_divideComplex, imaginary part

Percentage Accurate: 62.0% → 96.2%

Time: 13.5s

Alternatives: 11

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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right), \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma
  (log1p (expm1 (/ y.re (hypot y.re y.im))))
  (/ x.im (hypot y.re y.im))
  (* x.re (/ (/ y.im (hypot y.im y.re)) (- (hypot y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(log1p(expm1((y_46_re / hypot(y_46_re, y_46_im)))), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re * ((y_46_im / hypot(y_46_im, y_46_re)) / -hypot(y_46_im, y_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(log1p(expm1(Float64(y_46_re / hypot(y_46_re, y_46_im)))), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) / Float64(-hypot(y_46_im, y_46_re)))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Log[1 + N[(Exp[N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

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

   1. fma-neg 58.3%

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

   2. distribute-rgt-neg-out 58.3%

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

   3. +-commutative 58.3%

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

   4. fma-define 58.3%

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

3. Simplified 58.3%

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

   1. distribute-rgt-neg-out 58.3%

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

   2. fma-neg 58.3%

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

   3. fma-undefine 58.3%

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

   4. +-commutative 58.3%

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

   5. div-sub 57.4%

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

   6. *-commutative 57.4%

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

   7. add-sqr-sqrt 57.4%

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

   8. times-frac 57.0%

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

   9. fma-neg 57.0%

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

   10. hypot-define 57.0%

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

   11. hypot-define 72.3%

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

   12. associate-/l* 77.8%

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

   13. add-sqr-sqrt 77.8%

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

   14. pow2 77.8%

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

6. Applied egg-rr 77.8%

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

   1. *-un-lft-identity 77.8%

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

   2. unpow2 77.8%

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

   3. times-frac 95.2%

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

8. Applied egg-rr 95.2%

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

   1. associate-*l/ 95.2%

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

   2. *-lft-identity 95.2%

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

   3. hypot-undefine 77.8%

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

   4. unpow2 77.8%

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

   5. unpow2 77.8%

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

   6. +-commutative 77.8%

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

   7. unpow2 77.8%

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

   8. unpow2 77.8%

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

   9. hypot-define 95.2%

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

   10. hypot-undefine 77.8%

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

   11. unpow2 77.8%

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

   12. unpow2 77.8%

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

   13. +-commutative 77.8%

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

   14. unpow2 77.8%

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

   15. unpow2 77.8%

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

   16. hypot-define 95.2%

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

10. Simplified 95.2%

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

    1. log1p-expm1-u 95.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \]

12. Applied egg-rr 95.2%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \]

13. Final simplification 95.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right), \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
14. Add Preprocessing

Alternative 2: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ (- x.im (* y.im (/ x.re y.re))) (hypot y.im y.re))
          (/ y.re (hypot y.im y.re)))))
   (if (<= y.re -4e+20)
     t_0
     (if (<= y.re -2.4e-115)
       (fma
        (/ y.re (hypot y.re y.im))
        (/ x.im (hypot y.re y.im))
        (* (/ y.im (pow (hypot y.re y.im) 2.0)) (- x.re)))
       (if (<= y.re 9.5e-53) (/ (- (* x.im (/ y.re y.im)) x.re) 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 = ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / hypot(y_46_im, y_46_re)) * (y_46_re / hypot(y_46_im, y_46_re));
	double tmp;
	if (y_46_re <= -4e+20) {
		tmp = t_0;
	} else if (y_46_re <= -2.4e-115) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((y_46_im / pow(hypot(y_46_re, y_46_im), 2.0)) * -x_46_re));
	} else if (y_46_re <= 9.5e-53) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / 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(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / hypot(y_46_im, y_46_re)) * Float64(y_46_re / hypot(y_46_im, y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4e+20)
		tmp = t_0;
	elseif (y_46_re <= -2.4e-115)
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(y_46_im / (hypot(y_46_re, y_46_im) ^ 2.0)) * Float64(-x_46_re)));
	elseif (y_46_re <= 9.5e-53)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4e+20], t$95$0, If[LessEqual[y$46$re, -2.4e-115], 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[(y$46$im / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-53], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4e20 or 9.5000000000000008e-53 < y.re

