_divideComplex, imaginary part

Percentage Accurate: 61.7% → 96.2%

Time: 15.2s

Alternatives: 13

Speedup: 1.1×

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: 61.7% 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: 96.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma
  (/ y.re (hypot y.re y.im))
  (/ x.im (hypot y.re y.im))
  (* x.re (/ (/ y.im (hypot y.im y.re)) (- (hypot y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re * ((y_46_im / hypot(y_46_im, y_46_re)) / -hypot(y_46_im, y_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) / Float64(-hypot(y_46_im, y_46_re)))))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. fma-neg 62.6%

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

   2. distribute-rgt-neg-out 62.6%

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

   3. +-commutative 62.6%

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

   4. fma-define 62.6%

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

3. Simplified 62.6%

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

   1. distribute-rgt-neg-out 62.6%

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

   2. fma-neg 62.6%

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

   3. fma-undefine 62.6%

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

   4. +-commutative 62.6%

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

   5. div-sub 58.6%

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

   6. *-commutative 58.6%

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

   7. add-sqr-sqrt 58.6%

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

   8. times-frac 60.3%

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

   9. fma-neg 60.3%

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

   10. hypot-define 60.3%

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

   11. hypot-define 73.7%

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

   12. associate-/l* 76.7%

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

   13. add-sqr-sqrt 76.7%

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

   14. pow2 76.7%

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

6. Applied egg-rr 76.7%

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

   1. *-un-lft-identity 76.7%

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

   2. unpow2 76.7%

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

   3. times-frac 95.5%

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

8. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-lft-identity 95.5%

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

   3. hypot-undefine 76.7%

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

   4. unpow2 76.7%

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

   5. unpow2 76.7%

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

   6. +-commutative 76.7%

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

   7. unpow2 76.7%

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

   8. unpow2 76.7%

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

   9. hypot-define 95.5%

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

   10. hypot-undefine 76.7%

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

   11. unpow2 76.7%

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

   12. unpow2 76.7%

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

   13. +-commutative 76.7%

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

   14. unpow2 76.7%

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

   15. unpow2 76.7%

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

   16. hypot-define 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

    \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
12. Add Preprocessing

Alternative 2: 93.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ t_3 := \mathsf{fma}\left(t\_0, t\_1, \frac{x.re}{-y.im}\right)\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+111}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{+136}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (hypot y.re y.im)))
        (t_1 (/ x.im (hypot y.re y.im)))
        (t_2 (fma t_0 t_1 (* x.re (/ y.im (- (pow (hypot y.re y.im) 2.0))))))
        (t_3 (fma t_0 t_1 (/ x.re (- y.im)))))
   (if (<= y.im -5.5e+111)
     t_3
     (if (<= y.im -1.6e-120)
       t_2
       (if (<= y.im 1.6e-168)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 6.9e+136) t_2 t_3))))))

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 / hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / hypot(y_46_re, y_46_im);
	double t_2 = fma(t_0, t_1, (x_46_re * (y_46_im / -pow(hypot(y_46_re, y_46_im), 2.0))));
	double t_3 = fma(t_0, t_1, (x_46_re / -y_46_im));
	double tmp;
	if (y_46_im <= -5.5e+111) {
		tmp = t_3;
	} else if (y_46_im <= -1.6e-120) {
		tmp = t_2;
	} else if (y_46_im <= 1.6e-168) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 6.9e+136) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	t_2 = fma(t_0, t_1, Float64(x_46_re * Float64(y_46_im / Float64(-(hypot(y_46_re, y_46_im) ^ 2.0)))))
	t_3 = fma(t_0, t_1, Float64(x_46_re / Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_im <= -5.5e+111)
		tmp = t_3;
	elseif (y_46_im <= -1.6e-120)
		tmp = t_2;
	elseif (y_46_im <= 1.6e-168)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 6.9e+136)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[(x$46$re * N[(y$46$im / (-N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1 + N[(x$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.5e+111], t$95$3, If[LessEqual[y$46$im, -1.6e-120], t$95$2, If[LessEqual[y$46$im, 1.6e-168], 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$46$im, 6.9e+136], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.4999999999999998e111 or 6.9e136 < y.im

