_divideComplex, imaginary part

Percentage Accurate: 61.7% → 97.4%

Time: 18.9s

Alternatives: 14

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: 97.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}{x.re}}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma
  (/ y.re (hypot y.re y.im))
  (/ x.im (hypot y.re y.im))
  (/ -1.0 (* (hypot y.im y.re) (/ (/ (hypot y.im y.re) y.im) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-1.0 / (hypot(y_46_im, y_46_re) * ((hypot(y_46_im, y_46_re) / y_46_im) / x_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(-1.0 / Float64(hypot(y_46_im, y_46_re) * Float64(Float64(hypot(y_46_im, y_46_re) / y_46_im) / x_46_re))))
end

Wolfram

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

Derivation

1. Initial program 59.8%

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

   1. div-sub 58.8%

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

   2. *-commutative 58.8%

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

   3. fma-define 58.8%

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

   4. add-sqr-sqrt 58.8%

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

   5. times-frac 60.7%

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

   6. fma-neg 60.7%

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

   7. fma-define 60.7%

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

   8. hypot-define 60.7%

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

   9. fma-define 60.7%

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

   10. hypot-define 74.4%

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

   11. associate-/l* 77.5%

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

   12. fma-define 77.5%

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

   13. add-sqr-sqrt 77.5%

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

   14. pow2 77.5%

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

4. Applied egg-rr 77.5%

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

   1. *-un-lft-identity 77.5%

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

   2. unpow2 77.5%

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

   3. times-frac 96.8%

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

   4. hypot-undefine 77.5%

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

   5. +-commutative 77.5%

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

   6. hypot-define 96.8%

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

   7. hypot-undefine 77.5%

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

   8. +-commutative 77.5%

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

   9. hypot-define 96.8%

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

6. Applied egg-rr 96.8%

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

   1. associate-*r* 97.8%

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

   2. clear-num 97.8%

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

   3. un-div-inv 97.9%

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

   4. un-div-inv 98.0%

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

8. Applied egg-rr 98.0%

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

   1. clear-num 97.8%

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

   2. inv-pow 97.8%

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

10. Applied egg-rr 97.8%

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

    1. unpow-1 97.8%

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

    2. associate-/r/ 98.8%

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

12. Simplified 98.8%

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

13. Final simplification 98.8%

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

Alternative 2: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, x.re \cdot \left(\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-1}{y.re}\right)\right)\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{x.im}{y.im}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (hypot y.re y.im)))
        (t_1 (/ x.im (hypot y.re y.im)))
        (t_2 (fma t_0 t_1 (* x.re (/ y.im (- (pow (hypot y.re y.im) 2.0)))))))
   (if (<= y.im -2.8e+84)
     (fma t_0 t_1 (* x.re (/ -1.0 y.im)))
     (if (<= y.im -5e-305)
       t_2
       (if (<= y.im 5.3e-191)
         (fma t_0 t_1 (* x.re (* (/ y.im (hypot y.im y.re)) (/ -1.0 y.re))))
         (if (<= y.im 2e+150)
           t_2
           (fma
            t_0
            (/ x.im y.im)
            (/
             (/ x.re (hypot y.im y.re))
             (/ (hypot y.im 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 / hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / hypot(y_46_re, y_46_im);
	double t_2 = fma(t_0, t_1, (x_46_re * (y_46_im / -pow(hypot(y_46_re, y_46_im), 2.0))));
	double tmp;
	if (y_46_im <= -2.8e+84) {
		tmp = fma(t_0, t_1, (x_46_re * (-1.0 / y_46_im)));
	} else if (y_46_im <= -5e-305) {
		tmp = t_2;
	} else if (y_46_im <= 5.3e-191) {
		tmp = fma(t_0, t_1, (x_46_re * ((y_46_im / hypot(y_46_im, y_46_re)) * (-1.0 / y_46_re))));
	} else if (y_46_im <= 2e+150) {
		tmp = t_2;
	} else {
		tmp = fma(t_0, (x_46_im / y_46_im), ((x_46_re / hypot(y_46_im, y_46_re)) / (hypot(y_46_im, 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(y_46_re / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	t_2 = fma(t_0, t_1, Float64(x_46_re * Float64(y_46_im / Float64(-(hypot(y_46_re, y_46_im) ^ 2.0)))))
	tmp = 0.0
	if (y_46_im <= -2.8e+84)
		tmp = fma(t_0, t_1, Float64(x_46_re * Float64(-1.0 / y_46_im)));
	elseif (y_46_im <= -5e-305)
		tmp = t_2;
	elseif (y_46_im <= 5.3e-191)
		tmp = fma(t_0, t_1, Float64(x_46_re * Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(-1.0 / y_46_re))));
	elseif (y_46_im <= 2e+150)
		tmp = t_2;
	else
		tmp = fma(t_0, Float64(x_46_im / y_46_im), Float64(Float64(x_46_re / hypot(y_46_im, y_46_re)) / Float64(hypot(y_46_im, y_46_re) / Float64(-y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[(x$46$re * N[(y$46$im / (-N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.8e+84], N[(t$95$0 * t$95$1 + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5e-305], t$95$2, If[LessEqual[y$46$im, 5.3e-191], N[(t$95$0 * t$95$1 + N[(x$46$re * N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+150], t$95$2, N[(t$95$0 * N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / (-y$46$im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.79999999999999982e84

