_divideComplex, imaginary part

Percentage Accurate: 60.8% → 82.8%

Time: 8.6s

Alternatives: 9

Speedup: 1.5×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 60.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ t_1 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (- (* (/ x.re (pow y.re 4.0)) y.im) (/ x.im (pow y.re 3.0)))
           y.im
           (/ (/ (- x.re) y.re) y.re))
          y.im
          (/ x.im y.re)))
        (t_1
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))))
   (if (<= y.re -3.3e+85)
     t_0
     (if (<= y.re -1.6e-104)
       t_1
       (if (<= y.re 9.8e-107)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1e+107) t_1 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(fma((((x_46_re / pow(y_46_re, 4.0)) * y_46_im) - (x_46_im / pow(y_46_re, 3.0))), y_46_im, ((-x_46_re / y_46_re) / y_46_re)), y_46_im, (x_46_im / y_46_re));
	double t_1 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -3.3e+85) {
		tmp = t_0;
	} else if (y_46_re <= -1.6e-104) {
		tmp = t_1;
	} else if (y_46_re <= 9.8e-107) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1e+107) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(fma(Float64(Float64(Float64(x_46_re / (y_46_re ^ 4.0)) * y_46_im) - Float64(x_46_im / (y_46_re ^ 3.0))), y_46_im, Float64(Float64(Float64(-x_46_re) / y_46_re) / y_46_re)), y_46_im, Float64(x_46_im / y_46_re))
	t_1 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -3.3e+85)
		tmp = t_0;
	elseif (y_46_re <= -1.6e-104)
		tmp = t_1;
	elseif (y_46_re <= 9.8e-107)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1e+107)
		tmp = t_1;
	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[(N[(N[(N[(x$46$re / N[Power[y$46$re, 4.0], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision] - N[(x$46$im / N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$im + N[(N[((-x$46$re) / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] * y$46$im + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.3e+85], t$95$0, If[LessEqual[y$46$re, -1.6e-104], t$95$1, If[LessEqual[y$46$re, 9.8e-107], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e+107], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.2999999999999999e85 or 9.9999999999999997e106 < y.re

   1. Initial program 38.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{x.re}{{y.re}^{2}} + y.im \cdot \left(\frac{x.re \cdot y.im}{{y.re}^{4}} - \frac{x.im}{{y.re}^{3}}\right), y.im, \frac{x.im}{y.re}\right)} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)} \]

   if -3.2999999999999999e85 < y.re < -1.59999999999999994e-104 or 9.79999999999999959e-107 < y.re < 9.9999999999999997e106

   1. Initial program 78.2%

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

   if -1.59999999999999994e-104 < y.re < 9.79999999999999959e-107

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+107}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x.re}{{y.re}^{4}} \cdot y.im - \frac{x.im}{{y.re}^{3}}, y.im, \frac{\frac{-x.re}{y.re}}{y.re}\right), y.im, \frac{x.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ y.re y.im) (/ x.im y.im) (/ (- x.re) y.im))))
   (if (<= y.im -1.7e+79)
     t_0
     (if (<= y.im -1.55e-156)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 2.4e-41) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / y_46_im), (x_46_im / y_46_im), (-x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.7e+79) {
		tmp = t_0;
	} else if (y_46_im <= -1.55e-156) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 2.4e-41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / y_46_im), Float64(x_46_im / y_46_im), Float64(Float64(-x_46_re) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.7e+79)
		tmp = t_0;
	elseif (y_46_im <= -1.55e-156)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 2.4e-41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_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[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+79], t$95$0, If[LessEqual[y$46$im, -1.55e-156], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e-41], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.70000000000000016e79 or 2.40000000000000022e-41 < y.im

   1. Initial program 41.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 78.6

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

   7. Applied rewrites 78.6%

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

   if -1.70000000000000016e79 < y.im < -1.5499999999999999e-156

   1. Initial program 84.5%

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

   if -1.5499999999999999e-156 < y.im < 2.40000000000000022e-41

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ (- x.im) y.im) y.re x.re) (- y.im))))
   (if (<= y.im -1.7e+79)
     t_0
     (if (<= y.im -1.55e-156)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 2.4e-41) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((-x_46_im / y_46_im), y_46_re, x_46_re) / -y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+79) {
		tmp = t_0;
	} else if (y_46_im <= -1.55e-156) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 2.4e-41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(Float64(-x_46_im) / y_46_im), y_46_re, x_46_re) / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.7e+79)
		tmp = t_0;
	elseif (y_46_im <= -1.55e-156)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 2.4e-41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_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[(N[((-x$46$im) / y$46$im), $MachinePrecision] * y$46$re + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+79], t$95$0, If[LessEqual[y$46$im, -1.55e-156], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e-41], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.70000000000000016e79 or 2.40000000000000022e-41 < y.im

   1. Initial program 41.9%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.3%

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

   9. Taylor expanded in y.im around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re + -1 \cdot \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-*l/ N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 78.6

