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -1.2e+147)
     t_2
     (if (<= y.im -0.00012)
       t_1
       (if (<= y.im 1.45e-101)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.6e+69) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(x_46_im, (y_46_im / t_0), ((x_46_re * y_46_re) / t_0));
	double t_2 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e+147) {
		tmp = t_2;
	} else if (y_46_im <= -0.00012) {
		tmp = t_1;
	} else if (y_46_im <= 1.45e-101) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.6e+69) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(x_46_im, Float64(y_46_im / t_0), Float64(Float64(x_46_re * y_46_re) / t_0))
	t_2 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+147)
		tmp = t_2;
	elseif (y_46_im <= -0.00012)
		tmp = t_1;
	elseif (y_46_im <= 1.45e-101)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.6e+69)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(N[(x$46$re * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e+147], t$95$2, If[LessEqual[y$46$im, -0.00012], t$95$1, If[LessEqual[y$46$im, 1.45e-101], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+69], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.20000000000000001e147 or 1.59999999999999992e69 < y.im

   1. Initial program 35.8%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.20000000000000001e147 < y.im < -1.20000000000000003e-4 or 1.45e-101 < y.im < 1.59999999999999992e69

   1. Initial program 81.8%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -1.20000000000000003e-4 < y.im < 1.45e-101

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 2: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -3.8e+42)
     t_0
     (if (<= y.im 1.45e-101)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (if (<= y.im 7.2e+68)
         (/ (+ (* x.re y.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 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -3.8e+42) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e-101) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 7.2e+68) {
		tmp = ((x_46_re * y_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(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.8e+42)
		tmp = t_0;
	elseif (y_46_im <= 1.45e-101)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 7.2e+68)
		tmp = Float64(Float64(Float64(x_46_re * y_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

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$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+42], t$95$0, If[LessEqual[y$46$im, 1.45e-101], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+68], N[(N[(N[(x$46$re * y$46$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.im < -3.7999999999999998e42 or 7.1999999999999998e68 < y.im

   1. Initial program 43.3%

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

   3. Taylor expanded in y.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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -3.7999999999999998e42 < y.im < 1.45e-101

   1. Initial program 69.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 1.45e-101 < y.im < 7.1999999999999998e68

   1. Initial program 83.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.00022:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e+147)
   (/ x.im y.im)
   (if (<= y.im -0.00022)
     (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
     (if (<= y.im 2.3e-41)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -0.00022) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_im <= 2.3e-41) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e+147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -0.00022)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_im <= 2.3e-41)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e+147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -0.00022], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3e-41], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2500000000000001e147 or 2.3000000000000001e-41 < y.im

   1. Initial program 43.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if -1.2500000000000001e147 < y.im < -2.20000000000000008e-4

   1. Initial program 79.7%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

   if -2.20000000000000008e-4 < y.im < 2.3000000000000001e-41

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.00022:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -3.8e+42)
     t_0
     (if (<= y.im 2.3e-41) (/ (fma x.im (/ y.im y.re) x.re) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -3.8e+42) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-41) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.8e+42)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-41)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+42], t$95$0, If[LessEqual[y$46$im, 2.3e-41], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.7999999999999998e42 or 2.3000000000000001e-41 < y.im

   1. Initial program 48.5%

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

   3. Taylor expanded in y.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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -3.7999999999999998e42 < y.im < 2.3000000000000001e-41

   1. Initial program 71.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 5: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.00012:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.25e+147)
   (/ x.im y.im)
   (if (<= y.im -0.00012)
     (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
     (if (<= y.im 2.25e-41) (/ x.re y.re) (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.25e+147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -0.00012) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_im <= 2.25e-41) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.25e+147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -0.00012)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_im <= 2.25e-41)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.25e+147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -0.00012], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.25e-41], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2500000000000001e147 or 2.25e-41 < y.im

   1. Initial program 43.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if -1.2500000000000001e147 < y.im < -1.20000000000000003e-4

   1. Initial program 79.7%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

   if -1.20000000000000003e-4 < y.im < 2.25e-41

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.00012:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -0.00022:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -0.00022)
   (/ x.im y.im)
   (if (<= y.im 2.25e-41) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -0.00022) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.25e-41) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-0.00022d0)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2.25d-41) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -0.00022) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.25e-41) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -0.00022:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2.25e-41:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -0.00022)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.25e-41)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -0.00022)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2.25e-41)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -0.00022], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.25e-41], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.20000000000000008e-4 or 2.25e-41 < y.im

   1. Initial program 51.6%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

   if -2.20000000000000008e-4 < y.im < 2.25e-41

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

Alternative 7: 42.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.7%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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. lower-/.f64 43.5

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

5. Applied rewrites 43.5%

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

Reproduce

herbie shell --seed 2024214 
(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))))