   1. Initial program 45.5%

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

      1. fma-neg 45.5%

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

      2. distribute-rgt-neg-out 45.5%

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

      3. +-commutative 45.5%

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

      4. fma-define 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in y.re around inf 45.5%

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

      1. mul-1-neg 45.5%

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

      2. unsub-neg 45.5%

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

      3. *-commutative 45.5%

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

      4. associate-/l* 45.5%

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

   7. Simplified 45.5%

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

      1. *-commutative 45.5%

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

      2. add-sqr-sqrt 45.5%

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

      3. fma-undefine 45.5%

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

      4. hypot-undefine 45.5%

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

      5. fma-undefine 45.5%

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

      6. hypot-undefine 45.5%

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

      7. times-frac 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -4e20 < y.re < -2.40000000000000021e-115

   1. Initial program 87.6%

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

      1. fma-neg 87.6%

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

      2. distribute-rgt-neg-out 87.6%

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

      3. +-commutative 87.6%

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

      4. fma-define 87.6%

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

   3. Simplified 87.6%

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

      1. distribute-rgt-neg-out 87.6%

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

      2. fma-neg 87.6%

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

      3. fma-undefine 87.6%

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

      4. +-commutative 87.6%

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

      5. div-sub 87.6%

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

      6. *-commutative 87.6%

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

      7. add-sqr-sqrt 87.6%

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

      8. times-frac 87.7%

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

      9. fma-neg 87.7%

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

      10. hypot-define 87.7%

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

      11. hypot-define 87.7%

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

      12. associate-/l* 93.7%

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

      13. add-sqr-sqrt 93.7%

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

      14. pow2 93.7%

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

   6. Applied egg-rr 93.7%

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

   if -2.40000000000000021e-115 < y.re < 9.5000000000000008e-53

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. distribute-rgt-neg-out 65.2%