   1. Initial program 50.9%

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

      1. div-sub 50.9%

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

      2. *-commutative 50.9%

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

      3. fma-define 50.9%

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

      4. add-sqr-sqrt 50.9%

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

      5. times-frac 51.3%

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

      6. fma-neg 51.3%

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

      7. fma-define 51.3%

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

      8. hypot-define 51.3%

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

      9. fma-define 51.3%

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

      10. hypot-define 68.6%

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

      11. associate-/l* 75.8%

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

      12. fma-define 75.8%

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

      13. add-sqr-sqrt 75.8%

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

      14. pow2 75.8%

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

   4. Applied egg-rr 75.8%

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

   5. Taylor expanded in y.im around inf 99.8%

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

   if -2.79999999999999982e84 < y.im < -4.99999999999999985e-305 or 5.29999999999999985e-191 < y.im < 1.99999999999999996e150

   1. Initial program 76.3%

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

      1. div-sub 75.4%

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

      2. *-commutative 75.4%

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

      3. fma-define 75.4%

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

      4. add-sqr-sqrt 75.4%

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

      5. times-frac 78.6%

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

      6. fma-neg 78.6%

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

      7. fma-define 78.6%

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

      8. hypot-define 78.6%

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

      9. fma-define 78.6%

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

      10. hypot-define 90.6%

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

      11. associate-/l* 92.7%

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

      12. fma-define 92.7%

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

      13. add-sqr-sqrt 92.7%

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

      14. pow2 92.7%

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

   4. Applied egg-rr 92.7%

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

   if -4.99999999999999985e-305 < y.im < 5.29999999999999985e-191

   1. Initial program 71.9%

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

      1. div-sub 66.6%

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

      2. *-commutative 66.6%

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

      3. fma-define 66.6%

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

      4. add-sqr-sqrt 66.6%

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

      5. times-frac 68.5%

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

      6. fma-neg 68.5%

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

      7. fma-define 68.5%

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

      8. hypot-define 68.5%

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

      9. fma-define 68.5%

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

      10. hypot-define 80.0%

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

      11. associate-/l* 80.2%

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

      12. fma-define 80.2%

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

      13. add-sqr-sqrt 80.2%

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

      14. pow2 80.2%

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

   4. Applied egg-rr 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. unpow2 80.2%

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

      3. times-frac 99.9%

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

      4. hypot-undefine 80.2%

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

      5. +-commutative 80.2%

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

      6. hypot-define 99.9%

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

      7. hypot-undefine 80.2%

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

      8. +-commutative 80.2%

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

      9. hypot-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y.im around 0 99.9%

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

   if 1.99999999999999996e150 < y.im

   1. Initial program 21.7%

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

      1. div-sub 21.7%

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

      2. *-commutative 21.7%

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

      3. fma-define 21.7%

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

      4. add-sqr-sqrt 21.7%

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

      5. times-frac 21.8%

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

      6. fma-neg 21.8%

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

      7. fma-define 21.8%

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

      8. hypot-define 21.8%

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

      9. fma-define 21.8%

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

      10. hypot-define 37.7%

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

      11. associate-/l* 40.9%

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

      12. fma-define 40.9%

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

      13. add-sqr-sqrt 40.9%

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

      14. pow2 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. unpow2 40.9%

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

      3. times-frac 95.3%

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

      4. hypot-undefine 40.9%

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

      5. +-commutative 40.9%

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

      6. hypot-define 95.3%

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

      7. hypot-undefine 40.9%

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

      8. +-commutative 40.9%

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

      9. hypot-define 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. associate-*r* 99.5%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y.re around 0 91.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \left(\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-1}{y.re}\right)\right)\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{y.im}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

