_divideComplex, imaginary part

Percentage Accurate: 61.7% → 97.7%

Time: 15.4s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - (y_46_im * (x_46_re / Math.hypot(y_46_re, y_46_im)))) / Math.hypot(y_46_re, y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - (y_46_im * (x_46_re / math.hypot(y_46_re, y_46_im)))) / math.hypot(y_46_re, y_46_im)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. div-sub 58.6%

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

   2. *-un-lft-identity 58.6%

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

   3. add-sqr-sqrt 58.6%

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

   4. times-frac 58.6%

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

   5. fmm-def 59.0%

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

   6. hypot-define 59.0%

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

   7. hypot-define 63.3%

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

   8. associate-/l* 65.5%

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

   9. add-sqr-sqrt 65.5%

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

   10. pow2 65.5%

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

   11. hypot-define 65.5%

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

4. Applied egg-rr 65.5%

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

   1. fmm-undef 65.1%

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

   2. *-commutative 65.1%

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

   3. associate-/l* 79.1%

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

   4. associate-*r/ 75.7%

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

   5. *-commutative 75.7%

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

   6. associate-/l* 74.6%

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

6. Simplified 74.6%

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

   1. *-un-lft-identity 74.6%

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

   2. unpow2 74.6%

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

   3. times-frac 83.9%

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

8. Applied egg-rr 83.9%

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

   1. associate-*l/ 83.9%

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

   2. *-lft-identity 83.9%

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

   3. hypot-undefine 74.7%

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

   4. unpow2 74.7%

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

   5. unpow2 74.7%

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

   6. +-commutative 74.7%

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

   7. unpow2 74.7%

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

   8. unpow2 74.7%

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

   9. hypot-define 83.9%

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

   10. hypot-undefine 74.7%

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

   11. unpow2 74.7%

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

   12. unpow2 74.7%

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

   13. +-commutative 74.7%

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

   14. unpow2 74.7%

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

   15. unpow2 74.7%

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

   16. hypot-define 83.9%

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

10. Simplified 83.9%

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

    1. associate-*l/ 84.1%

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

    2. *-un-lft-identity 84.1%

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

    3. associate-*r/ 97.3%

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

    4. hypot-undefine 79.0%

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

    5. +-commutative 79.0%

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

    6. hypot-undefine 97.3%

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

    7. sub-div 97.3%

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

    8. hypot-undefine 79.0%

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

    9. +-commutative 79.0%

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

    10. hypot-undefine 97.3%

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

12. Applied egg-rr 97.3%

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

Alternative 2: 84.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.6e-22)
   (/ (- (- x.im) (* y.im (/ x.re (hypot y.re y.im)))) (hypot y.re y.im))
   (if (<= y.re 9.8e-111)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (if (<= y.re 1e+139)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (- x.im (* y.im (/ x.re y.re))) (hypot y.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 <= -3.6e-22) {
		tmp = (-x_46_im - (y_46_im * (x_46_re / hypot(y_46_re, y_46_im)))) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 9.8e-111) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1e+139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

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 <= -3.6e-22) {
		tmp = (-x_46_im - (y_46_im * (x_46_re / Math.hypot(y_46_re, y_46_im)))) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 9.8e-111) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1e+139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / Math.hypot(y_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 <= -3.6e-22:
		tmp = (-x_46_im - (y_46_im * (x_46_re / math.hypot(y_46_re, y_46_im)))) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= 9.8e-111:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 1e+139:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / math.hypot(y_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 <= -3.6e-22)
		tmp = Float64(Float64(Float64(-x_46_im) - Float64(y_46_im * Float64(x_46_re / hypot(y_46_re, y_46_im)))) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 9.8e-111)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1e+139)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / hypot(y_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 <= -3.6e-22)
		tmp = (-x_46_im - (y_46_im * (x_46_re / hypot(y_46_re, y_46_im)))) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= 9.8e-111)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1e+139)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.6e-22], N[(N[((-x$46$im) - N[(y$46$im * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.8e-111], 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, 1e+139], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.5999999999999998e-22