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

      3. +-commutative 65.2%

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

      4. fma-define 65.2%

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

   3. Simplified 65.2%

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

      1. distribute-rgt-neg-out 65.2%

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

      2. fma-neg 65.2%

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

      3. fma-undefine 65.2%

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

      4. +-commutative 65.2%

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

      5. div-sub 62.8%

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

      6. *-commutative 62.8%

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

      7. add-sqr-sqrt 62.8%

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

      8. times-frac 57.9%

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

      9. fma-neg 57.9%

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

      10. hypot-define 57.9%

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

      11. hypot-define 59.9%

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

      12. associate-/l* 65.4%

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

      13. add-sqr-sqrt 65.4%

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

      14. pow2 65.4%

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

   6. Applied egg-rr 65.4%

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

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

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

      1. associate-/l* 91.9%

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

   9. Simplified 91.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.3%

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

   1. fma-neg 58.3%

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

   2. distribute-rgt-neg-out 58.3%

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

   3. +-commutative 58.3%

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

   4. fma-define 58.3%

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

3. Simplified 58.3%

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

   1. distribute-rgt-neg-out 58.3%

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

   2. fma-neg 58.3%

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

   3. fma-undefine 58.3%

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

   4. +-commutative 58.3%

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

   5. div-sub 57.4%

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

   6. *-commutative 57.4%

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

   7. add-sqr-sqrt 57.4%

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

   8. times-frac 57.0%

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

   9. fma-neg 57.0%

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

   10. hypot-define 57.0%

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

   11. hypot-define 72.3%

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

   12. associate-/l* 77.8%

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

   13. add-sqr-sqrt 77.8%

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

   14. pow2 77.8%

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

6. Applied egg-rr 77.8%

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

   1. *-un-lft-identity 77.8%

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

   2. unpow2 77.8%

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

   3. times-frac 95.2%

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

8. Applied egg-rr 95.2%

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

   1. associate-*l/ 95.2%

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

   2. *-lft-identity 95.2%

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

   3. hypot-undefine 77.8%

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

   4. unpow2 77.8%

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

   5. unpow2 77.8%

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

   6. +-commutative 77.8%

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

   7. unpow2 77.8%

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

   8. unpow2 77.8%

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

   9. hypot-define 95.2%

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

   10. hypot-undefine 77.8%

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

   11. unpow2 77.8%

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

   12. unpow2 77.8%

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

   13. +-commutative 77.8%

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

   14. unpow2 77.8%

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

   15. unpow2 77.8%

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

   16. hypot-define 95.2%

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

10. Simplified 95.2%

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

11. Final simplification 95.2%

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

Alternative 4: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-115} \lor \neg \left(y.re \leq 9 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.4e-115) (not (<= y.re 9e-53)))
   (*
    (/ (- x.im (* y.im (/ x.re y.re))) (hypot y.im y.re))
    (/ y.re (hypot y.im y.re)))
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.4e-115) || !(y_46_re <= 9e-53)) {
		tmp = ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / hypot(y_46_im, y_46_re)) * (y_46_re / hypot(y_46_im, y_46_re));
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.4e-115) || !(y_46_re <= 9e-53)) {
		tmp = ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / Math.hypot(y_46_im, y_46_re)) * (y_46_re / Math.hypot(y_46_im, y_46_re));
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.4e-115) or not (y_46_re <= 9e-53):
		tmp = ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / math.hypot(y_46_im, y_46_re)) * (y_46_re / math.hypot(y_46_im, y_46_re))
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.4e-115) || !(y_46_re <= 9e-53))
		tmp = Float64(Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / hypot(y_46_im, y_46_re)) * Float64(y_46_re / hypot(y_46_im, y_46_re)));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.4e-115) || ~((y_46_re <= 9e-53)))
		tmp = ((x_46_im - (y_46_im * (x_46_re / y_46_re))) / hypot(y_46_im, y_46_re)) * (y_46_re / hypot(y_46_im, y_46_re));
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.4e-115], N[Not[LessEqual[y$46$re, 9e-53]], $MachinePrecision]], N[(N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.3999999999999998e-115 or 8.9999999999999997e-53 < y.re

   1. Initial program 54.1%

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

      1. fma-neg 54.1%

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

      2. distribute-rgt-neg-out 54.1%

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

      3. +-commutative 54.1%

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

      4. fma-define 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in y.re around inf 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

      3. *-commutative 54.0%

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

      4. associate-/l* 52.2%

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

   7. Simplified 52.2%

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

      1. *-commutative 52.2%

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

      2. add-sqr-sqrt 52.2%

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

      3. fma-undefine 52.2%

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

      4. hypot-undefine 52.2%

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

      5. fma-undefine 52.2%

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

      6. hypot-undefine 52.2%

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

      7. times-frac 93.0%

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

   9. Applied egg-rr 93.0%

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

   if -3.3999999999999998e-115 < y.re < 8.9999999999999997e-53

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. distribute-rgt-neg-out 65.2%