   1. div-sub 58.8%

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

   2. *-commutative 58.8%

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

   3. fma-define 58.8%

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

   4. add-sqr-sqrt 58.8%

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

   5. times-frac 60.7%

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

   6. fma-neg 60.7%

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

   7. fma-define 60.7%

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

   8. hypot-define 60.7%

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

   9. fma-define 60.7%

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

   10. hypot-define 74.4%

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

   11. associate-/l* 77.5%

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

   12. fma-define 77.5%

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

   13. add-sqr-sqrt 77.5%

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

   14. pow2 77.5%

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

4. Applied egg-rr 77.5%

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

   1. *-un-lft-identity 77.5%

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

   2. unpow2 77.5%

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

   3. times-frac 96.8%

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

   4. hypot-undefine 77.5%

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

   5. +-commutative 77.5%

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

   6. hypot-define 96.8%

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

   7. hypot-undefine 77.5%

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

   8. +-commutative 77.5%

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

   9. hypot-define 96.8%

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

6. Applied egg-rr 96.8%

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

   1. associate-*r* 97.8%

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

   2. clear-num 97.8%

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

   3. un-div-inv 97.9%

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

   4. un-div-inv 98.0%

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

8. Applied egg-rr 98.0%

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

9. Final simplification 98.0%

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

Alternative 4: 80.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1850000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, \frac{\frac{x.re}{y.re}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (hypot y.re y.im)))
        (t_1 (/ x.im (hypot y.re y.im)))
        (t_2 (fma t_0 t_1 (* x.re (/ -1.0 y.im)))))
   (if (<= y.im -1850000000.0)
     t_2
     (if (<= y.im 3.5e-263)
       (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
       (if (<= y.im 3e-173)
         (fma t_0 t_1 (/ (/ x.re y.re) (/ (hypot y.im y.re) (- y.im))))
         (if (<= y.im 1.65e+35)
           (/ (fma x.im y.re (* y.im (- x.re))) (fma y.im y.im (* y.re y.re)))
           t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re / hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / hypot(y_46_re, y_46_im);
	double t_2 = fma(t_0, t_1, (x_46_re * (-1.0 / y_46_im)));
	double tmp;
	if (y_46_im <= -1850000000.0) {
		tmp = t_2;
	} else if (y_46_im <= 3.5e-263) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 3e-173) {
		tmp = fma(t_0, t_1, ((x_46_re / y_46_re) / (hypot(y_46_im, y_46_re) / -y_46_im)));
	} else if (y_46_im <= 1.65e+35) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	t_2 = fma(t_0, t_1, Float64(x_46_re * Float64(-1.0 / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1850000000.0)
		tmp = t_2;
	elseif (y_46_im <= 3.5e-263)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 3e-173)
		tmp = fma(t_0, t_1, Float64(Float64(x_46_re / y_46_re) / Float64(hypot(y_46_im, y_46_re) / Float64(-y_46_im))));
	elseif (y_46_im <= 1.65e+35)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1850000000.0], t$95$2, If[LessEqual[y$46$im, 3.5e-263], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3e-173], N[(t$95$0 * t$95$1 + N[(N[(x$46$re / y$46$re), $MachinePrecision] / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / (-y$46$im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+35], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.85e9 or 1.6500000000000001e35 < y.im

   1. Initial program 41.5%

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

      1. div-sub 41.5%

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

      2. *-commutative 41.5%

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

      3. fma-define 41.5%

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

      4. add-sqr-sqrt 41.5%

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

      5. times-frac 44.0%

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

      6. fma-neg 44.0%

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

      7. fma-define 44.0%

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

      8. hypot-define 44.0%

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

      9. fma-define 44.0%

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

      10. hypot-define 60.6%

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

      11. associate-/l* 66.0%

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

      12. fma-define 66.0%

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

      13. add-sqr-sqrt 66.0%

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

      14. pow2 66.0%

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

   4. Applied egg-rr 66.0%

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

   5. Taylor expanded in y.im around inf 90.8%

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

   if -1.85e9 < y.im < 3.49999999999999969e-263

   1. Initial program 79.6%

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

   3. Taylor expanded in y.re around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

   if 3.49999999999999969e-263 < y.im < 3.0000000000000001e-173

   1. Initial program 67.7%

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

      1. div-sub 60.4%

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

      2. *-commutative 60.4%

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

      3. fma-define 60.4%

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

      4. add-sqr-sqrt 60.4%

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

      5. times-frac 60.7%

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

      6. fma-neg 60.7%

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

      7. fma-define 60.7%

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

      8. hypot-define 60.8%

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

      9. fma-define 60.8%

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

      10. hypot-define 68.8%

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

      11. associate-/l* 69.1%

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

      12. fma-define 69.1%

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

      13. add-sqr-sqrt 69.1%

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

      14. pow2 69.1%

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

   4. Applied egg-rr 69.1%

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

      1. *-un-lft-identity 69.1%

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

      2. unpow2 69.1%

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

      3. times-frac 99.8%

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

      4. hypot-undefine 69.1%

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

      5. +-commutative 69.1%

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

      6. hypot-define 99.8%

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

      7. hypot-undefine 69.1%

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

      8. +-commutative 69.1%

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

      9. hypot-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r* 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.8%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y.im around 0 94.6%