   1. Initial program 36.2%

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

      1. fma-neg 36.2%

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

      2. distribute-rgt-neg-out 36.2%

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

      3. +-commutative 36.2%

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

      4. fma-define 36.2%

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

   3. Simplified 36.2%

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

      1. distribute-rgt-neg-out 36.2%

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

      2. fma-neg 36.2%

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

      3. fma-undefine 36.2%

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

      4. +-commutative 36.2%

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

      5. div-sub 36.2%

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

      6. *-commutative 36.2%

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

      7. add-sqr-sqrt 36.2%

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

      8. times-frac 36.5%

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

      9. fma-neg 36.5%

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

      10. hypot-define 36.5%

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

      11. hypot-define 45.1%

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

      12. associate-/l* 50.0%

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

      13. add-sqr-sqrt 50.0%

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

      14. pow2 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in y.im around inf 96.4%

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

   if -5.4999999999999998e111 < y.im < -1.6e-120 or 1.60000000000000003e-168 < y.im < 6.9e136

   1. Initial program 77.8%

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

      1. fma-neg 77.8%

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

      2. distribute-rgt-neg-out 77.8%

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

      3. +-commutative 77.8%

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

      4. fma-define 77.8%

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

   3. Simplified 77.8%

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

      1. distribute-rgt-neg-out 77.8%

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

      2. fma-neg 77.8%

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

      3. fma-undefine 77.8%

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

      4. +-commutative 77.8%

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

      5. div-sub 76.9%

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

      6. *-commutative 76.9%

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

      7. add-sqr-sqrt 76.9%

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

      8. times-frac 77.5%

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

      9. fma-neg 77.5%

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

      10. hypot-define 77.5%

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

      11. hypot-define 90.1%

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

      12. associate-/l* 94.7%

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

      13. add-sqr-sqrt 94.7%

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

      14. pow2 94.7%

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

   6. Applied egg-rr 94.7%

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

   if -1.6e-120 < y.im < 1.60000000000000003e-168

   1. Initial program 68.4%

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

      1. fma-neg 68.4%

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

      2. distribute-rgt-neg-out 68.4%

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

      3. +-commutative 68.4%

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

      4. fma-define 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in y.re around inf 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

      3. distribute-lft-neg-in 91.8%

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

   7. Simplified 91.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ x.re (- y.im)))))
   (if (<= y.im -1.55e-34)
     t_0
     (if (<= y.im 4.8e-117)
       (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
       (if (<= y.im 1e+29)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re / -y_46_im));
	double tmp;
	if (y_46_im <= -1.55e-34) {
		tmp = t_0;
	} else if (y_46_im <= 4.8e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1e+29) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re / Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.55e-34)
		tmp = t_0;
	elseif (y_46_im <= 4.8e-117)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1e+29)
		tmp = 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)));
	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 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e-34], t$95$0, If[LessEqual[y$46$im, 4.8e-117], 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$46$im, 1e+29], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.5499999999999999e-34 or 9.99999999999999914e28 < y.im

   1. Initial program 50.8%

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

      1. fma-neg 50.8%

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

      2. distribute-rgt-neg-out 50.8%

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

      3. +-commutative 50.8%

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

      4. fma-define 50.8%

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

   3. Simplified 50.8%

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

      1. distribute-rgt-neg-out 50.8%

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

      2. fma-neg 50.8%

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

      3. fma-undefine 50.8%

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

      4. +-commutative 50.8%

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

      5. div-sub 50.8%

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

      6. *-commutative 50.8%

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

      7. add-sqr-sqrt 50.9%

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

      8. times-frac 52.7%

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

      9. fma-neg 52.7%

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

      10. hypot-define 52.7%

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

      11. hypot-define 61.9%

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

      12. associate-/l* 67.3%

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

      13. add-sqr-sqrt 67.3%

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

      14. pow2 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in y.im around inf 91.4%

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

   if -1.5499999999999999e-34 < y.im < 4.80000000000000028e-117

   1. Initial program 68.9%

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

      1. fma-neg 68.9%

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

      2. distribute-rgt-neg-out 68.9%

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

      3. +-commutative 68.9%

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

      4. fma-define 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in y.re around inf 91.0%