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

      3. +-commutative 65.2%

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

      4. fma-define 65.2%

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

   3. Simplified 65.2%

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

      1. distribute-rgt-neg-out 65.2%

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

      2. fma-neg 65.2%

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

      3. fma-undefine 65.2%

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

      4. +-commutative 65.2%

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

      5. div-sub 62.8%

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

      6. *-commutative 62.8%

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

      7. add-sqr-sqrt 62.8%

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

      8. times-frac 57.9%

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

      9. fma-neg 57.9%

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

      10. hypot-define 57.9%

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

      11. hypot-define 59.9%

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

      12. associate-/l* 65.4%

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

      13. add-sqr-sqrt 65.4%

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

      14. pow2 65.4%

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

   6. Applied egg-rr 65.4%

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

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

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

      1. associate-/l* 91.9%

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

   9. Simplified 91.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-115} \lor \neg \left(y.re \leq 9 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3.9e+85)
     (/ (- x.im (* y.im (/ x.re y.re))) y.re)
     (if (<= y.re -2.7e-114)
       t_0
       (if (<= y.re 1.2e-52)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 1.4e+89)
           t_0
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3.9e+85) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -2.7e-114) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-52) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+89) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-3.9d+85)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= (-2.7d-114)) then
        tmp = t_0
    else if (y_46re <= 1.2d-52) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46re <= 1.4d+89) then
        tmp = t_0
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3.9e+85) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -2.7e-114) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e-52) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.4e+89) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.9e+85:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= -2.7e-114:
		tmp = t_0
	elif y_46_re <= 1.2e-52:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 1.4e+89:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.9e+85)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= -2.7e-114)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-52)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.4e+89)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.9e+85)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= -2.7e-114)
		tmp = t_0;
	elseif (y_46_re <= 1.2e-52)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.4e+89)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.9e+85], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.7e-114], t$95$0, If[LessEqual[y$46$re, 1.2e-52], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+89], t$95$0, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.90000000000000033e85

   1. Initial program 38.7%

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

      1. fma-neg 38.7%

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

      2. distribute-rgt-neg-out 38.7%

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

      3. +-commutative 38.7%

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

      4. fma-define 38.7%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in y.re around inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. unsub-neg 69.5%

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

      3. *-commutative 69.5%

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

      4. associate-/l* 87.2%

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

   7. Simplified 87.2%

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

   if -3.90000000000000033e85 < y.re < -2.7e-114 or 1.2000000000000001e-52 < y.re < 1.3999999999999999e89

   1. Initial program 80.3%

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

   if -2.7e-114 < y.re < 1.2000000000000001e-52

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. distribute-rgt-neg-out 65.2%

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

      3. +-commutative 65.2%

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

      4. fma-define 65.2%

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

   3. Simplified 65.2%

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

      1. distribute-rgt-neg-out 65.2%

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

      2. fma-neg 65.2%

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

      3. fma-undefine 65.2%

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

      4. +-commutative 65.2%

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

      5. div-sub 62.8%

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

      6. *-commutative 62.8%

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

      7. add-sqr-sqrt 62.8%

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

      8. times-frac 57.9%

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

      9. fma-neg 57.9%

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

      10. hypot-define 57.9%

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

      11. hypot-define 59.9%

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

      12. associate-/l* 65.4%

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

      13. add-sqr-sqrt 65.4%

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

      14. pow2 65.4%

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

   6. Applied egg-rr 65.4%

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

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

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

      1. associate-/l* 91.9%

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

   9. Simplified 91.9%

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

   if 1.3999999999999999e89 < y.re

   1. Initial program 20.6%

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

      1. fma-neg 20.6%

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

      2. distribute-rgt-neg-out 20.6%

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

      3. +-commutative 20.6%

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

      4. fma-define 20.6%

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

   3. Simplified 20.6%

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

   5. Taylor expanded in y.re around inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

      3. distribute-lft-neg-in 82.7%

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

   7. Simplified 82.7%

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

      1. associate-/l* 90.4%

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

      2. add-sqr-sqrt 45.5%

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

      3. sqrt-unprod 72.1%

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

      4. sqr-neg 72.1%

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

      5. sqrt-unprod 37.1%

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

      6. add-sqr-sqrt 82.6%

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

      7. cancel-sign-sub 82.6%

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

      8. add-sqr-sqrt 45.5%

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

      9. sqrt-unprod 74.8%

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

      10. sqr-neg 74.8%

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

      11. sqrt-unprod 44.8%

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

      12. add-sqr-sqrt 90.4%

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

   9. Applied egg-rr 90.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-114}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1550000 \lor \neg \left(y.re \leq 3.7 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1550000.0) (not (<= y.re 3.7e-47)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1550000.0) || !(y_46_re <= 3.7e-47)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1550000.0d0)) .or. (.not. (y_46re <= 3.7d-47))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1550000.0) || !(y_46_re <= 3.7e-47)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1550000.0) or not (y_46_re <= 3.7e-47):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1550000.0) || !(y_46_re <= 3.7e-47))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1550000.0) || ~((y_46_re <= 3.7e-47)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1550000.0], N[Not[LessEqual[y$46$re, 3.7e-47]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.55e6 or 3.7e-47 < y.re

