_divideComplex, imaginary part

Percentage Accurate: 60.9% → 98.0%

Time: 16.1s

Alternatives: 15

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: 60.9% 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: 98.0% accurate, 0.0× speedup

Math

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

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))
  (/ (* (/ y.im (hypot y.im y.re)) (- x.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)), (((y_46_im / hypot(y_46_im, y_46_re)) * -x_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(Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(-x_46_re)) / 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[(N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

   1. div-sub 61.0%

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

   2. *-commutative 61.0%

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

   3. add-sqr-sqrt 61.0%

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

   4. times-frac 62.1%

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

   5. fma-neg 62.1%

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

   6. hypot-define 62.1%

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

   7. hypot-define 76.4%

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

   8. associate-/l* 79.7%

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

   9. add-sqr-sqrt 79.7%

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

   10. pow2 79.7%

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

   11. hypot-define 79.7%

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

4. Applied egg-rr 79.7%

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

   1. *-un-lft-identity 79.7%

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

   2. unpow2 79.7%

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

   3. times-frac 97.6%

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

   4. add-sqr-sqrt 47.5%

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

   5. sqrt-prod 59.3%

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

   6. sqr-neg 59.3%

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

   7. sqrt-unprod 31.1%

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

   8. add-sqr-sqrt 58.5%

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

   9. hypot-undefine 56.4%

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

   10. +-commutative 56.4%

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

   11. hypot-define 58.5%

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

   12. add-sqr-sqrt 31.1%

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

   13. sqrt-unprod 59.3%

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

   14. sqr-neg 59.3%

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

   15. sqrt-prod 47.5%

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

   16. add-sqr-sqrt 97.6%

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

   17. hypot-undefine 79.7%

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

   18. +-commutative 79.7%

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

   19. hypot-define 97.6%

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

6. Applied egg-rr 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*l/ 97.6%

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

   3. *-un-lft-identity 97.6%

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

   4. associate-*l/ 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 80.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.re \cdot x.im + \mathsf{fma}\left(x.re, -y.im, \mathsf{fma}\left(x.re, -y.im, y.im \cdot x.re\right)\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{+79}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re (/ x.im (hypot y.im y.re))) (hypot y.im y.re)))
        (t_1 (+ (* y.re y.re) (* y.im y.im)))
        (t_2
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ x.re (- y.im)))))
   (if (<= y.re -3.8e+133)
     t_0
     (if (<= y.re -1.9e-131)
       (/
        (+ (* y.re x.im) (fma x.re (- y.im) (fma x.re (- y.im) (* y.im x.re))))
        t_1)
       (if (<= y.re 8.2e-29)
         t_2
         (if (<= y.re 1.3e+44)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_1)
           (if (<= y.re 1.36e+79) t_2 t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re);
	double t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_2 = 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_re <= -3.8e+133) {
		tmp = t_0;
	} else if (y_46_re <= -1.9e-131) {
		tmp = ((y_46_re * x_46_im) + fma(x_46_re, -y_46_im, fma(x_46_re, -y_46_im, (y_46_im * x_46_re)))) / t_1;
	} else if (y_46_re <= 8.2e-29) {
		tmp = t_2;
	} else if (y_46_re <= 1.3e+44) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_1;
	} else if (y_46_re <= 1.36e+79) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re))
	t_1 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_2 = 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_re <= -3.8e+133)
		tmp = t_0;
	elseif (y_46_re <= -1.9e-131)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) + fma(x_46_re, Float64(-y_46_im), fma(x_46_re, Float64(-y_46_im), Float64(y_46_im * x_46_re)))) / t_1);
	elseif (y_46_re <= 8.2e-29)
		tmp = t_2;
	elseif (y_46_re <= 1.3e+44)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_1);
	elseif (y_46_re <= 1.36e+79)
		tmp = t_2;
	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[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$46$re, -3.8e+133], t$95$0, If[LessEqual[y$46$re, -1.9e-131], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * (-y$46$im) + N[(x$46$re * (-y$46$im) + N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 8.2e-29], t$95$2, If[LessEqual[y$46$re, 1.3e+44], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 1.36e+79], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.8000000000000002e133 or 1.36000000000000003e79 < y.re

   1. Initial program 34.7%

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

   3. Taylor expanded in x.im around inf 35.1%

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

      1. rem-square-sqrt 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. +-commutative 35.1%

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

      7. unpow2 35.1%

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

      8. unpow2 35.1%

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

      9. hypot-undefine 35.1%

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

      10. unpow2 35.1%

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

      11. *-commutative 35.1%

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

      12. associate-*r/ 36.1%

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

      13. hypot-undefine 36.1%

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

      14. unpow2 36.1%

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

      15. unpow2 36.1%

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

      16. +-commutative 36.1%

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

      17. unpow2 36.1%

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

      18. unpow2 36.1%

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

      19. hypot-define 36.1%

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

   5. Simplified 36.1%

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

      1. associate-*r/ 35.1%

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

      2. unpow2 35.1%

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

      3. hypot-undefine 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. unpow2 35.1%

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

      7. add-sqr-sqrt 35.1%

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

      8. +-commutative 35.1%

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

      9. unpow2 35.1%

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

      10. rem-square-sqrt 35.1%

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

      11. hypot-undefine 35.1%

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

      12. hypot-undefine 35.1%

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

      13. frac-times 79.9%

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

      14. associate-*l/ 79.9%

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

   7. Applied egg-rr 79.9%

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

   if -3.8000000000000002e133 < y.re < -1.89999999999999997e-131

   1. Initial program 90.6%

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

      1. prod-diff 90.6%

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

      2. *-commutative 90.6%

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

      3. fma-define 90.6%

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

      4. associate-+l+ 90.6%

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

      5. distribute-rgt-neg-in 90.6%

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

      6. fma-define 90.7%

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

      7. *-commutative 90.7%

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

      8. fma-undefine 90.6%

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

      9. distribute-lft-neg-in 90.6%

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

      10. *-commutative 90.6%

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

      11. distribute-rgt-neg-in 90.6%

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

      12. fma-define 90.7%

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

   4. Applied egg-rr 90.7%

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

   if -1.89999999999999997e-131 < y.re < 8.1999999999999996e-29 or 1.3e44 < y.re < 1.36000000000000003e79

   1. Initial program 66.5%

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

      1. div-sub 61.4%

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

      2. *-commutative 61.4%

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

      3. add-sqr-sqrt 61.3%

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

      4. times-frac 62.3%

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

      5. fma-neg 62.3%

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

      6. hypot-define 62.3%

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

      7. hypot-define 64.3%

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

      8. associate-/l* 70.1%

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

      9. add-sqr-sqrt 70.1%

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

      10. pow2 70.1%

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

      11. hypot-define 70.1%

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

   4. Applied egg-rr 70.1%

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

   5. Taylor expanded in y.im around inf 95.2%

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

   if 8.1999999999999996e-29 < y.re < 1.3e44

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. distribute-lft-neg-in 89.5%

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

      5. fma-define 89.5%

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

   4. Applied egg-rr 89.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.re \cdot x.im + \mathsf{fma}\left(x.re, -y.im, \mathsf{fma}\left(x.re, -y.im, y.im \cdot x.re\right)\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
      2e+281)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.im y.re (* x.re (- y.im))) (hypot y.re y.im)))
   (fma
    (/ y.re (hypot y.re y.im))
    (/ x.im (hypot y.re y.im))
    (/ x.re (- y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+281) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = 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));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 2e+281)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) / hypot(y_46_re, y_46_im)));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 2e+281], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e281