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

   if 3.0000000000000001e-173 < y.im < 1.6500000000000001e35

   1. Initial program 79.4%

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

      1. fma-neg 79.4%

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

      2. distribute-rgt-neg-out 79.4%

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

      3. +-commutative 79.4%

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

      4. fma-define 79.4%

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

   3. Simplified 79.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1850000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{x.re}{y.re}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}\\ t_1 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-x.im}{y.re}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_2, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_2, \frac{\frac{x.re}{y.re}}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (hypot y.im y.re) (- y.im)))
        (t_1 (/ y.re (hypot y.re y.im)))
        (t_2 (/ x.im (hypot y.re y.im))))
   (if (<= y.re -5.5e-125)
     (fma t_1 (/ (- x.im) y.re) (/ (/ x.re (hypot y.im y.re)) t_0))
     (if (<= y.re 7.4e+77)
       (fma t_1 t_2 (* x.re (/ -1.0 y.im)))
       (fma t_1 t_2 (/ (/ x.re y.re) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = hypot(y_46_im, y_46_re) / -y_46_im;
	double t_1 = y_46_re / hypot(y_46_re, y_46_im);
	double t_2 = x_46_im / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -5.5e-125) {
		tmp = fma(t_1, (-x_46_im / y_46_re), ((x_46_re / hypot(y_46_im, y_46_re)) / t_0));
	} else if (y_46_re <= 7.4e+77) {
		tmp = fma(t_1, t_2, (x_46_re * (-1.0 / y_46_im)));
	} else {
		tmp = fma(t_1, t_2, ((x_46_re / y_46_re) / t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(hypot(y_46_im, y_46_re) / Float64(-y_46_im))
	t_1 = Float64(y_46_re / hypot(y_46_re, y_46_im))
	t_2 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -5.5e-125)
		tmp = fma(t_1, Float64(Float64(-x_46_im) / y_46_re), Float64(Float64(x_46_re / hypot(y_46_im, y_46_re)) / t_0));
	elseif (y_46_re <= 7.4e+77)
		tmp = fma(t_1, t_2, Float64(x_46_re * Float64(-1.0 / y_46_im)));
	else
		tmp = fma(t_1, t_2, Float64(Float64(x_46_re / y_46_re) / t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / (-y$46$im)), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.5e-125], N[(t$95$1 * N[((-x$46$im) / y$46$re), $MachinePrecision] + N[(N[(x$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.4e+77], N[(t$95$1 * t$95$2 + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$2 + N[(N[(x$46$re / y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.4999999999999997e-125

   1. Initial program 66.4%

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

      1. div-sub 66.4%

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

      2. *-commutative 66.4%

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

      3. fma-define 66.4%

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

      4. add-sqr-sqrt 66.4%

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

      5. times-frac 69.0%

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

      6. fma-neg 69.0%

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

      7. fma-define 69.0%

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

      8. hypot-define 69.0%

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

      9. fma-define 69.0%

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

      10. hypot-define 83.5%

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

      11. associate-/l* 85.7%

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

      12. fma-define 85.7%

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

      13. add-sqr-sqrt 85.7%

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

      14. pow2 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. unpow2 85.7%

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

      3. times-frac 96.3%

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

      4. hypot-undefine 85.7%

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

      5. +-commutative 85.7%

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

      6. hypot-define 96.3%

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

      7. hypot-undefine 85.7%

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

      8. +-commutative 85.7%

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

      9. hypot-define 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. associate-*r* 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.8%