   1. Initial program 48.5%

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

      1. div-sub 48.5%

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

      2. *-un-lft-identity 48.5%

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

      3. add-sqr-sqrt 48.5%

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

      4. times-frac 48.4%

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

      5. fmm-def 48.4%

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

      6. hypot-define 48.4%

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

      7. hypot-define 53.2%

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

      8. associate-/l* 54.6%

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

      9. add-sqr-sqrt 54.6%

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

      10. pow2 54.6%

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

      11. hypot-define 54.6%

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

   4. Applied egg-rr 54.6%

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

      1. fmm-undef 54.6%

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

      2. *-commutative 54.6%

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

      3. associate-/l* 85.4%

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

      4. associate-*r/ 84.0%

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

      5. *-commutative 84.0%

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

      6. associate-/l* 85.5%

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

   6. Simplified 85.5%

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

      1. *-un-lft-identity 85.5%

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

      2. unpow2 85.5%

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

      3. times-frac 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-lft-identity 96.6%

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

      3. hypot-undefine 85.5%

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

      4. unpow2 85.5%

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

      5. unpow2 85.5%

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

      6. +-commutative 85.5%

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

      7. unpow2 85.5%

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

      8. unpow2 85.5%

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

      9. hypot-define 96.6%

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

      10. hypot-undefine 85.5%

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

      11. unpow2 85.5%

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

      12. unpow2 85.5%

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

      13. +-commutative 85.5%

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

      14. unpow2 85.5%

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

      15. unpow2 85.5%

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

      16. hypot-define 96.6%

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

   10. Simplified 96.6%

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

       1. associate-*l/ 96.9%

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

       2. *-un-lft-identity 96.9%

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

       3. associate-*r/ 99.8%

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

       4. hypot-undefine 85.8%

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

       5. +-commutative 85.8%

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

       6. hypot-undefine 99.8%

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

       7. sub-div 99.8%

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

       8. hypot-undefine 85.8%

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

       9. +-commutative 85.8%

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

       10. hypot-undefine 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y.re around -inf 90.5%

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

       1. neg-mul-1 90.5%

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

   15. Simplified 90.5%

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

   if -3.5999999999999998e-22 < y.re < 9.80000000000000038e-111

   1. Initial program 68.6%

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

      1. div-sub 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. add-sqr-sqrt 64.4%

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

      4. times-frac 64.3%

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

      5. fmm-def 65.3%

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

      6. hypot-define 65.3%

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

      7. hypot-define 65.5%

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

      8. associate-/l* 71.8%

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

      9. add-sqr-sqrt 71.8%

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

      10. pow2 71.8%

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

      11. hypot-define 71.8%

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

   4. Applied egg-rr 71.8%

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

      1. fmm-undef 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-/l* 69.9%

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

      4. associate-*r/ 63.6%

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

      5. *-commutative 63.6%

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

      6. associate-/l* 62.1%

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

   6. Simplified 62.1%

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

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

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

      1. associate-/l* 91.6%

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

   9. Simplified 91.6%

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

   if 9.80000000000000038e-111 < y.re < 1.00000000000000003e139

   1. Initial program 87.4%

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

   if 1.00000000000000003e139 < y.re

   1. Initial program 24.5%

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

      1. div-sub 24.5%

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

      2. *-un-lft-identity 24.5%

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

      3. add-sqr-sqrt 24.5%

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

      4. times-frac 24.5%

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

      5. fmm-def 24.5%

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

      6. hypot-define 24.5%

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

      7. hypot-define 45.8%

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

      8. associate-/l* 46.2%

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

      9. add-sqr-sqrt 46.2%

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

      10. pow2 46.2%

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

      11. hypot-define 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. fmm-undef 46.2%