   1. Initial program 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. add-sqr-sqrt 78.0%

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

      3. times-frac 78.1%

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

      4. hypot-define 78.1%

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

      5. fma-neg 78.1%

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

      6. distribute-rgt-neg-in 78.1%

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

      7. hypot-define 96.9%

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

   4. Applied egg-rr 96.9%

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

   if 2.0000000000000001e281 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 15.0%

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

      1. div-sub 10.0%

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

      2. *-commutative 10.0%

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

      3. add-sqr-sqrt 10.0%

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

      4. times-frac 11.2%

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

      5. fma-neg 11.2%

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

      6. hypot-define 11.2%

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

      7. hypot-define 44.3%

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

      8. associate-/l* 54.0%

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

      9. add-sqr-sqrt 54.0%

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

      10. pow2 54.0%

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

      11. hypot-define 54.0%

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

   4. Applied egg-rr 54.0%

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

   5. Taylor expanded in y.im around inf 73.3%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t\_0}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+79}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (/ (* y.re (/ x.im (hypot y.im y.re))) (hypot y.im y.re)))
        (t_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -4.2e+134)
     t_1
     (if (<= y.re -1.05e-139)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.re 2e-29)
         t_2
         (if (<= y.re 1.6e+44)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
           (if (<= y.re 1.25e+79) t_2 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re);
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -4.2e+134) {
		tmp = t_1;
	} else if (y_46_re <= -1.05e-139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_re <= 2e-29) {
		tmp = t_2;
	} else if (y_46_re <= 1.6e+44) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else if (y_46_re <= 1.25e+79) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re))
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e+134)
		tmp = t_1;
	elseif (y_46_re <= -1.05e-139)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_re <= 2e-29)
		tmp = t_2;
	elseif (y_46_re <= 1.6e+44)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	elseif (y_46_re <= 1.25e+79)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -4.2e+134], t$95$1, If[LessEqual[y$46$re, -1.05e-139], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2e-29], t$95$2, If[LessEqual[y$46$re, 1.6e+44], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+79], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.2000000000000002e134 or 1.25e79 < y.re

   1. Initial program 34.7%

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

   3. Taylor expanded in x.im around inf 35.1%

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

      1. rem-square-sqrt 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. +-commutative 35.1%

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

      7. unpow2 35.1%

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

      8. unpow2 35.1%

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

      9. hypot-undefine 35.1%

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

      10. unpow2 35.1%

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

      11. *-commutative 35.1%

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

      12. associate-*r/ 36.1%

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

      13. hypot-undefine 36.1%

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

      14. unpow2 36.1%

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

      15. unpow2 36.1%

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

      16. +-commutative 36.1%

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

      17. unpow2 36.1%

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

      18. unpow2 36.1%

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

      19. hypot-define 36.1%

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

   5. Simplified 36.1%

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

      1. associate-*r/ 35.1%

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

      2. unpow2 35.1%

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

      3. hypot-undefine 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. unpow2 35.1%

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

      7. add-sqr-sqrt 35.1%

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

      8. +-commutative 35.1%

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

      9. unpow2 35.1%

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

      10. rem-square-sqrt 35.1%

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

      11. hypot-undefine 35.1%

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

      12. hypot-undefine 35.1%

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

      13. frac-times 79.9%

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

      14. associate-*l/ 79.9%

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

   7. Applied egg-rr 79.9%

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

   if -4.2000000000000002e134 < y.re < -1.05000000000000004e-139

   1. Initial program 90.6%

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

   if -1.05000000000000004e-139 < y.re < 1.99999999999999989e-29 or 1.60000000000000002e44 < y.re < 1.25e79

   1. Initial program 66.5%

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

      1. div-sub 61.4%

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

      2. *-commutative 61.4%

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

      3. add-sqr-sqrt 61.3%

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

      4. times-frac 62.3%

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

      5. fma-neg 62.3%

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

      6. hypot-define 62.3%

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

      7. hypot-define 64.3%

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

      8. associate-/l* 70.1%

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

      9. add-sqr-sqrt 70.1%

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

      10. pow2 70.1%

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

      11. hypot-define 70.1%

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

   4. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 49.6%

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

      5. sqrt-prod 48.7%

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

      6. sqr-neg 48.7%

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

      7. sqrt-unprod 19.6%

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

      8. add-sqr-sqrt 37.1%

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

      9. hypot-undefine 31.7%

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

      10. +-commutative 31.7%

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

      11. hypot-define 37.1%

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

      12. add-sqr-sqrt 19.6%

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

      13. sqrt-unprod 48.7%

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

      14. sqr-neg 48.7%

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

      15. sqrt-prod 49.6%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.1%

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

      18. +-commutative 70.1%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in y.re around 0 83.3%

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

       1. neg-mul-1 83.3%

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

       2. associate-*r/ 83.9%

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

       3. +-commutative 83.9%

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

       4. unsub-neg 83.9%

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

       5. *-lft-identity 83.9%

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

       6. unpow2 83.9%

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

       7. times-frac 90.2%

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

       8. *-commutative 90.2%

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

       9. associate-*r/ 90.2%

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

       10. *-rgt-identity 90.2%

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

       11. associate-/l* 91.0%

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

       12. div-sub 92.1%

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

   11. Simplified 92.1%

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

   if 1.99999999999999989e-29 < y.re < 1.60000000000000002e44

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. distribute-lft-neg-in 89.5%

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

      5. fma-define 89.5%

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

   4. Applied egg-rr 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\frac{y.re \cdot x.im + \mathsf{fma}\left(x.re, -y.im, \mathsf{fma}\left(x.re, -y.im, y.im \cdot x.re\right)\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-28}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re (/ x.im (hypot y.im y.re))) (hypot y.im y.re)))
        (t_1 (+ (* y.re y.re) (* y.im y.im)))
        (t_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -9.2e+133)
     t_0
     (if (<= y.re -1.9e-136)
       (/
        (+ (* y.re x.im) (fma x.re (- y.im) (fma x.re (- y.im) (* y.im x.re))))
        t_1)
       (if (<= y.re 1.2e-28)
         t_2
         (if (<= y.re 1.6e+44)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_1)
           (if (<= y.re 1.7e+79) t_2 t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re);
	double t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -9.2e+133) {
		tmp = t_0;
	} else if (y_46_re <= -1.9e-136) {
		tmp = ((y_46_re * x_46_im) + fma(x_46_re, -y_46_im, fma(x_46_re, -y_46_im, (y_46_im * x_46_re)))) / t_1;
	} else if (y_46_re <= 1.2e-28) {
		tmp = t_2;
	} else if (y_46_re <= 1.6e+44) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_1;
	} else if (y_46_re <= 1.7e+79) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re))
	t_1 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -9.2e+133)
		tmp = t_0;
	elseif (y_46_re <= -1.9e-136)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) + fma(x_46_re, Float64(-y_46_im), fma(x_46_re, Float64(-y_46_im), Float64(y_46_im * x_46_re)))) / t_1);
	elseif (y_46_re <= 1.2e-28)
		tmp = t_2;
	elseif (y_46_re <= 1.6e+44)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_1);
	elseif (y_46_re <= 1.7e+79)
		tmp = t_2;
	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[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -9.2e+133], t$95$0, If[LessEqual[y$46$re, -1.9e-136], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * (-y$46$im) + N[(x$46$re * (-y$46$im) + N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 1.2e-28], t$95$2, If[LessEqual[y$46$re, 1.6e+44], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+79], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.1999999999999996e133 or 1.70000000000000016e79 < y.re

   1. Initial program 34.7%

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

   3. Taylor expanded in x.im around inf 35.1%

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

      1. rem-square-sqrt 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. +-commutative 35.1%