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

      1. associate-*r/ 91.0%

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

      2. neg-mul-1 91.0%

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

      3. distribute-lft-neg-in 91.0%

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

   7. Simplified 91.0%

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

   if 4.80000000000000028e-117 < y.im < 9.99999999999999914e28

   1. Initial program 87.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{-y.im}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.1e-12)
   (fma (/ y.re (hypot y.re y.im)) (/ x.im (- y.im)) (/ x.re (- y.im)))
   (if (<= y.im 1.7e-113)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (if (<= y.im 1.8e+29)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (- (* x.im (/ (/ y.re y.im) 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 <= -3.1e-12) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / -y_46_im), (x_46_re / -y_46_im));
	} else if (y_46_im <= 1.7e-113) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im * ((y_46_re / y_46_im) / 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 <= -3.1e-12)
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / Float64(-y_46_im)), Float64(x_46_re / Float64(-y_46_im)));
	elseif (y_46_im <= 1.7e-113)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = 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)));
	else
		tmp = Float64(Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) / 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, -3.1e-12], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $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-113], 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$46$im, 1.8e+29], 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], N[(N[(x$46$im * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.1000000000000001e-12

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

      1. distribute-rgt-neg-out 46.2%

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

      2. fma-neg 46.2%

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

      3. fma-undefine 46.1%

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

      4. +-commutative 46.1%

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

      5. div-sub 46.1%

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

      6. *-commutative 46.1%

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

      7. add-sqr-sqrt 46.2%

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

      8. times-frac 49.4%

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

      9. fma-neg 49.4%

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

      10. hypot-define 49.4%

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

      11. hypot-define 62.1%

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

      12. associate-/l* 64.3%

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

      13. add-sqr-sqrt 64.3%

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

      14. pow2 64.3%

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

   6. Applied egg-rr 64.3%

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

   7. Taylor expanded in y.im around inf 91.3%

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

   8. Taylor expanded in y.im around -inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   10. Simplified 84.2%

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

   if -3.1000000000000001e-12 < y.im < 1.7000000000000001e-113

   1. Initial program 69.4%

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

      1. fma-neg 69.4%

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

      2. distribute-rgt-neg-out 69.4%

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

      3. +-commutative 69.4%

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

      4. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in y.re around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. neg-mul-1 89.4%

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

      3. distribute-lft-neg-in 89.4%

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

   7. Simplified 89.4%

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

   if 1.7000000000000001e-113 < y.im < 1.79999999999999988e29

   1. Initial program 87.6%

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

   if 1.79999999999999988e29 < y.im

   1. Initial program 54.2%

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

      1. fma-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

      3. +-commutative 54.2%

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

      4. fma-define 54.1%

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

   3. Simplified 54.1%

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

      1. distribute-rgt-neg-out 54.1%

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

      2. fma-neg 54.1%

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

      3. fma-undefine 54.2%

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

      4. +-commutative 54.2%

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

      5. div-sub 54.2%

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

      6. *-commutative 54.2%

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

      7. add-sqr-sqrt 54.2%

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

      8. times-frac 54.4%

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

      9. fma-neg 54.4%

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

      10. hypot-define 54.4%

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

      11. hypot-define 58.0%

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

      12. associate-/l* 68.0%

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

      13. add-sqr-sqrt 68.0%

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

      14. pow2 68.0%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in y.re around 0 89.6%

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

      1. associate-/l* 89.7%

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

   9. Simplified 89.7%

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

       1. *-un-lft-identity 89.7%

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

       2. unpow2 89.7%

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

       3. times-frac 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. associate-*l/ 89.9%

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

       2. *-lft-identity 89.9%

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

   13. Simplified 89.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{-y.im}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5e-10)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 1.2e-117)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (if (<= y.im 1.8e+29)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (- (* x.im (/ (/ y.re y.im) 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 <= -5e-10) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.2e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5d-10)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 1.2d-117) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 1.8d+29) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46im * ((y_46re / y_46im) / y_46im)) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5e-10) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.2e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5e-10:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 1.2e-117:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 1.8e+29:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im * ((y_46_re / y_46_im) / 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 <= -5e-10)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 1.2e-117)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = 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)));
	else
		tmp = Float64(Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5e-10)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 1.2e-117)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.8e+29)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5e-10], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.2e-117], 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$46$im, 1.8e+29], 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], N[(N[(x$46$im * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.00000000000000031e-10

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in y.re around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. unpow2 75.1%