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

      2. *-commutative 46.2%

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

      3. associate-/l* 83.2%

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

      4. associate-*r/ 74.2%

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

      5. *-commutative 74.2%

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

      6. associate-/l* 83.2%

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

   6. Simplified 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 91.5%

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

   8. Applied egg-rr 91.5%

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

      1. associate-*l/ 91.4%

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

      2. *-lft-identity 91.4%

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

      3. hypot-undefine 83.2%

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

      4. unpow2 83.2%

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

      5. unpow2 83.2%

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

      6. +-commutative 83.2%

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

      7. unpow2 83.2%

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

      8. unpow2 83.2%

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

      9. hypot-define 91.4%

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

      10. hypot-undefine 83.2%

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

      11. unpow2 83.2%

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

      12. unpow2 83.2%

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

      13. +-commutative 83.2%

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

      14. unpow2 83.2%

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

      15. unpow2 83.2%

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

      16. hypot-define 91.4%

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

   10. Simplified 91.4%

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

       1. associate-*l/ 91.7%

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

       2. *-un-lft-identity 91.7%

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

       3. associate-*r/ 99.9%

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

       4. hypot-undefine 83.5%

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

       5. +-commutative 83.5%

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

       6. hypot-undefine 99.9%

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

       7. sub-div 99.9%

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

       8. hypot-undefine 83.5%

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

       9. +-commutative 83.5%

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

       10. hypot-undefine 99.9%

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

   12. Applied egg-rr 99.9%

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

   13. Taylor expanded in y.re around inf 82.9%

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

       1. mul-1-neg 82.9%

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

       2. sub-neg 82.9%

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

       3. *-commutative 82.9%

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

       4. associate-/l* 85.0%

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

   15. Simplified 85.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.im - y.im \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{t\_1}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

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.im (/ x.re y.re)))))
   (if (<= y.re -1.6e+67)
     (/ t_1 y.re)
     (if (<= y.re -9.2e-37)
       t_0
       (if (<= y.re 2.5e-111)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 3.4e+139) t_0 (/ t_1 (hypot y.re y.im))))))))

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_im * (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.6e+67) {
		tmp = t_1 / y_46_re;
	} else if (y_46_re <= -9.2e-37) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-111) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.4e+139) {
		tmp = t_0;
	} else {
		tmp = t_1 / hypot(y_46_re, y_46_im);
	}
	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 = ((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_im * (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -1.6e+67) {
		tmp = t_1 / y_46_re;
	} else if (y_46_re <= -9.2e-37) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-111) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.4e+139) {
		tmp = t_0;
	} else {
		tmp = t_1 / Math.hypot(y_46_re, y_46_im);
	}
	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_im * (x_46_re / y_46_re))
	tmp = 0
	if y_46_re <= -1.6e+67:
		tmp = t_1 / y_46_re
	elif y_46_re <= -9.2e-37:
		tmp = t_0
	elif y_46_re <= 2.5e-111:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 3.4e+139:
		tmp = t_0
	else:
		tmp = t_1 / math.hypot(y_46_re, y_46_im)
	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(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.6e+67)
		tmp = Float64(t_1 / y_46_re);
	elseif (y_46_re <= -9.2e-37)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-111)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.4e+139)
		tmp = t_0;
	else
		tmp = Float64(t_1 / hypot(y_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)
	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_im * (x_46_re / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -1.6e+67)
		tmp = t_1 / y_46_re;
	elseif (y_46_re <= -9.2e-37)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-111)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 3.4e+139)
		tmp = t_0;
	else
		tmp = t_1 / hypot(y_46_re, y_46_im);
	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[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e+67], N[(t$95$1 / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -9.2e-37], t$95$0, If[LessEqual[y$46$re, 2.5e-111], 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, 3.4e+139], t$95$0, N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.59999999999999991e67

   1. Initial program 39.2%

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

   3. Taylor expanded in y.re around inf 84.8%

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

      1. mul-1-neg 84.8%

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

      2. unsub-neg 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-/l* 88.2%

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

   5. Simplified 88.2%

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

   if -1.59999999999999991e67 < y.re < -9.1999999999999999e-37 or 2.5000000000000001e-111 < y.re < 3.4000000000000002e139