   1. Initial program 47.6%

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

      1. fma-neg 47.6%

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

      2. distribute-rgt-neg-out 47.6%

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

      3. +-commutative 47.6%

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

      4. fma-define 47.6%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in y.re around inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. unsub-neg 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-/l* 77.0%

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

   7. Simplified 77.0%

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

   if -1.55e6 < y.re < 3.7e-47

   1. Initial program 69.6%

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

      1. fma-neg 69.6%

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

      2. distribute-rgt-neg-out 69.6%

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

      3. +-commutative 69.6%

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

      4. fma-define 69.6%

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

   3. Simplified 69.6%

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

      1. distribute-rgt-neg-out 69.6%

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

      2. fma-neg 69.6%

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

      3. fma-undefine 69.6%

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

      4. +-commutative 69.6%

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

      5. div-sub 67.7%

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

      6. *-commutative 67.7%

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

      7. add-sqr-sqrt 67.7%

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

      8. times-frac 63.8%

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

      9. fma-neg 63.8%

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

      10. hypot-define 63.8%

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

      11. hypot-define 65.4%

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

      12. associate-/l* 71.3%

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

      13. add-sqr-sqrt 71.3%

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

      14. pow2 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in y.im around inf 86.3%

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

      1. associate-/l* 86.7%

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

   9. Simplified 86.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1550000 \lor \neg \left(y.re \leq 3.7 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-87} \lor \neg \left(y.re \leq 1.55 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.1e-87) (not (<= y.re 1.55e-52)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ x.re (- y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.1e-87) || !(y_46_re <= 1.55e-52)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.1d-87)) .or. (.not. (y_46re <= 1.55d-52))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = x_46re / -y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.1e-87) || !(y_46_re <= 1.55e-52)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.1e-87) or not (y_46_re <= 1.55e-52):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = x_46_re / -y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.1e-87) || !(y_46_re <= 1.55e-52))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.1e-87) || ~((y_46_re <= 1.55e-52)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.1e-87], N[Not[LessEqual[y$46$re, 1.55e-52]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.09999999999999998e-87 or 1.5499999999999999e-52 < y.re

   1. Initial program 52.0%

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

      1. fma-neg 52.0%

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

      2. distribute-rgt-neg-out 52.0%

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

      3. +-commutative 52.0%

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

      4. fma-define 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in y.re around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

      3. distribute-lft-neg-in 65.4%

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

   7. Simplified 65.4%

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

      1. associate-/l* 72.7%

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

      2. add-sqr-sqrt 30.8%

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

      3. sqrt-unprod 45.5%

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

      4. sqr-neg 45.5%

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

      5. sqrt-unprod 30.9%

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

      6. add-sqr-sqrt 54.5%

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

      7. cancel-sign-sub 54.5%

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

      8. add-sqr-sqrt 23.6%

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

      9. sqrt-unprod 48.9%

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

      10. sqr-neg 48.9%

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

      11. sqrt-unprod 41.8%

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

      12. add-sqr-sqrt 72.7%

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

   9. Applied egg-rr 72.7%

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

   if -3.09999999999999998e-87 < y.re < 1.5499999999999999e-52

   1. Initial program 67.2%

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

      1. fma-neg 67.2%

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

      2. distribute-rgt-neg-out 67.2%

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

      3. +-commutative 67.2%

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

      4. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y.re around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-87} \lor \neg \left(y.re \leq 1.55 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.6e-90)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (if (<= y.re 1.8e-52)
     (/ x.re (- y.im))
     (/ (- x.im (* y.im (/ x.re y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.6e-90) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.8e-52) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.6d-90)) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 1.8d-52) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.6e-90) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.8e-52) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.6e-90:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 1.8e-52:
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.6e-90)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.8e-52)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.6e-90)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.8e-52)
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.6e-90], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-52], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.60000000000000004e-90