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

      5. associate-/r* 76.7%

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

      6. div-sub 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-/l* 83.9%

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

   7. Simplified 83.9%

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

   if -5.00000000000000031e-10 < y.im < 1.20000000000000007e-117

   1. Initial program 69.4%

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

      1. fma-neg 69.4%

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

      2. distribute-rgt-neg-out 69.4%

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

      3. +-commutative 69.4%

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

      4. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in y.re around inf 89.4%

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

      1. associate-*r/ 89.4%

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

      2. neg-mul-1 89.4%

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

      3. distribute-lft-neg-in 89.4%

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

   7. Simplified 89.4%

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

   if 1.20000000000000007e-117 < y.im < 1.79999999999999988e29

   1. Initial program 87.6%

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

   if 1.79999999999999988e29 < y.im

   1. Initial program 54.2%

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

      1. fma-neg 54.2%

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

      2. distribute-rgt-neg-out 54.2%

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

      3. +-commutative 54.2%

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

      4. fma-define 54.1%

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

   3. Simplified 54.1%

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

      1. distribute-rgt-neg-out 54.1%

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

      2. fma-neg 54.1%

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

      3. fma-undefine 54.2%

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

      4. +-commutative 54.2%

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

      5. div-sub 54.2%

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

      6. *-commutative 54.2%

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

      7. add-sqr-sqrt 54.2%

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

      8. times-frac 54.4%

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

      9. fma-neg 54.4%

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

      10. hypot-define 54.4%

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

      11. hypot-define 58.0%

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

      12. associate-/l* 68.0%

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

      13. add-sqr-sqrt 68.0%

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

      14. pow2 68.0%

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

   6. Applied egg-rr 68.0%

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

   7. Taylor expanded in y.re around 0 89.6%

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

      1. associate-/l* 89.7%

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

   9. Simplified 89.7%

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

       1. *-un-lft-identity 89.7%

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

       2. unpow2 89.7%

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

       3. times-frac 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. associate-*l/ 89.9%

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

       2. *-lft-identity 89.9%

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

   13. Simplified 89.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.5e-9)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 1.02e-7)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (- (* x.im (/ (/ y.re y.im) 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 <= -5.5e-9) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.02e-7) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5.5d-9)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 1.02d-7) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = (x_46im * ((y_46re / y_46im) / y_46im)) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.5e-9) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.02e-7) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.5e-9:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 1.02e-7:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = (x_46_im * ((y_46_re / y_46_im) / 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 <= -5.5e-9)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 1.02e-7)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.5e-9)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 1.02e-7)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.5e-9], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.02e-7], 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[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.4999999999999996e-9

   1. Initial program 46.1%

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

      1. fma-neg 46.1%

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

      2. distribute-rgt-neg-out 46.1%

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

      3. +-commutative 46.1%

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

      4. fma-define 46.2%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in y.re around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. unpow2 75.1%

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

      5. associate-/r* 76.7%

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

      6. div-sub 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-/l* 83.9%

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

   7. Simplified 83.9%

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

   if -5.4999999999999996e-9 < y.im < 1.02e-7

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in y.re around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. neg-mul-1 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

   7. Simplified 84.6%

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

   if 1.02e-7 < y.im

   1. Initial program 60.8%

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

      1. fma-neg 60.8%

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

      2. distribute-rgt-neg-out 60.8%

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

      3. +-commutative 60.8%

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

      4. fma-define 60.8%

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

   3. Simplified 60.8%

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

      1. distribute-rgt-neg-out 60.8%

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

      2. fma-neg 60.8%

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

      3. fma-undefine 60.8%

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

      4. +-commutative 60.8%

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

      5. div-sub 60.8%

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

      6. *-commutative 60.8%

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

      7. add-sqr-sqrt 60.8%

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

      8. times-frac 61.0%

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

      9. fma-neg 61.0%

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

      10. hypot-define 61.0%

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

      11. hypot-define 64.1%

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

      12. associate-/l* 72.6%

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

      13. add-sqr-sqrt 72.6%

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

      14. pow2 72.6%

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

   6. Applied egg-rr 72.6%

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

   7. Taylor expanded in y.re around 0 87.9%

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

      1. associate-/l* 88.1%

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

   9. Simplified 88.1%

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

       1. *-un-lft-identity 88.1%

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

       2. unpow2 88.1%

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

       3. times-frac 88.2%

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

   11. Applied egg-rr 88.2%

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

       1. associate-*l/ 88.2%

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

       2. *-lft-identity 88.2%

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

   13. Simplified 88.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{-11} \lor \neg \left(y.im \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.8e-11) (not (<= y.im 5.2e-8)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.8e-11) || !(y_46_im <= 5.2e-8)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.8d-11)) .or. (.not. (y_46im <= 5.2d-8))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.8e-11) || !(y_46_im <= 5.2e-8)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.8e-11) or not (y_46_im <= 5.2e-8):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.8e-11) || !(y_46_im <= 5.2e-8))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.8e-11) || ~((y_46_im <= 5.2e-8)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.8e-11], N[Not[LessEqual[y$46$im, 5.2e-8]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.8e-11 or 5.2000000000000002e-8 < y.im