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

      7. unpow2 35.1%

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

      8. unpow2 35.1%

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

      9. hypot-undefine 35.1%

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

      10. unpow2 35.1%

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

      11. *-commutative 35.1%

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

      12. associate-*r/ 36.1%

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

      13. hypot-undefine 36.1%

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

      14. unpow2 36.1%

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

      15. unpow2 36.1%

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

      16. +-commutative 36.1%

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

      17. unpow2 36.1%

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

      18. unpow2 36.1%

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

      19. hypot-define 36.1%

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

   5. Simplified 36.1%

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

      1. associate-*r/ 35.1%

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

      2. unpow2 35.1%

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

      3. hypot-undefine 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-undefine 35.1%

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

      6. unpow2 35.1%

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

      7. add-sqr-sqrt 35.1%

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

      8. +-commutative 35.1%

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

      9. unpow2 35.1%

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

      10. rem-square-sqrt 35.1%

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

      11. hypot-undefine 35.1%

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

      12. hypot-undefine 35.1%

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

      13. frac-times 79.9%

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

      14. associate-*l/ 79.9%

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

   7. Applied egg-rr 79.9%

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

   if -9.1999999999999996e133 < y.re < -1.9000000000000001e-136

   1. Initial program 90.6%

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

      1. prod-diff 90.6%

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

      2. *-commutative 90.6%

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

      3. fma-define 90.6%

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

      4. associate-+l+ 90.6%

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

      5. distribute-rgt-neg-in 90.6%

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

      6. fma-define 90.7%

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

      7. *-commutative 90.7%

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

      8. fma-undefine 90.6%

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

      9. distribute-lft-neg-in 90.6%

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

      10. *-commutative 90.6%

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

      11. distribute-rgt-neg-in 90.6%

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

      12. fma-define 90.7%

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

   4. Applied egg-rr 90.7%

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

   if -1.9000000000000001e-136 < y.re < 1.2000000000000001e-28 or 1.60000000000000002e44 < y.re < 1.70000000000000016e79

   1. Initial program 66.5%

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

      1. div-sub 61.4%

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

      2. *-commutative 61.4%

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

      3. add-sqr-sqrt 61.3%

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

      4. times-frac 62.3%

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

      5. fma-neg 62.3%

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

      6. hypot-define 62.3%

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

      7. hypot-define 64.3%

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

      8. associate-/l* 70.1%

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

      9. add-sqr-sqrt 70.1%

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

      10. pow2 70.1%

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

      11. hypot-define 70.1%

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

   4. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 49.6%

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

      5. sqrt-prod 48.7%

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

      6. sqr-neg 48.7%

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

      7. sqrt-unprod 19.6%

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

      8. add-sqr-sqrt 37.1%

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

      9. hypot-undefine 31.7%

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

      10. +-commutative 31.7%

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

      11. hypot-define 37.1%

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

      12. add-sqr-sqrt 19.6%

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

      13. sqrt-unprod 48.7%

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

      14. sqr-neg 48.7%

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

      15. sqrt-prod 49.6%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.1%

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

      18. +-commutative 70.1%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in y.re around 0 83.3%

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

       1. neg-mul-1 83.3%

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

       2. associate-*r/ 83.9%

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

       3. +-commutative 83.9%

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

       4. unsub-neg 83.9%

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

       5. *-lft-identity 83.9%

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

       6. unpow2 83.9%

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

       7. times-frac 90.2%

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

       8. *-commutative 90.2%

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

       9. associate-*r/ 90.2%

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

       10. *-rgt-identity 90.2%

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

       11. associate-/l* 91.0%

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

       12. div-sub 92.1%

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

   11. Simplified 92.1%

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

   if 1.2000000000000001e-28 < y.re < 1.60000000000000002e44

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. distribute-lft-neg-in 89.5%

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

      5. fma-define 89.5%

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

   4. Applied egg-rr 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\frac{y.re \cdot x.im + \mathsf{fma}\left(x.re, -y.im, \mathsf{fma}\left(x.re, -y.im, y.im \cdot x.re\right)\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.12 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t\_0}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -2.12e+133)
     (- (/ x.im y.re) (* x.re (* y.im (pow y.re -2.0))))
     (if (<= y.re -6e-131)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.re 2e-29)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 2.5e+40)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
           (/ x.im (* (hypot y.im y.re) (/ (hypot 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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -2.12e+133) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im * pow(y_46_re, -2.0)));
	} else if (y_46_re <= -6e-131) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_re <= 2e-29) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.5e+40) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else {
		tmp = x_46_im / (hypot(y_46_im, y_46_re) * (hypot(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)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.12e+133)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im * (y_46_re ^ -2.0))));
	elseif (y_46_re <= -6e-131)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_re <= 2e-29)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.5e+40)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	else
		tmp = Float64(x_46_im / Float64(hypot(y_46_im, y_46_re) * Float64(hypot(y_46_im, y_46_re) / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.12e+133], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im * N[Power[y$46$re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6e-131], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2e-29], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+40], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$im / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.12e133

   1. Initial program 30.2%

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

   3. Taylor expanded in y.re around inf 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

      1. div-inv 77.3%

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

      2. pow-flip 79.1%

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

      3. metadata-eval 79.1%

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

   7. Applied egg-rr 79.1%

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

   if -2.12e133 < y.re < -5.99999999999999992e-131

   1. Initial program 90.6%

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

   if -5.99999999999999992e-131 < y.re < 1.99999999999999989e-29

   1. Initial program 69.7%

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

      1. div-sub 64.0%

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

      2. *-commutative 64.0%

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

      3. add-sqr-sqrt 64.0%

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

      4. times-frac 65.1%

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

      5. fma-neg 65.1%

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

      6. hypot-define 65.1%

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

      7. hypot-define 65.1%

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

      8. associate-/l* 71.4%

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

      9. add-sqr-sqrt 71.4%

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

      10. pow2 71.4%

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

      11. hypot-define 71.4%

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

   4. Applied egg-rr 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. unpow2 71.4%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 49.1%

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

      5. sqrt-prod 51.2%

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

      6. sqr-neg 51.2%

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

      7. sqrt-unprod 19.3%

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

      8. add-sqr-sqrt 35.2%

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

      9. hypot-undefine 29.2%

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

      10. +-commutative 29.2%

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

      11. hypot-define 35.2%

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

      12. add-sqr-sqrt 19.3%

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

      13. sqrt-unprod 51.2%

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

      14. sqr-neg 51.2%

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

      15. sqrt-prod 49.1%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 71.3%

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

      18. +-commutative 71.3%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in y.re around 0 86.8%

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

       1. neg-mul-1 86.8%

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

       2. associate-*r/ 87.3%

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

       3. +-commutative 87.3%

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

       4. unsub-neg 87.3%

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

       5. *-lft-identity 87.3%

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

       6. unpow2 87.3%

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

       7. times-frac 92.1%

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

       8. *-commutative 92.1%

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

       9. associate-*r/ 92.2%

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

       10. *-rgt-identity 92.2%

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

       11. associate-/l* 92.2%

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

       12. div-sub 93.3%

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

   11. Simplified 93.3%

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

   if 1.99999999999999989e-29 < y.re < 2.50000000000000002e40

   1. Initial program 89.0%

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

      1. sub-neg 89.0%

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

      2. +-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. distribute-lft-neg-in 89.0%