   1. Initial program 85.6%

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

   if -9.1999999999999999e-37 < y.re < 2.5000000000000001e-111

   1. Initial program 67.3%

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

      1. div-sub 62.9%

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

      2. *-un-lft-identity 62.9%

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

      3. add-sqr-sqrt 62.9%

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

      4. times-frac 62.9%

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

      5. fmm-def 63.9%

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

      6. hypot-define 63.9%

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

      7. hypot-define 64.1%

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

      8. associate-/l* 70.6%

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

      9. add-sqr-sqrt 70.6%

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

      10. pow2 70.6%

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

      11. hypot-define 70.6%

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

   4. Applied egg-rr 70.6%

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

      1. fmm-undef 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-/l* 68.6%

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

      4. associate-*r/ 62.1%

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

      5. *-commutative 62.1%

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

      6. associate-/l* 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in y.im around inf 92.0%

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

      1. associate-/l* 92.2%

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

   9. Simplified 92.2%

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

   if 3.4000000000000002e139 < y.re

   1. Initial program 24.5%

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

      1. div-sub 24.5%

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

      2. *-un-lft-identity 24.5%

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

      3. add-sqr-sqrt 24.5%

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

      4. times-frac 24.5%

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

      5. fmm-def 24.5%

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

      6. hypot-define 24.5%

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

      7. hypot-define 45.8%

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

      8. associate-/l* 46.2%

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

      9. add-sqr-sqrt 46.2%

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

      10. pow2 46.2%

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

      11. hypot-define 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. fmm-undef 46.2%