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

      4. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y.re around -inf 89.3%

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

       1. associate-*r/ 89.3%

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

       2. neg-mul-1 89.3%

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

   11. Simplified 89.3%

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

   if -5.4999999999999997e-125 < y.re < 7.3999999999999999e77

   1. Initial program 63.1%

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

      1. div-sub 60.9%

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

      2. *-commutative 60.9%

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

      3. fma-define 60.9%

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

      4. add-sqr-sqrt 60.9%

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

      5. times-frac 60.1%

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

      6. fma-neg 60.1%

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

      7. fma-define 60.1%

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

      8. hypot-define 60.1%

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

      9. fma-define 60.1%

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

      10. hypot-define 63.5%

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

      11. associate-/l* 68.5%

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

      12. fma-define 68.5%

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

      13. add-sqr-sqrt 68.5%

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

      14. pow2 68.5%

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

   4. Applied egg-rr 68.5%

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

   5. Taylor expanded in y.im around inf 86.6%

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

   if 7.3999999999999999e77 < y.re

   1. Initial program 38.1%

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

      1. div-sub 38.1%

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

      2. *-commutative 38.1%

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

      3. fma-define 38.1%

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

      4. add-sqr-sqrt 38.1%

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

      5. times-frac 44.9%

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

      6. fma-neg 44.9%

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

      7. fma-define 44.9%

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

      8. hypot-define 44.9%

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

      9. fma-define 44.9%

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

      10. hypot-define 81.1%

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

      11. associate-/l* 81.5%

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

      12. fma-define 81.5%

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

      13. add-sqr-sqrt 81.5%

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

      14. pow2 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. unpow2 81.5%

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

      3. times-frac 94.8%

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

      4. hypot-undefine 81.5%

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

      5. +-commutative 81.5%

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

      6. hypot-define 94.8%

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

      7. hypot-undefine 81.5%

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

      8. +-commutative 81.5%

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

      9. hypot-define 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. associate-*r* 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y.im around 0 88.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.im}{y.re}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{x.re}{y.re}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{-y.im}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{if}\;y.im \leq -550000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-255}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (* x.re (/ -1.0 y.im)))))
   (if (<= y.im -550000.0)
     t_0
     (if (<= y.im 3.9e-255)
       (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
       (if (<= y.im 6.8e+41)
         (/ (fma x.im y.re (* y.im (- x.re))) (fma y.im y.im (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re * (-1.0 / y_46_im)));
	double tmp;
	if (y_46_im <= -550000.0) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-255) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 6.8e+41) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(-1.0 / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -550000.0)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-255)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 6.8e+41)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -550000.0], t$95$0, If[LessEqual[y$46$im, 3.9e-255], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.8e+41], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.5e5 or 6.79999999999999996e41 < y.im

   1. Initial program 41.5%

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

      1. div-sub 41.5%

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

      2. *-commutative 41.5%

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

      3. fma-define 41.5%

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

      4. add-sqr-sqrt 41.5%

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

      5. times-frac 44.0%

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

      6. fma-neg 44.0%

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

      7. fma-define 44.0%

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

      8. hypot-define 44.0%

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

      9. fma-define 44.0%

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

      10. hypot-define 60.6%

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

      11. associate-/l* 66.0%

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

      12. fma-define 66.0%

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

      13. add-sqr-sqrt 66.0%

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

      14. pow2 66.0%

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

   4. Applied egg-rr 66.0%

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

   5. Taylor expanded in y.im around inf 90.8%

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

   if -5.5e5 < y.im < 3.9000000000000001e-255

   1. Initial program 78.3%

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

   3. Taylor expanded in y.re around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

      4. associate-/l* 87.9%

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

   5. Simplified 87.9%

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

   if 3.9000000000000001e-255 < y.im < 6.79999999999999996e41

   1. Initial program 77.3%

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

      1. fma-neg 77.4%

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

      2. distribute-rgt-neg-out 77.4%

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

      3. +-commutative 77.4%

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

      4. fma-define 77.4%

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

   3. Simplified 77.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -550000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-255}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-251}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ (/ y.re y.im) (/ y.im x.im)) (/ x.re y.im))))
   (if (<= y.im -3.4e+69)
     t_0
     (if (<= y.im 3e-251)
       (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
       (if (<= y.im 9.8e+132)
         (/ (fma x.im y.re (* y.im (- x.re))) (fma y.im y.im (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -3.4e+69) {
		tmp = t_0;
	} else if (y_46_im <= 3e-251) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 9.8e+132) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.4e+69)
		tmp = t_0;
	elseif (y_46_im <= 3e-251)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 9.8e+132)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.4e+69], t$95$0, If[LessEqual[y$46$im, 3e-251], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.8e+132], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.39999999999999986e69 or 9.8000000000000003e132 < y.im

   1. Initial program 35.3%

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

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

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-commutative 74.6%

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

      5. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. pow2 75.0%

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

      3. times-frac 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. associate-*r* 82.4%

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

      2. clear-num 82.5%

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

      3. un-div-inv 82.5%

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

      4. un-div-inv 82.5%

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

   9. Applied egg-rr 82.5%

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

   if -3.39999999999999986e69 < y.im < 2.9999999999999999e-251

   1. Initial program 76.3%

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

   3. Taylor expanded in y.re around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if 2.9999999999999999e-251 < y.im < 9.8000000000000003e132