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

      5. fma-define 89.1%

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

   4. Applied egg-rr 89.1%

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

   if 2.50000000000000002e40 < y.re

   1. Initial program 39.1%

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

   3. Taylor expanded in x.im around inf 39.7%

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

      1. rem-square-sqrt 39.7%

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

      2. +-commutative 39.7%

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

      3. unpow2 39.7%

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

      4. unpow2 39.7%

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

      5. hypot-undefine 39.7%

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

      6. +-commutative 39.7%

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

      7. unpow2 39.7%

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

      8. unpow2 39.7%

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

      9. hypot-undefine 39.7%

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

      10. unpow2 39.7%

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

      11. *-commutative 39.7%

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

      12. associate-*r/ 40.7%

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

      13. hypot-undefine 40.7%

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

      14. unpow2 40.7%

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

      15. unpow2 40.7%

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

      16. +-commutative 40.7%

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

      17. unpow2 40.7%

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

      18. unpow2 40.7%

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

      19. hypot-define 40.7%

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

   5. Simplified 40.7%

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

      1. associate-*r/ 39.7%

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

      2. unpow2 39.7%

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

      3. hypot-undefine 39.7%

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

      4. unpow2 39.7%

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

      5. hypot-undefine 39.7%

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

      6. unpow2 39.7%

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

      7. add-sqr-sqrt 39.7%

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

      8. +-commutative 39.7%

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

      9. unpow2 39.7%

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

      10. rem-square-sqrt 39.7%

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

      11. hypot-undefine 39.7%

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

      12. hypot-undefine 39.7%

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

      13. frac-times 76.5%

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

      14. clear-num 76.5%

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

      15. frac-times 74.8%

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

      16. *-un-lft-identity 74.8%

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

   7. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.re} \cdot \mathsf{hypot}\left(y.im, y.re\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.12 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -2.12 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (* x.re (* y.im (pow y.re -2.0)))))
        (t_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -2.12e+133)
     t_1
     (if (<= y.re -5.5e-129)
       t_0
       (if (<= y.re 2.4e-29)
         t_2
         (if (<= y.re 1.75e+44) t_0 (if (<= y.re 1e+88) t_2 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im * pow(y_46_re, -2.0)));
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.12e+133) {
		tmp = t_1;
	} else if (y_46_re <= -5.5e-129) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-29) {
		tmp = t_2;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_0;
	} else if (y_46_re <= 1e+88) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (x_46re * (y_46im * (y_46re ** (-2.0d0))))
    t_2 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46re <= (-2.12d+133)) then
        tmp = t_1
    else if (y_46re <= (-5.5d-129)) then
        tmp = t_0
    else if (y_46re <= 2.4d-29) then
        tmp = t_2
    else if (y_46re <= 1.75d+44) then
        tmp = t_0
    else if (y_46re <= 1d+88) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im * Math.pow(y_46_re, -2.0)));
	double t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.12e+133) {
		tmp = t_1;
	} else if (y_46_re <= -5.5e-129) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-29) {
		tmp = t_2;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_0;
	} else if (y_46_re <= 1e+88) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im * math.pow(y_46_re, -2.0)))
	t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_re <= -2.12e+133:
		tmp = t_1
	elif y_46_re <= -5.5e-129:
		tmp = t_0
	elif y_46_re <= 2.4e-29:
		tmp = t_2
	elif y_46_re <= 1.75e+44:
		tmp = t_0
	elif y_46_re <= 1e+88:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im * (y_46_re ^ -2.0))))
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.12e+133)
		tmp = t_1;
	elseif (y_46_re <= -5.5e-129)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-29)
		tmp = t_2;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_0;
	elseif (y_46_re <= 1e+88)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im * (y_46_re ^ -2.0)));
	t_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -2.12e+133)
		tmp = t_1;
	elseif (y_46_re <= -5.5e-129)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-29)
		tmp = t_2;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_0;
	elseif (y_46_re <= 1e+88)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im * N[Power[y$46$re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -2.12e+133], t$95$1, If[LessEqual[y$46$re, -5.5e-129], t$95$0, If[LessEqual[y$46$re, 2.4e-29], t$95$2, If[LessEqual[y$46$re, 1.75e+44], t$95$0, If[LessEqual[y$46$re, 1e+88], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.12e133 or 9.99999999999999959e87 < y.re

   1. Initial program 34.0%

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

   3. Taylor expanded in y.re around inf 73.6%

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

      1. +-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. associate-/l* 77.8%

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

   5. Simplified 77.8%

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

      1. div-inv 77.8%

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

      2. pow-flip 78.9%

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

      3. metadata-eval 78.9%

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

   7. Applied egg-rr 78.9%

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

   if -2.12e133 < y.re < -5.50000000000000023e-129 or 2.39999999999999992e-29 < y.re < 1.75e44

   1. Initial program 90.3%

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

   if -5.50000000000000023e-129 < y.re < 2.39999999999999992e-29 or 1.75e44 < y.re < 9.99999999999999959e87

   1. Initial program 65.9%

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

      1. div-sub 60.9%

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

      2. *-commutative 60.9%

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

      3. add-sqr-sqrt 60.9%

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

      4. times-frac 61.9%

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

      5. fma-neg 61.9%

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

      6. hypot-define 61.9%

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

      7. hypot-define 64.6%

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

      8. associate-/l* 70.3%

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

      9. add-sqr-sqrt 70.3%

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

      10. pow2 70.3%

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

      11. hypot-define 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. unpow2 70.3%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 48.7%

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

      5. sqrt-prod 47.9%

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

      6. sqr-neg 47.9%

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

      7. sqrt-unprod 21.7%

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

      8. add-sqr-sqrt 38.6%

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

      9. hypot-undefine 33.3%

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

      10. +-commutative 33.3%

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

      11. hypot-define 38.6%

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

      12. add-sqr-sqrt 21.7%

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

      13. sqrt-unprod 47.9%

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

      14. sqr-neg 47.9%

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

      15. sqrt-prod 48.7%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.3%

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

      18. +-commutative 70.3%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. associate-*r/ 82.7%

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

       3. +-commutative 82.7%

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

       4. unsub-neg 82.7%

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

       5. *-lft-identity 82.7%

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

       6. unpow2 82.7%

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

       7. times-frac 88.7%

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

       8. *-commutative 88.7%

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

       9. associate-*r/ 88.7%

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

       10. *-rgt-identity 88.7%

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

       11. associate-/l* 90.4%

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

       12. div-sub 91.4%

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

   11. Simplified 91.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.12 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{+88}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-136}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t\_2}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_2}\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im))
        (t_1 (- (/ x.im y.re) (* x.re (* y.im (pow y.re -2.0)))))
        (t_2 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -2.2e+135)
     t_1
     (if (<= y.re -1.1e-136)
       (/ (- (* y.re x.im) (* y.im x.re)) t_2)
       (if (<= y.re 3.3e-29)
         t_0
         (if (<= y.re 1.75e+44)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_2)
           (if (<= y.re 9.6e+87) t_0 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im * pow(y_46_re, -2.0)));
	double t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -2.2e+135) {
		tmp = t_1;
	} else if (y_46_re <= -1.1e-136) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_2;
	} else if (y_46_re <= 3.3e-29) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e+44) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_2;
	} else if (y_46_re <= 9.6e+87) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im * (y_46_re ^ -2.0))))
	t_2 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.2e+135)
		tmp = t_1;
	elseif (y_46_re <= -1.1e-136)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_2);
	elseif (y_46_re <= 3.3e-29)
		tmp = t_0;
	elseif (y_46_re <= 1.75e+44)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_2);
	elseif (y_46_re <= 9.6e+87)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im * N[Power[y$46$re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+135], t$95$1, If[LessEqual[y$46$re, -1.1e-136], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 3.3e-29], t$95$0, If[LessEqual[y$46$re, 1.75e+44], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 9.6e+87], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1999999999999999e135 or 9.59999999999999926e87 < y.re

