_divideComplex, imaginary part

Percentage Accurate: 62.5% → 82.6%

Time: 10.5s

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: 62.5% 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: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(y.im, \frac{-x.re}{t\_0}, \frac{y.re \cdot x.im}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+74}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 y.im (/ (- x.re) t_0) (/ (* y.re x.im) t_0)))
        (t_2 (/ (fma (- y.im) (/ x.re y.re) x.im) y.re)))
   (if (<= y.re -3.7e+74)
     t_2
     (if (<= y.re -3.5e-58)
       t_1
       (if (<= y.re 1.2e-71)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 5.2e+53) 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(y_46_im, (-x_46_re / t_0), ((y_46_re * x_46_im) / t_0));
	double t_2 = fma(-y_46_im, (x_46_re / y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -3.7e+74) {
		tmp = t_2;
	} else if (y_46_re <= -3.5e-58) {
		tmp = t_1;
	} else if (y_46_re <= 1.2e-71) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 5.2e+53) {
		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(y_46_im, Float64(Float64(-x_46_re) / t_0), Float64(Float64(y_46_re * x_46_im) / t_0))
	t_2 = Float64(fma(Float64(-y_46_im), Float64(x_46_re / y_46_re), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.7e+74)
		tmp = t_2;
	elseif (y_46_re <= -3.5e-58)
		tmp = t_1;
	elseif (y_46_re <= 1.2e-71)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 5.2e+53)
		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[(y$46$im * N[((-x$46$re) / t$95$0), $MachinePrecision] + N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-y$46$im) * N[(x$46$re / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.7e+74], t$95$2, If[LessEqual[y$46$re, -3.5e-58], t$95$1, If[LessEqual[y$46$re, 1.2e-71], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+53], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.7000000000000001e74 or 5.19999999999999996e53 < y.re

   1. Initial program 33.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 79.4

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

   5. Simplified 79.4%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 83.1

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

   7. Applied egg-rr 83.1%

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

   if -3.7000000000000001e74 < y.re < -3.4999999999999999e-58 or 1.2e-71 < y.re < 5.19999999999999996e53

   1. Initial program 79.4%

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

   3. Taylor expanded in x.im around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. unpow2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 86.8

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

   5. Simplified 86.8%

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

   if -3.4999999999999999e-58 < y.re < 1.2e-71

   1. Initial program 65.6%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 84.7

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

   8. Simplified 84.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{-x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \frac{-x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8.8 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (fma (- y.im) (/ x.re y.re) x.im) y.re)))
   (if (<= y.re -1.7e+82)
     t_1
     (if (<= y.re -8.8e-91)
       t_0
       (if (<= y.re 8.6e-164)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 2e+53) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma(-y_46_im, (x_46_re / y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -1.7e+82) {
		tmp = t_1;
	} else if (y_46_re <= -8.8e-91) {
		tmp = t_0;
	} else if (y_46_re <= 8.6e-164) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 2e+53) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(Float64(-y_46_im), Float64(x_46_re / y_46_re), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.7e+82)
		tmp = t_1;
	elseif (y_46_re <= -8.8e-91)
		tmp = t_0;
	elseif (y_46_re <= 8.6e-164)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 2e+53)
		tmp = t_0;
	else
		tmp = t_1;
	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[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-y$46$im) * N[(x$46$re / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.7e+82], t$95$1, If[LessEqual[y$46$re, -8.8e-91], t$95$0, If[LessEqual[y$46$re, 8.6e-164], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2e+53], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.69999999999999997e82 or 2e53 < y.re

   1. Initial program 32.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 79.0

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

   5. Simplified 79.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. /-lowering-/.f64 82.7

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

   7. Applied egg-rr 82.7%

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

   if -1.69999999999999997e82 < y.re < -8.8000000000000003e-91 or 8.5999999999999996e-164 < y.re < 2e53