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y.re around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. unpow2 81.2%

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

      5. associate-/r* 82.1%

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

      6. div-sub 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-/l* 85.9%

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

   7. Simplified 85.9%

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

   if -5.8e-11 < y.im < 5.2000000000000002e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in y.re around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. neg-mul-1 84.6%

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

      3. distribute-lft-neg-in 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{-11} \lor \neg \left(y.im \leq 5.2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{-9} \lor \neg \left(y.im \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.4e-9) (not (<= y.im 4.8e-8)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e-9) || !(y_46_im <= 4.8e-8)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.4d-9)) .or. (.not. (y_46im <= 4.8d-8))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e-9) || !(y_46_im <= 4.8e-8)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.4e-9) or not (y_46_im <= 4.8e-8):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.4e-9) || !(y_46_im <= 4.8e-8))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.4e-9) || ~((y_46_im <= 4.8e-8)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.4e-9], N[Not[LessEqual[y$46$im, 4.8e-8]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.39999999999999992e-9 or 4.79999999999999997e-8 < y.im

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y.re around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

      4. unpow2 81.2%

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

      5. associate-/r* 82.1%

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

      6. div-sub 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-/l* 85.9%

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

   7. Simplified 85.9%

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

   if -1.39999999999999992e-9 < y.im < 4.79999999999999997e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in y.re around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{-9} \lor \neg \left(y.im \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-11} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.2e-11) (not (<= y.im 5.5e-8)))
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e-11) || !(y_46_im <= 5.5e-8)) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-8.2d-11)) .or. (.not. (y_46im <= 5.5d-8))) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e-11) || !(y_46_im <= 5.5e-8)) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.2e-11) or not (y_46_im <= 5.5e-8):
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.2e-11) || !(y_46_im <= 5.5e-8))
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -8.2e-11) || ~((y_46_im <= 5.5e-8)))
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -8.2e-11], N[Not[LessEqual[y$46$im, 5.5e-8]], $MachinePrecision]], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.2000000000000001e-11 or 5.5000000000000003e-8 < y.im

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

      1. distribute-rgt-neg-out 53.1%

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

      2. fma-neg 53.1%

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

      3. fma-undefine 53.1%

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

      4. +-commutative 53.1%

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

      5. div-sub 53.1%

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

      6. *-commutative 53.1%

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

      7. add-sqr-sqrt 53.2%

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

      8. times-frac 54.9%

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

      9. fma-neg 54.9%

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

      10. hypot-define 54.9%

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

      11. hypot-define 63.1%

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

      12. associate-/l* 68.3%

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

      13. add-sqr-sqrt 68.3%

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

      14. pow2 68.3%

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

   6. Applied egg-rr 68.3%

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

   7. Taylor expanded in y.im around inf 82.1%

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

      1. associate-/l* 85.2%

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

   9. Simplified 85.2%

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

   if -8.2000000000000001e-11 < y.im < 5.5000000000000003e-8

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in y.re around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-11} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{-10} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.6e-10) (not (<= y.im 3.4e-7)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e-10) || !(y_46_im <= 3.4e-7)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.6d-10)) .or. (.not. (y_46im <= 3.4d-7))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e-10) || !(y_46_im <= 3.4e-7)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.6e-10) or not (y_46_im <= 3.4e-7):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.6e-10) || !(y_46_im <= 3.4e-7))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.6e-10) || ~((y_46_im <= 3.4e-7)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.6e-10], N[Not[LessEqual[y$46$im, 3.4e-7]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.60000000000000031e-10 or 3.39999999999999974e-7 < y.im

   1. Initial program 53.1%

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

      1. fma-neg 53.1%

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

      2. distribute-rgt-neg-out 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

   3. Simplified 53.1%

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

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

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

      1. associate-*r/ 71.0%

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

      2. neg-mul-1 71.0%

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

   7. Simplified 71.0%

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

   if -5.60000000000000031e-10 < y.im < 3.39999999999999974e-7

   1. Initial program 72.0%

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

      1. fma-neg 72.0%

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

      2. distribute-rgt-neg-out 72.0%

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

      3. +-commutative 72.0%

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

      4. fma-define 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in y.re around inf 84.6%