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

      2. *-commutative 46.2%

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

      3. associate-/l* 83.2%

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

      4. associate-*r/ 74.2%

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

      5. *-commutative 74.2%

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

      6. associate-/l* 83.2%

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

   6. Simplified 83.2%

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

      1. *-un-lft-identity 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 91.5%

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

   8. Applied egg-rr 91.5%

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

      1. associate-*l/ 91.4%

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

      2. *-lft-identity 91.4%

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

      3. hypot-undefine 83.2%

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

      4. unpow2 83.2%

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

      5. unpow2 83.2%

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

      6. +-commutative 83.2%

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

      7. unpow2 83.2%

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

      8. unpow2 83.2%

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

      9. hypot-define 91.4%

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

      10. hypot-undefine 83.2%

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

      11. unpow2 83.2%

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

      12. unpow2 83.2%

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

      13. +-commutative 83.2%

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

      14. unpow2 83.2%

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

      15. unpow2 83.2%

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

      16. hypot-define 91.4%

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

   10. Simplified 91.4%

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

       1. associate-*l/ 91.7%

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

       2. *-un-lft-identity 91.7%

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

       3. associate-*r/ 99.9%

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

       4. hypot-undefine 83.5%

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

       5. +-commutative 83.5%

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

       6. hypot-undefine 99.9%

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

       7. sub-div 99.9%

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

       8. hypot-undefine 83.5%

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

       9. +-commutative 83.5%

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

       10. hypot-undefine 99.9%

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

   12. Applied egg-rr 99.9%

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

   13. Taylor expanded in y.re around inf 82.9%

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

       1. mul-1-neg 82.9%

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

       2. sub-neg 82.9%

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

       3. *-commutative 82.9%

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

       4. associate-/l* 85.0%

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

   15. Simplified 85.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -1.14e+67)
     t_1
     (if (<= y.re -8e-37)
       t_0
       (if (<= y.re 1.2e-108)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 1.65e+143) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.14e+67) {
		tmp = t_1;
	} else if (y_46_re <= -8e-37) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-108) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.65e+143) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    if (y_46re <= (-1.14d+67)) then
        tmp = t_1
    else if (y_46re <= (-8d-37)) then
        tmp = t_0
    else if (y_46re <= 1.2d-108) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46re <= 1.65d+143) then
        tmp = t_0
    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_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.14e+67) {
		tmp = t_1;
	} else if (y_46_re <= -8e-37) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-108) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.65e+143) {
		tmp = t_0;
	} 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_im * (x_46_re / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -1.14e+67:
		tmp = t_1
	elif y_46_re <= -8e-37:
		tmp = t_0
	elif y_46_re <= 1.2e-108:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 1.65e+143:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.14e+67)
		tmp = t_1;
	elseif (y_46_re <= -8e-37)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-108)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.65e+143)
		tmp = t_0;
	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_im * (x_46_re / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.14e+67)
		tmp = t_1;
	elseif (y_46_re <= -8e-37)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-108)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.65e+143)
		tmp = t_0;
	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 - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.14e+67], t$95$1, If[LessEqual[y$46$re, -8e-37], t$95$0, If[LessEqual[y$46$re, 1.2e-108], 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, 1.65e+143], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1400000000000001e67 or 1.65e143 < y.re

   1. Initial program 33.7%

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

   3. Taylor expanded in y.re around inf 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-/l* 86.9%

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

   5. Simplified 86.9%

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

   if -1.1400000000000001e67 < y.re < -8.00000000000000053e-37 or 1.20000000000000009e-108 < y.re < 1.65e143

   1. Initial program 85.6%

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

   if -8.00000000000000053e-37 < y.re < 1.20000000000000009e-108

   1. Initial program 67.3%

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

      1. div-sub 62.9%

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

      2. *-un-lft-identity 62.9%

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

      3. add-sqr-sqrt 62.9%

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

      4. times-frac 62.9%

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

      5. fmm-def 63.9%

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

      6. hypot-define 63.9%

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

      7. hypot-define 64.1%

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

      8. associate-/l* 70.6%

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

      9. add-sqr-sqrt 70.6%

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

      10. pow2 70.6%

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

      11. hypot-define 70.6%

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

   4. Applied egg-rr 70.6%

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

      1. fmm-undef 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-/l* 68.6%

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

      4. associate-*r/ 62.1%

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

      5. *-commutative 62.1%

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

      6. associate-/l* 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in y.im around inf 92.0%

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

      1. associate-/l* 92.2%

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

   9. Simplified 92.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+143}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+70} \lor \neg \left(y.re \leq 2.5 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.9e+70) (not (<= y.re 2.5e-79)))
   (/ (- x.im (* y.im (/ x.re y.re))) 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 <= -2.9e+70) || !(y_46_re <= 2.5e-79)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - 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 <= (-2.9d+70)) .or. (.not. (y_46re <= 2.5d-79))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / 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 <= -2.9e+70) || !(y_46_re <= 2.5e-79)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - 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 <= -2.9e+70) or not (y_46_re <= 2.5e-79):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - 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 <= -2.9e+70) || !(y_46_re <= 2.5e-79))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / 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 <= -2.9e+70) || ~((y_46_re <= 2.5e-79)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - 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, -2.9e+70], N[Not[LessEqual[y$46$re, 2.5e-79]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 < -2.8999999999999998e70 or 2.5e-79 < y.re

   1. Initial program 50.5%

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

   3. Taylor expanded in y.re around inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unsub-neg 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 81.8%

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

   5. Simplified 81.8%

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

   if -2.8999999999999998e70 < y.re < 2.5e-79

   1. Initial program 70.1%

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

      1. div-sub 66.8%

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

      2. *-un-lft-identity 66.8%

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

      3. add-sqr-sqrt 66.8%

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

      4. times-frac 66.7%

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

      5. fmm-def 67.5%

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

      6. hypot-define 67.5%

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

      7. hypot-define 67.6%

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

      8. associate-/l* 73.3%

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

      9. add-sqr-sqrt 73.3%

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

      10. pow2 73.3%

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

      11. hypot-define 73.3%

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

   4. Applied egg-rr 73.3%

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

      1. fmm-undef 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/l* 73.4%

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

      4. associate-*r/ 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-/l* 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in y.im around inf 83.2%