   1. Initial program 34.0%

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

   3. Taylor expanded in y.re around inf 73.6%

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

      1. +-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. associate-/l* 77.8%

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

   5. Simplified 77.8%

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

      1. div-inv 77.8%

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

      2. pow-flip 78.9%

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

      3. metadata-eval 78.9%

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

   7. Applied egg-rr 78.9%

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

   if -2.1999999999999999e135 < y.re < -1.1000000000000001e-136

   1. Initial program 90.6%

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

   if -1.1000000000000001e-136 < y.re < 3.30000000000000028e-29 or 1.75e44 < y.re < 9.59999999999999926e87

   1. Initial program 65.9%

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

      1. div-sub 60.9%

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

      2. *-commutative 60.9%

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

      3. add-sqr-sqrt 60.9%

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

      4. times-frac 61.9%

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

      5. fma-neg 61.9%

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

      6. hypot-define 61.9%

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

      7. hypot-define 64.6%

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

      8. associate-/l* 70.3%

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

      9. add-sqr-sqrt 70.3%

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

      10. pow2 70.3%

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

      11. hypot-define 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. unpow2 70.3%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 48.7%

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

      5. sqrt-prod 47.9%

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

      6. sqr-neg 47.9%

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

      7. sqrt-unprod 21.7%

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

      8. add-sqr-sqrt 38.6%

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

      9. hypot-undefine 33.3%

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

      10. +-commutative 33.3%

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

      11. hypot-define 38.6%

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

      12. add-sqr-sqrt 21.7%

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

      13. sqrt-unprod 47.9%

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

      14. sqr-neg 47.9%

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

      15. sqrt-prod 48.7%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.3%

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

      18. +-commutative 70.3%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. associate-*r/ 82.7%

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

       3. +-commutative 82.7%

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

       4. unsub-neg 82.7%

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

       5. *-lft-identity 82.7%

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

       6. unpow2 82.7%

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

       7. times-frac 88.7%

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

       8. *-commutative 88.7%

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

       9. associate-*r/ 88.7%

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

       10. *-rgt-identity 88.7%

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

       11. associate-/l* 90.4%

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

       12. div-sub 91.4%

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

   11. Simplified 91.4%

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

   if 3.30000000000000028e-29 < y.re < 1.75e44

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. distribute-lft-neg-in 89.5%

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

      5. fma-define 89.5%

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

   4. Applied egg-rr 89.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-136}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im))
        (t_1
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.3e+135)
     (/ x.im y.re)
     (if (<= y.re -1.8e-135)
       t_1
       (if (<= y.re 2e-29)
         t_0
         (if (<= y.re 1.75e+44)
           t_1
           (if (<= y.re 9.6e+87) t_0 (/ x.im (hypot y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.3e+135) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.8e-135) {
		tmp = t_1;
	} else if (y_46_re <= 2e-29) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_1;
	} else if (y_46_re <= 9.6e+87) {
		tmp = t_0;
	} else {
		tmp = x_46_im / hypot(y_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.3e+135) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.8e-135) {
		tmp = t_1;
	} else if (y_46_re <= 2e-29) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_1;
	} else if (y_46_re <= 9.6e+87) {
		tmp = t_0;
	} else {
		tmp = x_46_im / Math.hypot(y_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.3e+135:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -1.8e-135:
		tmp = t_1
	elif y_46_re <= 2e-29:
		tmp = t_0
	elif y_46_re <= 1.75e+44:
		tmp = t_1
	elif y_46_re <= 9.6e+87:
		tmp = t_0
	else:
		tmp = x_46_im / math.hypot(y_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	t_1 = 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)))
	tmp = 0.0
	if (y_46_re <= -1.3e+135)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.8e-135)
		tmp = t_1;
	elseif (y_46_re <= 2e-29)
		tmp = t_0;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_1;
	elseif (y_46_re <= 9.6e+87)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / hypot(y_46_im, y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.3e+135)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -1.8e-135)
		tmp = t_1;
	elseif (y_46_re <= 2e-29)
		tmp = t_0;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_1;
	elseif (y_46_re <= 9.6e+87)
		tmp = t_0;
	else
		tmp = x_46_im / hypot(y_46_im, y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$re, -1.3e+135], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.8e-135], t$95$1, If[LessEqual[y$46$re, 2e-29], t$95$0, If[LessEqual[y$46$re, 1.75e+44], t$95$1, If[LessEqual[y$46$re, 9.6e+87], t$95$0, N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.3e135

   1. Initial program 30.2%

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

   3. Taylor expanded in y.re around inf 77.3%

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

   if -1.3e135 < y.re < -1.79999999999999989e-135 or 1.99999999999999989e-29 < y.re < 1.75e44

   1. Initial program 90.3%

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

   if -1.79999999999999989e-135 < y.re < 1.99999999999999989e-29 or 1.75e44 < y.re < 9.59999999999999926e87

   1. Initial program 65.9%

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

      1. div-sub 60.9%

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

      2. *-commutative 60.9%

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

      3. add-sqr-sqrt 60.9%

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

      4. times-frac 61.9%

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

      5. fma-neg 61.9%

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

      6. hypot-define 61.9%

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

      7. hypot-define 64.6%

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

      8. associate-/l* 70.3%

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

      9. add-sqr-sqrt 70.3%

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

      10. pow2 70.3%

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

      11. hypot-define 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. unpow2 70.3%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 48.7%

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

      5. sqrt-prod 47.9%

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

      6. sqr-neg 47.9%

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

      7. sqrt-unprod 21.7%

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

      8. add-sqr-sqrt 38.6%

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

      9. hypot-undefine 33.3%

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

      10. +-commutative 33.3%

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

      11. hypot-define 38.6%

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

      12. add-sqr-sqrt 21.7%

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

      13. sqrt-unprod 47.9%

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

      14. sqr-neg 47.9%

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

      15. sqrt-prod 48.7%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.3%

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

      18. +-commutative 70.3%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. associate-*r/ 82.7%