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

      1. mul-1-neg 84.6%

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

      2. unsub-neg 84.6%

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

      3. associate-/l* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{-10} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{-12} \lor \neg \left(y.im \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.1e-12) (not (<= y.im 2.3e-68)))
   (/ 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_im <= -1.1e-12) || !(y_46_im <= 2.3e-68)) {
		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_46im <= (-1.1d-12)) .or. (.not. (y_46im <= 2.3d-68))) 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_im <= -1.1e-12) || !(y_46_im <= 2.3e-68)) {
		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_im <= -1.1e-12) or not (y_46_im <= 2.3e-68):
		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_im <= -1.1e-12) || !(y_46_im <= 2.3e-68))
		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_im <= -1.1e-12) || ~((y_46_im <= 2.3e-68)))
		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[Or[LessEqual[y$46$im, -1.1e-12], N[Not[LessEqual[y$46$im, 2.3e-68]], $MachinePrecision]], 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.im < -1.09999999999999996e-12 or 2.29999999999999997e-68 < y.im

   1. Initial program 55.9%

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

      1. fma-neg 55.9%

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

      2. distribute-rgt-neg-out 55.9%

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

      3. +-commutative 55.9%

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

      4. fma-define 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in y.re around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   7. Simplified 67.4%

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

   if -1.09999999999999996e-12 < y.im < 2.29999999999999997e-68

   1. Initial program 71.0%

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

      1. fma-neg 71.0%

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

      2. distribute-rgt-neg-out 71.0%

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

      3. +-commutative 71.0%

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

      4. fma-define 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in y.re around inf 64.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{-12} \lor \neg \left(y.im \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+104} \lor \neg \left(y.im \leq 1.55 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8e+104) (not (<= y.im 1.55e+104)))
   (/ 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_im <= -8e+104) || !(y_46_im <= 1.55e+104)) {
		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_46im <= (-8d+104)) .or. (.not. (y_46im <= 1.55d+104))) 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_im <= -8e+104) || !(y_46_im <= 1.55e+104)) {
		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_im <= -8e+104) or not (y_46_im <= 1.55e+104):
		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_im <= -8e+104) || !(y_46_im <= 1.55e+104))
		tmp = 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_im <= -8e+104) || ~((y_46_im <= 1.55e+104)))
		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[Or[LessEqual[y$46$im, -8e+104], N[Not[LessEqual[y$46$im, 1.55e+104]], $MachinePrecision]], 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.im < -8e104 or 1.55000000000000008e104 < y.im

   1. Initial program 38.5%

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

      1. fma-neg 38.5%

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

      2. distribute-rgt-neg-out 38.5%

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

      3. +-commutative 38.5%

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

      4. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in y.re around 0 80.6%

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

      1. associate-*r/ 80.6%

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

      2. neg-mul-1 80.6%

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

   7. Simplified 80.6%

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

      1. neg-sub0 80.6%

         \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]

      2. sub-neg 80.6%

         \[\leadsto \frac{\color{blue}{0 + \left(-x.re\right)}}{y.im} \]

      3. add-sqr-sqrt 32.8%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-x.re} \cdot \sqrt{-x.re}}}{y.im} \]

      4. sqrt-unprod 36.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}}{y.im} \]

      5. sqr-neg 36.5%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{x.re \cdot x.re}}}{y.im} \]

      6. sqrt-unprod 13.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{x.re} \cdot \sqrt{x.re}}}{y.im} \]

      7. add-sqr-sqrt 25.2%

         \[\leadsto \frac{0 + \color{blue}{x.re}}{y.im} \]

   9. Applied egg-rr 25.2%

      \[\leadsto \frac{\color{blue}{0 + x.re}}{y.im} \]
   10. Step-by-step derivation

       1. +-lft-identity 25.2%

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

   11. Simplified 25.2%

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

   if -8e104 < y.im < 1.55000000000000008e104

   1. Initial program 74.1%

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

      1. fma-neg 74.1%

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

      2. distribute-rgt-neg-out 74.1%

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

      3. +-commutative 74.1%

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

      4. fma-define 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y.re around inf 50.6%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+104} \lor \neg \left(y.im \leq 1.55 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.9% accurate, 5.0× 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 62.6%

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

   1. fma-neg 62.6%

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

   2. distribute-rgt-neg-out 62.6%

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

   3. +-commutative 62.6%

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

   4. fma-define 62.6%

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

3. Simplified 62.6%

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

5. Taylor expanded in y.re around inf 37.4%

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

Reproduce

herbie shell --seed 2024132 
(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))))