   1. Initial program 79.0%

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

   if -8.8000000000000003e-91 < y.re < 8.5999999999999996e-164

   1. Initial program 62.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 59.7

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

   5. Simplified 59.7%

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

   6. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 92.2

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

   8. Simplified 92.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -8.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, \frac{x.re}{y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* y.re x.im) (* y.im x.re)) (* y.re y.re))))
   (if (<= y.re -5.5e+123)
     (/ x.im y.re)
     (if (<= y.re -3.2e-77)
       t_0
       (if (<= y.re 1.9e-30)
         (/ x.re (- y.im))
         (if (<= y.re 9.4e+92) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -5.5e+123) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-77) {
		tmp = t_0;
	} else if (y_46_re <= 1.9e-30) {
		tmp = x_46_re / -y_46_im;
	} else if (y_46_re <= 9.4e+92) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / (y_46re * y_46re)
    if (y_46re <= (-5.5d+123)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-3.2d-77)) then
        tmp = t_0
    else if (y_46re <= 1.9d-30) then
        tmp = x_46re / -y_46im
    else if (y_46re <= 9.4d+92) then
        tmp = t_0
    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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -5.5e+123) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-77) {
		tmp = t_0;
	} else if (y_46_re <= 1.9e-30) {
		tmp = x_46_re / -y_46_im;
	} else if (y_46_re <= 9.4e+92) {
		tmp = t_0;
	} 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):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re)
	tmp = 0
	if y_46_re <= -5.5e+123:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -3.2e-77:
		tmp = t_0
	elif y_46_re <= 1.9e-30:
		tmp = x_46_re / -y_46_im
	elif y_46_re <= 9.4e+92:
		tmp = t_0
	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)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -5.5e+123)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.2e-77)
		tmp = t_0;
	elseif (y_46_re <= 1.9e-30)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	elseif (y_46_re <= 9.4e+92)
		tmp = t_0;
	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)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.5e+123)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -3.2e-77)
		tmp = t_0;
	elseif (y_46_re <= 1.9e-30)
		tmp = x_46_re / -y_46_im;
	elseif (y_46_re <= 9.4e+92)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.5e+123], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.2e-77], t$95$0, If[LessEqual[y$46$re, 1.9e-30], N[(x$46$re / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 9.4e+92], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.5000000000000002e123 or 9.4000000000000001e92 < y.re

   1. Initial program 25.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 77.5

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

   5. Simplified 77.5%

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

   if -5.5000000000000002e123 < y.re < -3.2e-77 or 1.9000000000000002e-30 < y.re < 9.4000000000000001e92

   1. Initial program 80.2%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 61.4

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

   5. Simplified 61.4%

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

   if -3.2e-77 < y.re < 1.9000000000000002e-30

   1. Initial program 65.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 65.6

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

   5. Simplified 65.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.05 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.re (fma y.im y.im (* y.re y.re))))))
   (if (<= y.re -2.9e+82)
     (/ x.im y.re)
     (if (<= y.re -3.05e-93)
       t_0
       (if (<= y.re 3.4e-43)
         (/ x.re (- y.im))
         (if (<= y.re 1.22e+154) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_re <= -2.9e+82) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.05e-93) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-43) {
		tmp = x_46_re / -y_46_im;
	} else if (y_46_re <= 1.22e+154) {
		tmp = t_0;
	} 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)
	t_0 = Float64(x_46_im * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.9e+82)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.05e-93)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-43)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	elseif (y_46_re <= 1.22e+154)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(x$46$im * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e+82], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.05e-93], t$95$0, If[LessEqual[y$46$re, 3.4e-43], N[(x$46$re / (-y$46$im)), $MachinePrecision], If[LessEqual[y$46$re, 1.22e+154], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9000000000000001e82 or 1.22e154 < y.re

   1. Initial program 24.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 77.7

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

   5. Simplified 77.7%

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

   if -2.9000000000000001e82 < y.re < -3.04999999999999985e-93 or 3.4000000000000001e-43 < y.re < 1.22e154

   1. Initial program 78.3%

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

   3. Taylor expanded in x.re around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2} + {y.re}^{2}}} \]
   7. 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. *-lowering-*.f64 N/A