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

       3. +-commutative 82.7%

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

       4. unsub-neg 82.7%

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

       5. *-lft-identity 82.7%

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

       6. unpow2 82.7%

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

       7. times-frac 88.7%

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

       8. *-commutative 88.7%

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

       9. associate-*r/ 88.7%

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

       10. *-rgt-identity 88.7%

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

       11. associate-/l* 90.4%

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

       12. div-sub 91.4%

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

   11. Simplified 91.4%

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

   if 9.59999999999999926e87 < y.re

   1. Initial program 37.7%

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

   3. Taylor expanded in x.im around inf 38.1%

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

      1. rem-square-sqrt 38.1%

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

      2. +-commutative 38.1%

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

      3. unpow2 38.1%

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

      4. unpow2 38.1%

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

      5. hypot-undefine 38.1%

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

      6. +-commutative 38.1%

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

      7. unpow2 38.1%

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

      8. unpow2 38.1%

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

      9. hypot-undefine 38.1%

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

      10. unpow2 38.1%

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

      11. *-commutative 38.1%

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

      12. associate-*r/ 39.1%

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

      13. hypot-undefine 39.1%

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

      14. unpow2 39.1%

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

      15. unpow2 39.1%

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

      16. +-commutative 39.1%

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

      17. unpow2 39.1%

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

      18. unpow2 39.1%

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

      19. hypot-define 39.1%

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

   5. Simplified 39.1%

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

      1. associate-*r/ 38.1%

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

      2. unpow2 38.1%

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

      3. hypot-undefine 38.1%

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

      4. unpow2 38.1%

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

      5. hypot-undefine 38.1%

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

      6. unpow2 38.1%

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

      7. add-sqr-sqrt 38.1%

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

      8. +-commutative 38.1%

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

      9. unpow2 38.1%

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

      10. rem-square-sqrt 38.1%

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

      11. hypot-undefine 38.1%

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

      12. hypot-undefine 38.1%

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

      13. frac-times 80.5%

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

      14. associate-*l/ 80.5%

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

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in y.re around inf 78.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re \cdot \left(-y.re\right) - y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -2.9e+92)
     (/ x.im y.re)
     (if (<= y.re -3.1e-84)
       t_0
       (if (<= y.re -6.6e-128)
         (/ (* y.im x.re) (- (* y.re (- y.re)) (* y.im y.im)))
         (if (<= y.re 7.2e-26)
           t_1
           (if (<= y.re 2.7e+39)
             t_0
             (if (<= y.re 3.8e+89) t_1 (/ x.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.9e+92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.1e-84) {
		tmp = t_0;
	} else if (y_46_re <= -6.6e-128) {
		tmp = (y_46_im * x_46_re) / ((y_46_re * -y_46_re) - (y_46_im * y_46_im));
	} else if (y_46_re <= 7.2e-26) {
		tmp = t_1;
	} else if (y_46_re <= 2.7e+39) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+89) {
		tmp = t_1;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46re <= (-2.9d+92)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-3.1d-84)) then
        tmp = t_0
    else if (y_46re <= (-6.6d-128)) then
        tmp = (y_46im * x_46re) / ((y_46re * -y_46re) - (y_46im * y_46im))
    else if (y_46re <= 7.2d-26) then
        tmp = t_1
    else if (y_46re <= 2.7d+39) then
        tmp = t_0
    else if (y_46re <= 3.8d+89) then
        tmp = t_1
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -2.9e+92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.1e-84) {
		tmp = t_0;
	} else if (y_46_re <= -6.6e-128) {
		tmp = (y_46_im * x_46_re) / ((y_46_re * -y_46_re) - (y_46_im * y_46_im));
	} else if (y_46_re <= 7.2e-26) {
		tmp = t_1;
	} else if (y_46_re <= 2.7e+39) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+89) {
		tmp = t_1;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_re <= -2.9e+92:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -3.1e-84:
		tmp = t_0
	elif y_46_re <= -6.6e-128:
		tmp = (y_46_im * x_46_re) / ((y_46_re * -y_46_re) - (y_46_im * y_46_im))
	elif y_46_re <= 7.2e-26:
		tmp = t_1
	elif y_46_re <= 2.7e+39:
		tmp = t_0
	elif y_46_re <= 3.8e+89:
		tmp = t_1
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.9e+92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.1e-84)
		tmp = t_0;
	elseif (y_46_re <= -6.6e-128)
		tmp = Float64(Float64(y_46_im * x_46_re) / Float64(Float64(y_46_re * Float64(-y_46_re)) - Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 7.2e-26)
		tmp = t_1;
	elseif (y_46_re <= 2.7e+39)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+89)
		tmp = t_1;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -2.9e+92)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -3.1e-84)
		tmp = t_0;
	elseif (y_46_re <= -6.6e-128)
		tmp = (y_46_im * x_46_re) / ((y_46_re * -y_46_re) - (y_46_im * y_46_im));
	elseif (y_46_re <= 7.2e-26)
		tmp = t_1;
	elseif (y_46_re <= 2.7e+39)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+89)
		tmp = t_1;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e+92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-84], t$95$0, If[LessEqual[y$46$re, -6.6e-128], N[(N[(y$46$im * x$46$re), $MachinePrecision] / N[(N[(y$46$re * (-y$46$re)), $MachinePrecision] - N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-26], t$95$1, If[LessEqual[y$46$re, 2.7e+39], t$95$0, If[LessEqual[y$46$re, 3.8e+89], t$95$1, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.9000000000000001e92 or 3.80000000000000023e89 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around inf 78.2%

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

   if -2.9000000000000001e92 < y.re < -3.10000000000000002e-84 or 7.2000000000000003e-26 < y.re < 2.70000000000000003e39

   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. Taylor expanded in x.im around inf 65.4%

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

      1. *-commutative 65.4%

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

   5. Simplified 65.4%

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

   if -3.10000000000000002e-84 < y.re < -6.6e-128

   1. Initial program 99.6%

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

   3. Taylor expanded in x.im around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-rgt-neg-out 85.5%

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

   5. Simplified 85.5%

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

   if -6.6e-128 < y.re < 7.2000000000000003e-26 or 2.70000000000000003e39 < y.re < 3.80000000000000023e89

   1. Initial program 66.8%

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

      1. div-sub 62.0%

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

      2. *-commutative 62.0%

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

      3. add-sqr-sqrt 62.0%

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

      4. times-frac 62.9%

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

      5. fma-neg 62.9%

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

      6. hypot-define 62.9%

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

      7. hypot-define 65.6%

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

      8. associate-/l* 71.1%

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

      9. add-sqr-sqrt 71.1%

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

      10. pow2 71.1%

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

      11. hypot-define 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 49.2%

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

      5. sqrt-prod 48.4%

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

      6. sqr-neg 48.4%

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

      7. sqrt-unprod 21.1%

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

      8. add-sqr-sqrt 38.4%

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

      9. hypot-undefine 33.3%

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

      10. +-commutative 33.3%

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

      11. hypot-define 38.4%

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

      12. add-sqr-sqrt 21.1%

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

      13. sqrt-unprod 48.4%

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

      14. sqr-neg 48.4%

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

      15. sqrt-prod 49.2%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 71.1%

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

      18. +-commutative 71.1%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around 0 81.7%

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

       1. neg-mul-1 81.7%

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

       2. associate-*r/ 82.2%

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

       3. +-commutative 82.2%

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

       4. unsub-neg 82.2%

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

       5. *-lft-identity 82.2%

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

       6. unpow2 82.2%

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

       7. times-frac 88.1%

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

       8. *-commutative 88.1%

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

       9. associate-*r/ 88.1%

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

       10. *-rgt-identity 88.1%

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

       11. associate-/l* 89.8%

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

       12. div-sub 90.8%

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

   11. Simplified 90.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{y.im \cdot x.re}{y.re \cdot \left(-y.re\right) - y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im))
        (t_1
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.25e+135)
     (/ x.im y.re)
     (if (<= y.re -5.2e-132)
       t_1
       (if (<= y.re 2.4e-29)
         t_0
         (if (<= y.re 1.75e+44)
           t_1
           (if (<= y.re 9.6e+87) t_0 (/ x.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.25e+135) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -5.2e-132) {
		tmp = t_1;
	} else if (y_46_re <= 2.4e-29) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_1;
	} else if (y_46_re <= 9.6e+87) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    t_1 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.25d+135)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-5.2d-132)) then
        tmp = t_1
    else if (y_46re <= 2.4d-29) then
        tmp = t_0
    else if (y_46re <= 1.75d+44) then
        tmp = t_1
    else if (y_46re <= 9.6d+87) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.25e+135) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -5.2e-132) {
		tmp = t_1;
	} else if (y_46_re <= 2.4e-29) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e+44) {
		tmp = t_1;
	} else if (y_46_re <= 9.6e+87) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.25e+135:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -5.2e-132:
		tmp = t_1
	elif y_46_re <= 2.4e-29:
		tmp = t_0
	elif y_46_re <= 1.75e+44:
		tmp = t_1
	elif y_46_re <= 9.6e+87:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	t_1 = 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)))
	tmp = 0.0
	if (y_46_re <= -2.25e+135)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -5.2e-132)
		tmp = t_1;
	elseif (y_46_re <= 2.4e-29)
		tmp = t_0;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_1;
	elseif (y_46_re <= 9.6e+87)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.25e+135)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -5.2e-132)
		tmp = t_1;
	elseif (y_46_re <= 2.4e-29)
		tmp = t_0;
	elseif (y_46_re <= 1.75e+44)
		tmp = t_1;
	elseif (y_46_re <= 9.6e+87)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$re, -2.25e+135], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-132], t$95$1, If[LessEqual[y$46$re, 2.4e-29], t$95$0, If[LessEqual[y$46$re, 1.75e+44], t$95$1, If[LessEqual[y$46$re, 9.6e+87], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.25000000000000004e135 or 9.59999999999999926e87 < y.re