   1. Initial program 76.2%

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

      1. fma-neg 76.2%

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

      2. distribute-rgt-neg-out 76.2%

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

      3. +-commutative 76.2%

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

      4. fma-define 76.2%

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

   3. Simplified 76.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-251}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ (/ y.re y.im) (/ y.im x.im)) (/ x.re y.im))))
   (if (<= y.im -3.8e+68)
     t_0
     (if (<= y.im 2.6e-250)
       (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
       (if (<= y.im 8.5e+132)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -3.8e+68) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-250) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 8.5e+132) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re / y_46im) / (y_46im / x_46im)) - (x_46re / y_46im)
    if (y_46im <= (-3.8d+68)) then
        tmp = t_0
    else if (y_46im <= 2.6d-250) then
        tmp = (x_46im / y_46re) - (x_46re * (y_46im / (y_46re ** 2.0d0)))
    else if (y_46im <= 8.5d+132) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -3.8e+68) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-250) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 8.5e+132) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -3.8e+68:
		tmp = t_0
	elif y_46_im <= 2.6e-250:
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 8.5e+132:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.8e+68)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-250)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 8.5e+132)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -3.8e+68)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-250)
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 8.5e+132)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = t_0;
	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 / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+68], t$95$0, If[LessEqual[y$46$im, 2.6e-250], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.5e+132], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.8000000000000001e68 or 8.49999999999999969e132 < y.im

   1. Initial program 35.3%

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

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

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-commutative 74.6%

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

      5. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. pow2 75.0%

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

      3. times-frac 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. associate-*r* 82.4%

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

      2. clear-num 82.5%

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

      3. un-div-inv 82.5%

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

      4. un-div-inv 82.5%

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

   9. Applied egg-rr 82.5%

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

   if -3.8000000000000001e68 < y.im < 2.60000000000000008e-250

   1. Initial program 76.3%

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

   3. Taylor expanded in y.re around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if 2.60000000000000008e-250 < y.im < 8.49999999999999969e132

   1. Initial program 76.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-250}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+63} \lor \neg \left(y.re \leq -2.5 \cdot 10^{+22}\right) \land \left(y.re \leq -33000 \lor \neg \left(y.re \leq 1.65 \cdot 10^{+78}\right)\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.3e+63)
         (and (not (<= y.re -2.5e+22))
              (or (<= y.re -33000.0) (not (<= y.re 1.65e+78)))))
   (/ x.im y.re)
   (- (/ (/ y.re y.im) (/ y.im x.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 <= -2.3e+63) || (!(y_46_re <= -2.5e+22) && ((y_46_re <= -33000.0) || !(y_46_re <= 1.65e+78)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_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 <= (-2.3d+63)) .or. (.not. (y_46re <= (-2.5d+22))) .and. (y_46re <= (-33000.0d0)) .or. (.not. (y_46re <= 1.65d+78))) then
        tmp = x_46im / y_46re
    else
        tmp = ((y_46re / y_46im) / (y_46im / x_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 <= -2.3e+63) || (!(y_46_re <= -2.5e+22) && ((y_46_re <= -33000.0) || !(y_46_re <= 1.65e+78)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_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 <= -2.3e+63) or (not (y_46_re <= -2.5e+22) and ((y_46_re <= -33000.0) or not (y_46_re <= 1.65e+78))):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_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 <= -2.3e+63) || (!(y_46_re <= -2.5e+22) && ((y_46_re <= -33000.0) || !(y_46_re <= 1.65e+78))))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.3e+63) || (~((y_46_re <= -2.5e+22)) && ((y_46_re <= -33000.0) || ~((y_46_re <= 1.65e+78)))))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_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, -2.3e+63], And[N[Not[LessEqual[y$46$re, -2.5e+22]], $MachinePrecision], Or[LessEqual[y$46$re, -33000.0], N[Not[LessEqual[y$46$re, 1.65e+78]], $MachinePrecision]]]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.29999999999999993e63 or -2.4999999999999998e22 < y.re < -33000 or 1.65e78 < y.re

   1. Initial program 48.6%

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

   3. Taylor expanded in y.re around inf 74.0%

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

   if -2.29999999999999993e63 < y.re < -2.4999999999999998e22 or -33000 < y.re < 1.65e78