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

      1. associate-/l* 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+70} \lor \neg \left(y.re \leq 2.5 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-20} \lor \neg \left(y.re \leq 2.2 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.8e-20) (not (<= y.re 2.2e-79)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ 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 <= -2.8e-20) || !(y_46_re <= 2.2e-79)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = 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 <= (-2.8d-20)) .or. (.not. (y_46re <= 2.2d-79))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = 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 <= -2.8e-20) || !(y_46_re <= 2.2e-79)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = 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 <= -2.8e-20) or not (y_46_re <= 2.2e-79):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = 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 <= -2.8e-20) || !(y_46_re <= 2.2e-79))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -2.8e-20) || ~((y_46_re <= 2.2e-79)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = 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, -2.8e-20], N[Not[LessEqual[y$46$re, 2.2e-79]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.8000000000000003e-20 or 2.1999999999999999e-79 < y.re

   1. Initial program 53.2%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unsub-neg 75.0%

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

      3. *-commutative 75.0%

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

      4. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

   if -2.8000000000000003e-20 < y.re < 2.1999999999999999e-79

   1. Initial program 70.0%

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

   3. Taylor expanded in y.re around 0 74.6%

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

      1. associate-*r/ 74.6%

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

      2. neg-mul-1 74.6%

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

   5. Simplified 74.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-20} \lor \neg \left(y.re \leq 2.2 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+67} \lor \neg \left(y.re \leq 2.35 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1e+67) (not (<= y.re 2.35e-79)))
   (/ x.im y.re)
   (/ 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 <= -1e+67) || !(y_46_re <= 2.35e-79)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = 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 <= (-1d+67)) .or. (.not. (y_46re <= 2.35d-79))) then
        tmp = x_46im / y_46re
    else
        tmp = 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 <= -1e+67) || !(y_46_re <= 2.35e-79)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = 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 <= -1e+67) or not (y_46_re <= 2.35e-79):
		tmp = x_46_im / y_46_re
	else:
		tmp = 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 <= -1e+67) || !(y_46_re <= 2.35e-79))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -1e+67) || ~((y_46_re <= 2.35e-79)))
		tmp = x_46_im / y_46_re;
	else
		tmp = 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, -1e+67], N[Not[LessEqual[y$46$re, 2.35e-79]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.99999999999999983e66 or 2.3500000000000001e-79 < y.re

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around inf 68.5%

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

   if -9.99999999999999983e66 < y.re < 2.3500000000000001e-79

   1. Initial program 71.2%

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

   3. Taylor expanded in y.re around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+67} \lor \neg \left(y.re \leq 2.35 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 9.2 \cdot 10^{+228}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 9.2e+228) (/ x.im y.re) (/ x.re y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 9.2e+228) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 9.2d+228) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 9.2e+228) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 9.2e+228:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 9.2e+228)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 9.2e+228)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 9.2e+228], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 9.20000000000000052e228

   1. Initial program 61.5%

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

   3. Taylor expanded in y.re around inf 45.4%

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

   if 9.20000000000000052e228 < y.im

   1. Initial program 48.4%

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

   3. Taylor expanded in y.re around 0 83.5%

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

      1. associate-*r/ 83.5%

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

      2. neg-mul-1 83.5%

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

   5. Simplified 83.5%

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

      1. add-sqr-sqrt 30.7%

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

      2. sqrt-unprod 57.9%

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

      3. sqr-neg 57.9%

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

      4. sqrt-unprod 27.1%

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

      5. add-sqr-sqrt 49.8%

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

      6. *-un-lft-identity 49.8%

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

   7. Applied egg-rr 49.8%

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

      1. *-lft-identity 49.8%

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

   9. Simplified 49.8%

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

Alternative 9: 42.5% 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 60.3%

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

3. Taylor expanded in y.re around inf 42.4%

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

Reproduce

herbie shell --seed 2024180 
(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))))