   1. Initial program 34.0%

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

   3. Taylor expanded in y.re around inf 77.7%

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

   if -2.25000000000000004e135 < y.re < -5.2000000000000002e-132 or 2.39999999999999992e-29 < y.re < 1.75e44

   1. Initial program 90.3%

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

   if -5.2000000000000002e-132 < y.re < 2.39999999999999992e-29 or 1.75e44 < y.re < 9.59999999999999926e87

   1. Initial program 65.9%

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

      1. div-sub 60.9%

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

      2. *-commutative 60.9%

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

      3. add-sqr-sqrt 60.9%

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

      4. times-frac 61.9%

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

      5. fma-neg 61.9%

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

      6. hypot-define 61.9%

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

      7. hypot-define 64.6%

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

      8. associate-/l* 70.3%

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

      9. add-sqr-sqrt 70.3%

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

      10. pow2 70.3%

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

      11. hypot-define 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. unpow2 70.3%

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

      3. times-frac 99.8%

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

      4. add-sqr-sqrt 48.7%

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

      5. sqrt-prod 47.9%

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

      6. sqr-neg 47.9%

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

      7. sqrt-unprod 21.7%

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

      8. add-sqr-sqrt 38.6%

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

      9. hypot-undefine 33.3%

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

      10. +-commutative 33.3%

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

      11. hypot-define 38.6%

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

      12. add-sqr-sqrt 21.7%

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

      13. sqrt-unprod 47.9%

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

      14. sqr-neg 47.9%

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

      15. sqrt-prod 48.7%

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

      16. add-sqr-sqrt 99.8%

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

      17. hypot-undefine 70.3%

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

      18. +-commutative 70.3%

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

      19. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around 0 82.1%

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

       1. neg-mul-1 82.1%

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

       2. associate-*r/ 82.7%

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

       3. +-commutative 82.7%

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

       4. unsub-neg 82.7%

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

       5. *-lft-identity 82.7%

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

       6. unpow2 82.7%

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

       7. times-frac 88.7%

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

       8. *-commutative 88.7%

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

       9. associate-*r/ 88.7%

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

       10. *-rgt-identity 88.7%

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

       11. associate-/l* 90.4%

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

       12. div-sub 91.4%

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

   11. Simplified 91.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.re -4.2e+90)
     (/ x.im y.re)
     (if (<= y.re 4.7e-26)
       t_0
       (if (<= y.re 5.3e+39)
         (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 1.6e+88) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -4.2e+90) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 4.7e-26) {
		tmp = t_0;
	} else if (y_46_re <= 5.3e+39) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.6e+88) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46re <= (-4.2d+90)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 4.7d-26) then
        tmp = t_0
    else if (y_46re <= 5.3d+39) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1.6d+88) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -4.2e+90) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 4.7e-26) {
		tmp = t_0;
	} else if (y_46_re <= 5.3e+39) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.6e+88) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_re <= -4.2e+90:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 4.7e-26:
		tmp = t_0
	elif y_46_re <= 5.3e+39:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.6e+88:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e+90)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 4.7e-26)
		tmp = t_0;
	elseif (y_46_re <= 5.3e+39)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.6e+88)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -4.2e+90)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 4.7e-26)
		tmp = t_0;
	elseif (y_46_re <= 5.3e+39)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.6e+88)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -4.2e+90], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.7e-26], t$95$0, If[LessEqual[y$46$re, 5.3e+39], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e+88], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.19999999999999961e90 or 1.5999999999999999e88 < y.re

   1. Initial program 39.4%

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

   3. Taylor expanded in y.re around inf 78.5%

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

   if -4.19999999999999961e90 < y.re < 4.69999999999999989e-26 or 5.29999999999999979e39 < y.re < 1.5999999999999999e88

   1. Initial program 73.3%

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

      1. div-sub 69.9%

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

      2. *-commutative 69.9%

         \[\leadsto \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. add-sqr-sqrt 69.9%

         \[\leadsto \frac{y.re \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. times-frac 70.7%

         \[\leadsto \color{blue}{\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      5. fma-neg 70.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]

      6. hypot-define 70.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]

      7. hypot-define 73.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]

      8. associate-/l* 77.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}}\right) \]

      9. add-sqr-sqrt 77.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\right) \]

      10. pow2 77.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}\right) \]

      11. hypot-define 77.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}\right) \]

   4. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 77.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{\color{blue}{1 \cdot y.im}}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right) \]

      2. unpow2 77.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{1 \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}}\right) \]

      3. times-frac 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\right) \]

      4. add-sqr-sqrt 46.5%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{y.im} \cdot \sqrt{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      5. sqrt-prod 51.4%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{y.im \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      6. sqr-neg 51.4%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt{\color{blue}{\left(-y.im\right) \cdot \left(-y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      7. sqrt-unprod 26.1%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{-y.im} \cdot \sqrt{-y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      8. add-sqr-sqrt 44.9%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{-y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      9. hypot-undefine 41.4%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      10. +-commutative 41.4%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      11. hypot-define 44.9%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      12. add-sqr-sqrt 26.1%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{-y.im} \cdot \sqrt{-y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      13. sqrt-unprod 51.4%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      14. sqr-neg 51.4%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt{\color{blue}{y.im \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      15. sqrt-prod 46.5%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{y.im} \cdot \sqrt{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      16. add-sqr-sqrt 99.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      17. hypot-undefine 77.6%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\right)\right) \]

      18. +-commutative 77.6%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}}\right)\right) \]

      19. hypot-define 99.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \cdot x.re}\right) \]

      2. associate-*l/ 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{1 \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot x.re\right) \]

      3. *-un-lft-identity 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{\color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re\right) \]

      4. associate-*l/ 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\right) \]

   9. Taylor expanded in y.re around 0 72.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   10. Step-by-step derivation

       1. neg-mul-1 72.5%

          \[\leadsto \color{blue}{\left(-\frac{x.re}{y.im}\right)} + \frac{x.im \cdot y.re}{{y.im}^{2}} \]

       2. associate-*r/ 73.3%

          \[\leadsto \left(-\frac{x.re}{y.im}\right) + \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}}} \]

       3. +-commutative 73.3%

          \[\leadsto \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}} + \left(-\frac{x.re}{y.im}\right)} \]

       4. unsub-neg 73.3%

          \[\leadsto \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}} - \frac{x.re}{y.im}} \]

       5. *-lft-identity 73.3%

          \[\leadsto x.im \cdot \frac{\color{blue}{1 \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]

       6. unpow2 73.3%

          \[\leadsto x.im \cdot \frac{1 \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]

       7. times-frac 77.5%

          \[\leadsto x.im \cdot \color{blue}{\left(\frac{1}{y.im} \cdot \frac{y.re}{y.im}\right)} - \frac{x.re}{y.im} \]

       8. *-commutative 77.5%

          \[\leadsto x.im \cdot \color{blue}{\left(\frac{y.re}{y.im} \cdot \frac{1}{y.im}\right)} - \frac{x.re}{y.im} \]