   1. Initial program 67.7%

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

   3. Taylor expanded in y.re around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. unsub-neg 71.1%

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

      4. *-commutative 71.1%

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

      5. associate-/l* 68.6%

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

   5. Simplified 68.6%

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

      1. *-un-lft-identity 68.6%

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

      2. pow2 68.6%

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

      3. times-frac 72.9%

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

   7. Applied egg-rr 72.9%

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

      1. associate-*r* 76.3%

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

      2. clear-num 76.3%

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

      3. un-div-inv 76.9%

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

      4. un-div-inv 76.9%

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

   9. Applied egg-rr 76.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+63} \lor \neg \left(y.re \leq -2.5 \cdot 10^{+22}\right) \land \left(y.re \leq -33000 \lor \neg \left(y.re \leq 1.65 \cdot 10^{+78}\right)\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -0.0052:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ (/ y.re y.im) (/ y.im x.im)) (/ x.re y.im))))
   (if (<= y.re -4.5e+63)
     (/ x.im y.re)
     (if (<= y.re -2.7e+22)
       t_0
       (if (<= y.re -0.0052)
         (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 7.8e+77) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -4.5e+63) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.7e+22) {
		tmp = t_0;
	} else if (y_46_re <= -0.0052) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.8e+77) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re / y_46im) / (y_46im / x_46im)) - (x_46re / y_46im)
    if (y_46re <= (-4.5d+63)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-2.7d+22)) then
        tmp = t_0
    else if (y_46re <= (-0.0052d0)) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 7.8d+77) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -4.5e+63) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.7e+22) {
		tmp = t_0;
	} else if (y_46_re <= -0.0052) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.8e+77) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_re <= -4.5e+63:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -2.7e+22:
		tmp = t_0
	elif y_46_re <= -0.0052:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 7.8e+77:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.5e+63)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -2.7e+22)
		tmp = t_0;
	elseif (y_46_re <= -0.0052)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 7.8e+77)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -4.5e+63)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -2.7e+22)
		tmp = t_0;
	elseif (y_46_re <= -0.0052)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 7.8e+77)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.5e+63], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.7e+22], t$95$0, If[LessEqual[y$46$re, -0.0052], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+77], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.50000000000000017e63 or 7.7999999999999995e77 < y.re

   1. Initial program 46.1%

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

   3. Taylor expanded in y.re around inf 72.7%

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

   if -4.50000000000000017e63 < y.re < -2.7000000000000002e22 or -0.0051999999999999998 < y.re < 7.7999999999999995e77

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. mul-1-neg 71.4%

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

      3. unsub-neg 71.4%

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

      4. *-commutative 71.4%

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

      5. associate-/l* 68.8%

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

   5. Simplified 68.8%

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

      1. *-un-lft-identity 68.8%

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

      2. pow2 68.8%

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

      3. times-frac 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. associate-*r* 76.6%

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

      2. clear-num 76.6%

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

      3. un-div-inv 77.3%

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

      4. un-div-inv 77.3%

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

   9. Applied egg-rr 77.3%

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

   if -2.7000000000000002e22 < y.re < -0.0051999999999999998

   1. Initial program 85.7%

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

   3. Taylor expanded in x.im around inf 86.7%

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

      1. *-commutative 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -0.0052:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq -0.0019:\\ \;\;\;\;\frac{y.re \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -6.2e+59)
     (/ x.im y.re)
     (if (<= y.re -2.6e+22)
       (/ (* y.im (- x.re)) t_0)
       (if (<= y.re -0.0019)
         (/ (* y.re x.im) t_0)
         (if (<= y.re 1.25e+78)
           (- (/ (/ y.re y.im) (/ y.im x.im)) (/ x.re y.im))
           (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -6.2e+59) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.6e+22) {
		tmp = (y_46_im * -x_46_re) / t_0;
	} else if (y_46_re <= -0.0019) {
		tmp = (y_46_re * x_46_im) / t_0;
	} else if (y_46_re <= 1.25e+78) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46re <= (-6.2d+59)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-2.6d+22)) then
        tmp = (y_46im * -x_46re) / t_0
    else if (y_46re <= (-0.0019d0)) then
        tmp = (y_46re * x_46im) / t_0
    else if (y_46re <= 1.25d+78) then
        tmp = ((y_46re / y_46im) / (y_46im / x_46im)) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -6.2e+59) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.6e+22) {
		tmp = (y_46_im * -x_46_re) / t_0;
	} else if (y_46_re <= -0.0019) {
		tmp = (y_46_re * x_46_im) / t_0;
	} else if (y_46_re <= 1.25e+78) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_re <= -6.2e+59:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -2.6e+22:
		tmp = (y_46_im * -x_46_re) / t_0
	elif y_46_re <= -0.0019:
		tmp = (y_46_re * x_46_im) / t_0
	elif y_46_re <= 1.25e+78:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -6.2e+59)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -2.6e+22)
		tmp = Float64(Float64(y_46_im * Float64(-x_46_re)) / t_0);
	elseif (y_46_re <= -0.0019)
		tmp = Float64(Float64(y_46_re * x_46_im) / t_0);
	elseif (y_46_re <= 1.25e+78)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -6.2e+59)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -2.6e+22)
		tmp = (y_46_im * -x_46_re) / t_0;
	elseif (y_46_re <= -0.0019)
		tmp = (y_46_re * x_46_im) / t_0;
	elseif (y_46_re <= 1.25e+78)
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e+59], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.6e+22], N[(N[(y$46$im * (-x$46$re)), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, -0.0019], N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+78], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.20000000000000029e59 or 1.24999999999999996e78 < y.re