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

   11. Applied rewrites 78.6%

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

   if -1.70000000000000016e79 < y.im < -1.5499999999999999e-156

   1. Initial program 84.5%

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

   if -1.5499999999999999e-156 < y.im < 2.40000000000000022e-41

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ (- x.im) y.im) y.re x.re) (- y.im))))
   (if (<= y.im -1.1e+43)
     t_0
     (if (<= y.im 2.4e-41) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((-x_46_im / y_46_im), y_46_re, x_46_re) / -y_46_im;
	double tmp;
	if (y_46_im <= -1.1e+43) {
		tmp = t_0;
	} else if (y_46_im <= 2.4e-41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(Float64(-x_46_im) / y_46_im), y_46_re, x_46_re) / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.1e+43)
		tmp = t_0;
	elseif (y_46_im <= 2.4e-41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_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[(N[((-x$46$im) / y$46$im), $MachinePrecision] * y$46$re + x$46$re), $MachinePrecision] / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -1.1e+43], t$95$0, If[LessEqual[y$46$im, 2.4e-41], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.1e43 or 2.40000000000000022e-41 < y.im

   1. Initial program 44.3%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.0

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

   5. Applied rewrites 41.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in y.im around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re + -1 \cdot \frac{x.im \cdot y.re}{y.im}}{y.im}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-*l/ N/A

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

       7. associate-*l* N/A

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

       8. lower-fma.f64 N/A

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

       9. associate-*r/ N/A

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

       10. lower-/.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. mul-1-neg N/A

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

       14. lower-neg.f64 78.7

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

   11. Applied rewrites 78.7%

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

   if -1.1e43 < y.im < 2.40000000000000022e-41

   1. Initial program 73.9%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-x.im}{y.im}, y.re, x.re\right)}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))
   (if (<= y.re -1.02e-22)
     t_0
     (if (<= y.re 8.5e+20) (/ (- (/ (* x.im y.re) y.im) x.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_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.02e-22) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e+20) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    if (y_46re <= (-1.02d-22)) then
        tmp = t_0
    else if (y_46re <= 8.5d+20) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / 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_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.02e-22) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e+20) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -1.02e-22:
		tmp = t_0
	elif y_46_re <= 8.5e+20:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.02e-22)
		tmp = t_0;
	elseif (y_46_re <= 8.5e+20)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / 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_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.02e-22)
		tmp = t_0;
	elseif (y_46_re <= 8.5e+20)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / 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$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.02e-22], t$95$0, If[LessEqual[y$46$re, 8.5e+20], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.02000000000000002e-22 or 8.5e20 < y.re

   1. Initial program 52.6%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   if -1.02000000000000002e-22 < y.re < 8.5e20

   1. Initial program 67.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -1.35e+43)
     t_0
     (if (<= y.im 5.2e-41) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -1.35e+43) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-1.35d+43)) then
        tmp = t_0
    else if (y_46im <= 5.2d-41) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -1.35e+43) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -1.35e+43:
		tmp = t_0
	elif y_46_im <= 5.2e-41:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.35e+43)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.35e+43)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-41)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.35e+43], t$95$0, If[LessEqual[y$46$im, 5.2e-41], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.3500000000000001e43 or 5.1999999999999999e-41 < y.im

   1. Initial program 44.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.5

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

   5. Applied rewrites 65.5%

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

   if -1.3500000000000001e43 < y.im < 5.1999999999999999e-41

   1. Initial program 73.9%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+43}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.85e+39)
   (/ x.im y.re)
   (if (<= y.re -1.45e-47)
     (* (/ y.re (fma y.re y.re (* y.im y.im))) x.im)
     (if (<= y.re 7.5e+20) (/ (- x.re) y.im) (/ x.im 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 <= -1.85e+39) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.45e-47) {
		tmp = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else if (y_46_re <= 7.5e+20) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / 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 <= -1.85e+39)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.45e-47)
		tmp = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	elseif (y_46_re <= 7.5e+20)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.85e+39], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.45e-47], N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.5e+20], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.85000000000000006e39 or 7.5e20 < y.re

   1. Initial program 49.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -1.85000000000000006e39 < y.re < -1.45e-47

   1. Initial program 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.1

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

   7. Applied rewrites 74.1%

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

   if -1.45e-47 < y.re < 7.5e20

   1. Initial program 66.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.3e-22)
   (/ x.im y.re)
   (if (<= y.re 7.5e+20) (/ (- x.re) y.im) (/ x.im 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 <= -1.3e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 7.5e+20) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / 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 <= (-1.3d-22)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 7.5d+20) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / 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 <= -1.3e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 7.5e+20) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / 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 <= -1.3e-22:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 7.5e+20:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / 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 <= -1.3e-22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 7.5e+20)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_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)
	tmp = 0.0;
	if (y_46_re <= -1.3e-22)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 7.5e+20)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / 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, -1.3e-22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.5e+20], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.3e-22 or 7.5e20 < y.re

   1. Initial program 52.6%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -1.3e-22 < y.re < 7.5e20

   1. Initial program 67.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.3% accurate, 3.2× speedup

Math

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

FPCore

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

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_re;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 43.9

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

5. Applied rewrites 43.9%

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

Reproduce

herbie shell --seed 2024283 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))