       9. associate-*r/ 77.5%

          \[\leadsto x.im \cdot \color{blue}{\frac{\frac{y.re}{y.im} \cdot 1}{y.im}} - \frac{x.re}{y.im} \]

       10. *-rgt-identity 77.5%

           \[\leadsto x.im \cdot \frac{\color{blue}{\frac{y.re}{y.im}}}{y.im} - \frac{x.re}{y.im} \]

       11. associate-/l* 80.1%

           \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]

       12. div-sub 80.9%

           \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}} \]

   11. Simplified 80.9%

       \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}} \]

   if 4.69999999999999989e-26 < y.re < 5.29999999999999979e39

   1. Initial program 87.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around inf 65.4%

      \[\leadsto \frac{\color{blue}{x.im \cdot y.re}}{y.re \cdot y.re + y.im \cdot y.im} \]
   4. Step-by-step derivation

      1. *-commutative 65.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} \]

   5. Simplified 65.4%

      \[\leadsto \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+91} \lor \neg \left(y.re \leq 9.6 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.45e+91) (not (<= y.re 9.6e+87)))
   (/ x.im y.re)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.45e+91) || !(y_46_re <= 9.6e+87)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / 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_46re <= (-1.45d+91)) .or. (.not. (y_46re <= 9.6d+87))) then
        tmp = x_46im / y_46re
    else
        tmp = ((x_46im * (y_46re / 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_re <= -1.45e+91) || !(y_46_re <= 9.6e+87)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / 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_re <= -1.45e+91) or not (y_46_re <= 9.6e+87):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.45e+91) || !(y_46_re <= 9.6e+87))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - 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_re <= -1.45e+91) || ~((y_46_re <= 9.6e+87)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / 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[Or[LessEqual[y$46$re, -1.45e+91], N[Not[LessEqual[y$46$re, 9.6e+87]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.45000000000000007e91 or 9.59999999999999926e87 < y.re

   1. Initial program 39.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 78.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]

   if -1.45000000000000007e91 < y.re < 9.59999999999999926e87

   1. Initial program 74.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub 71.7%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

      2. *-commutative 71.7%

         \[\leadsto \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. add-sqr-sqrt 71.7%

         \[\leadsto \frac{y.re \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. times-frac 72.4%

         \[\leadsto \color{blue}{\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

      5. fma-neg 72.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]

      6. hypot-define 72.4%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]

      7. hypot-define 75.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]

      8. associate-/l* 79.3%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}}\right) \]

      9. add-sqr-sqrt 79.3%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\right) \]

      10. pow2 79.3%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}\right) \]

      11. hypot-define 79.3%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}\right) \]

   4. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 79.3%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{\color{blue}{1 \cdot y.im}}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right) \]

      2. unpow2 79.3%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{1 \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}}\right) \]

      3. times-frac 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\right) \]

      4. add-sqr-sqrt 47.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{y.im} \cdot \sqrt{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      5. sqrt-prod 51.6%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{y.im \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      6. sqr-neg 51.6%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt{\color{blue}{\left(-y.im\right) \cdot \left(-y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      7. sqrt-unprod 25.9%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\sqrt{-y.im} \cdot \sqrt{-y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      8. add-sqr-sqrt 47.5%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{-y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      9. hypot-undefine 44.3%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      10. +-commutative 44.3%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      11. hypot-define 47.5%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      12. add-sqr-sqrt 25.9%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{-y.im} \cdot \sqrt{-y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      13. sqrt-unprod 51.6%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      14. sqr-neg 51.6%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt{\color{blue}{y.im \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      15. sqrt-prod 47.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{\sqrt{y.im} \cdot \sqrt{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      16. add-sqr-sqrt 99.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\color{blue}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right) \]

      17. hypot-undefine 79.3%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\right)\right) \]

      18. +-commutative 79.3%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}}\right)\right) \]

      19. hypot-define 99.7%

          \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \cdot x.re}\right) \]

      2. associate-*l/ 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{1 \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot x.re\right) \]

      3. *-un-lft-identity 99.7%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{\color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re\right) \]

      4. associate-*l/ 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}\right) \]

   9. Taylor expanded in y.re around 0 67.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   10. Step-by-step derivation

       1. neg-mul-1 67.9%

          \[\leadsto \color{blue}{\left(-\frac{x.re}{y.im}\right)} + \frac{x.im \cdot y.re}{{y.im}^{2}} \]

       2. associate-*r/ 68.8%

          \[\leadsto \left(-\frac{x.re}{y.im}\right) + \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}}} \]

       3. +-commutative 68.8%

          \[\leadsto \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}} + \left(-\frac{x.re}{y.im}\right)} \]

       4. unsub-neg 68.8%

          \[\leadsto \color{blue}{x.im \cdot \frac{y.re}{{y.im}^{2}} - \frac{x.re}{y.im}} \]

       5. *-lft-identity 68.8%

          \[\leadsto x.im \cdot \frac{\color{blue}{1 \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]

       6. unpow2 68.8%

          \[\leadsto x.im \cdot \frac{1 \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]

       7. times-frac 72.5%

          \[\leadsto x.im \cdot \color{blue}{\left(\frac{1}{y.im} \cdot \frac{y.re}{y.im}\right)} - \frac{x.re}{y.im} \]

       8. *-commutative 72.5%

          \[\leadsto x.im \cdot \color{blue}{\left(\frac{y.re}{y.im} \cdot \frac{1}{y.im}\right)} - \frac{x.re}{y.im} \]

       9. associate-*r/ 72.5%

          \[\leadsto x.im \cdot \color{blue}{\frac{\frac{y.re}{y.im} \cdot 1}{y.im}} - \frac{x.re}{y.im} \]

       10. *-rgt-identity 72.5%

           \[\leadsto x.im \cdot \frac{\color{blue}{\frac{y.re}{y.im}}}{y.im} - \frac{x.re}{y.im} \]

       11. associate-/l* 75.4%

           \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]

       12. div-sub 76.7%

           \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}} \]

   11. Simplified 76.7%

       \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+91} \lor \neg \left(y.re \leq 9.6 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+61} \lor \neg \left(y.im \leq 1.56 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.02e+61) (not (<= y.im 1.56e-58)))
   (/ 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 <= -1.02e+61) || !(y_46_im <= 1.56e-58)) {
		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 <= (-1.02d+61)) .or. (.not. (y_46im <= 1.56d-58))) 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 <= -1.02e+61) || !(y_46_im <= 1.56e-58)) {
		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 <= -1.02e+61) or not (y_46_im <= 1.56e-58):
		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 <= -1.02e+61) || !(y_46_im <= 1.56e-58))
		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 <= -1.02e+61) || ~((y_46_im <= 1.56e-58)))
		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, -1.02e+61], N[Not[LessEqual[y$46$im, 1.56e-58]], $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 < -1.01999999999999999e61 or 1.56000000000000008e-58 < y.im

   1. Initial program 55.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0 65.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot x.re}{y.im}} \]

      2. neg-mul-1 65.3%

         \[\leadsto \frac{\color{blue}{-x.re}}{y.im} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{-x.re}{y.im}} \]

   if -1.01999999999999999e61 < y.im < 1.56000000000000008e-58

   1. Initial program 71.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around inf 70.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+61} \lor \neg \left(y.im \leq 1.56 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.4% 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 63.0%

   \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
2. Add Preprocessing

3. Taylor expanded in y.re around inf 44.4%

   \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]

4. Final simplification 44.4%

   \[\leadsto \frac{x.im}{y.re} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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))))