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

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

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 59.7

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

   8. Simplified 59.7%

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

   if -3.04999999999999985e-93 < y.re < 3.4000000000000001e-43

   1. Initial program 62.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 67.5

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

   5. Simplified 67.5%

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

Alternative 5: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -0.045:\\ \;\;\;\;\left(-y.im\right) \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -4.8e+148)
     t_0
     (if (<= y.im -0.045)
       (* (- y.im) (/ x.re (fma y.im y.im (* y.re y.re))))
       (if (<= y.im 5.7e+51) (/ (- 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 <= -4.8e+148) {
		tmp = t_0;
	} else if (y_46_im <= -0.045) {
		tmp = -y_46_im * (x_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_im <= 5.7e+51) {
		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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+148)
		tmp = t_0;
	elseif (y_46_im <= -0.045)
		tmp = Float64(Float64(-y_46_im) * Float64(x_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_im <= 5.7e+51)
		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[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+148], t$95$0, If[LessEqual[y$46$im, -0.045], N[((-y$46$im) * N[(x$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.7e+51], 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 < -4.79999999999999989e148 or 5.7000000000000002e51 < y.im

   1. Initial program 32.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 68.8

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

   5. Simplified 68.8%

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

   if -4.79999999999999989e148 < y.im < -0.044999999999999998

   1. Initial program 64.9%

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

   3. Taylor expanded in x.re around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 64.8

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 65.7

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

   8. Simplified 65.7%

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

   if -0.044999999999999998 < y.im < 5.7000000000000002e51

   1. Initial program 64.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -0.045:\\ \;\;\;\;\left(-y.im\right) \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.8e-10)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 5.1e+50)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (/ (fma x.im (/ y.re y.im) (- x.re)) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.8e-10) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 5.1e+50) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.8e-10)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 5.1e+50)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / 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, -6.8e-10], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.1e+50], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.8000000000000003e-10

   1. Initial program 47.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 77.3

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

   5. Simplified 77.3%

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

   if -6.8000000000000003e-10 < y.im < 5.0999999999999998e50

   1. Initial program 64.2%

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

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 81.0

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

   5. Simplified 81.0%

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

   if 5.0999999999999998e50 < y.im

   1. Initial program 35.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 35.4

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

   5. Simplified 35.4%

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

   6. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 80.8

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

   8. Simplified 80.8%

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

Alternative 7: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)))
   (if (<= y.im -1.85e-9)
     t_0
     (if (<= y.im 1.35e+49) (/ (- 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_re / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.85e-9) {
		tmp = t_0;
	} else if (y_46_im <= 1.35e+49) {
		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(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.85e-9)
		tmp = t_0;
	elseif (y_46_im <= 1.35e+49)
		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[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.85e-9], t$95$0, If[LessEqual[y$46$im, 1.35e+49], 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 < -1.85e-9 or 1.35000000000000005e49 < y.im

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around inf

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in y.im around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 78.4

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

   8. Simplified 78.4%

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

   if -1.85e-9 < y.im < 1.35000000000000005e49

   1. Initial program 64.2%

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

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 81.0

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

   5. Simplified 81.0%

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

Alternative 8: 64.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -160000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -160000000000.0)
   (/ x.im y.re)
   (if (<= y.re 7.2e-31) (/ 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 <= -160000000000.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 7.2e-31) {
		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 <= (-160000000000.0d0)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 7.2d-31) 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 <= -160000000000.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 7.2e-31) {
		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 <= -160000000000.0:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 7.2e-31:
		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 <= -160000000000.0)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 7.2e-31)
		tmp = Float64(x_46_re / Float64(-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 <= -160000000000.0)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 7.2e-31)
		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, -160000000000.0], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-31], 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 < -1.6e11 or 7.20000000000000007e-31 < y.re

   1. Initial program 43.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 67.3

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

   5. Simplified 67.3%

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

   if -1.6e11 < y.re < 7.20000000000000007e-31

   1. Initial program 67.2%

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

   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. /-lowering-/.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. neg-lowering-neg.f64 61.6

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

   5. Simplified 61.6%

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

Alternative 9: 43.0% 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 54.5%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 45.0

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

5. Simplified 45.0%

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

Reproduce

herbie shell --seed 2024204 
(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))))