   1. Initial program 46.1%

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

   3. Taylor expanded in y.re around inf 72.7%

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

   if -6.20000000000000029e59 < y.re < -2.6e22

   1. Initial program 90.3%

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

   3. Taylor expanded in x.im around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. neg-mul-1 70.8%

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

   5. Simplified 70.8%

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

   if -2.6e22 < y.re < -0.0019

   1. Initial program 85.7%

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

   3. Taylor expanded in x.im around inf 86.7%

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

      1. *-commutative 86.7%

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

   5. Simplified 86.7%

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

   if -0.0019 < y.re < 1.24999999999999996e78

   1. Initial program 66.3%

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

   3. Taylor expanded in y.re around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. *-commutative 72.0%

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

      5. associate-/l* 69.3%

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

   5. Simplified 69.3%

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

      1. *-un-lft-identity 69.3%

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

      2. pow2 69.3%

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

      3. times-frac 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. associate-*r* 77.7%

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

      2. clear-num 77.7%

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

      3. un-div-inv 78.4%

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

      4. un-div-inv 78.4%

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

   9. Applied egg-rr 78.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -0.0019:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+116} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.85e+116) (not (<= y.im 8.5e+132)))
   (- (/ (/ y.re y.im) (/ y.im x.im)) (/ x.re y.im))
   (/ (- (* y.re x.im) (* y.im x.re)) (+ (* 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) {
	double tmp;
	if ((y_46_im <= -1.85e+116) || !(y_46_im <= 8.5e+132)) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	} else {
		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));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.85d+116)) .or. (.not. (y_46im <= 8.5d+132))) then
        tmp = ((y_46re / y_46im) / (y_46im / x_46im)) - (x_46re / y_46im)
    else
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * 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 <= -1.85e+116) || !(y_46_im <= 8.5e+132)) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	} else {
		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));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.85e+116) or not (y_46_im <= 8.5e+132):
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im)
	else:
		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))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.85e+116) || !(y_46_im <= 8.5e+132))
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im));
	else
		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)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.85e+116) || ~((y_46_im <= 8.5e+132)))
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (x_46_re / y_46_im);
	else
		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));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.85e+116], N[Not[LessEqual[y$46$im, 8.5e+132]], $MachinePrecision]], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8500000000000001e116 or 8.49999999999999969e132 < y.im

   1. Initial program 29.5%

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

   3. Taylor expanded in y.re around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-/l* 74.3%

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

   5. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. pow2 74.3%

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

      3. times-frac 80.3%

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

   7. Applied egg-rr 80.3%

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

      1. associate-*r* 82.6%

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

      2. clear-num 82.6%

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

      3. un-div-inv 82.6%

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

      4. un-div-inv 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -1.8500000000000001e116 < y.im < 8.49999999999999969e132

   1. Initial program 77.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+116} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-35} \lor \neg \left(y.re \leq 1.25 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.7e-35) (not (<= y.re 1.25e+78)))
   (/ 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 <= -1.7e-35) || !(y_46_re <= 1.25e+78)) {
		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 <= (-1.7d-35)) .or. (.not. (y_46re <= 1.25d+78))) 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 <= -1.7e-35) || !(y_46_re <= 1.25e+78)) {
		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 <= -1.7e-35) or not (y_46_re <= 1.25e+78):
		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 <= -1.7e-35) || !(y_46_re <= 1.25e+78))
		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 <= -1.7e-35) || ~((y_46_re <= 1.25e+78)))
		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, -1.7e-35], N[Not[LessEqual[y$46$re, 1.25e+78]], $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 < -1.7000000000000001e-35 or 1.24999999999999996e78 < y.re

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf 66.0%

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

   if -1.7000000000000001e-35 < y.re < 1.24999999999999996e78

   1. Initial program 65.0%

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

   3. Taylor expanded in y.re around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-35} \lor \neg \left(y.re \leq 1.25 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.7% 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 59.8%

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

3. Taylor expanded in y.re around inf 43.1%

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

4. Final simplification 43.1%

   \[\leadsto \frac{x.im}{y.re} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))))