   1. Initial program 56.7%

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

      1. fma-neg 56.7%

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

      2. distribute-rgt-neg-out 56.7%

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

      3. +-commutative 56.7%

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

      4. fma-define 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in y.re around inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. neg-mul-1 62.5%

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

      3. distribute-lft-neg-in 62.5%

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

   7. Simplified 62.5%

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

      1. associate-/l* 72.4%

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

      2. add-sqr-sqrt 27.9%

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

      3. sqrt-unprod 41.4%

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

      4. sqr-neg 41.4%

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

      5. sqrt-unprod 33.1%

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

      6. add-sqr-sqrt 52.6%

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

      7. cancel-sign-sub 52.6%

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

      8. add-sqr-sqrt 19.5%

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

      9. sqrt-unprod 44.2%

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

      10. sqr-neg 44.2%

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

      11. sqrt-unprod 44.4%

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

      12. add-sqr-sqrt 72.4%

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

   9. Applied egg-rr 72.4%

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

   if -1.60000000000000004e-90 < y.re < 1.79999999999999994e-52

   1. Initial program 67.2%

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

      1. fma-neg 67.2%

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

      2. distribute-rgt-neg-out 67.2%

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

      3. +-commutative 67.2%

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

      4. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y.re around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   7. Simplified 73.2%

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

   if 1.79999999999999994e-52 < y.re

   1. Initial program 46.5%

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

      1. fma-neg 46.5%

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

      2. distribute-rgt-neg-out 46.5%

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

      3. +-commutative 46.5%

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

      4. fma-define 46.5%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in y.re around inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. unsub-neg 68.8%

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

      3. *-commutative 68.8%

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

      4. associate-/l* 73.0%

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

   7. Simplified 73.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-10} \lor \neg \left(y.re \leq 4.5 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.02e-10) (not (<= y.re 4.5e+84)))
   (/ 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 <= -1.02e-10) || !(y_46_re <= 4.5e+84)) {
		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 <= (-1.02d-10)) .or. (.not. (y_46re <= 4.5d+84))) 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 <= -1.02e-10) || !(y_46_re <= 4.5e+84)) {
		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 <= -1.02e-10) or not (y_46_re <= 4.5e+84):
		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 <= -1.02e-10) || !(y_46_re <= 4.5e+84))
		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 <= -1.02e-10) || ~((y_46_re <= 4.5e+84)))
		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, -1.02e-10], N[Not[LessEqual[y$46$re, 4.5e+84]], $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 < -1.01999999999999997e-10 or 4.4999999999999997e84 < y.re

   1. Initial program 39.3%

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

      1. fma-neg 39.3%

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

      2. distribute-rgt-neg-out 39.3%

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

      3. +-commutative 39.3%

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

      4. fma-define 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in y.re around inf 70.4%

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

   if -1.01999999999999997e-10 < y.re < 4.4999999999999997e84

   1. Initial program 70.9%

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

      1. fma-neg 70.9%

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

      2. distribute-rgt-neg-out 70.9%

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

      3. +-commutative 70.9%

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

      4. fma-define 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in y.re around 0 60.9%

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

      1. associate-*r/ 60.9%

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

      2. neg-mul-1 60.9%

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

   7. Simplified 60.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-10} \lor \neg \left(y.re \leq 4.5 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 1.65 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 1.65e+112) (/ x.im y.re) (/ x.re y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 1.65e+112) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 1.65d+112) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 1.65e+112) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 1.65e+112:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 1.65e+112)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 1.65e+112)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 1.65e+112], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.64999999999999995e112

   1. Initial program 62.7%

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

      1. fma-neg 62.7%

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

      2. distribute-rgt-neg-out 62.7%

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

      3. +-commutative 62.7%

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

      4. fma-define 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in y.re around inf 45.1%

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

   if 1.64999999999999995e112 < y.im

   1. Initial program 40.9%

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

      1. fma-neg 40.9%

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

      2. distribute-rgt-neg-out 40.9%

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

      3. +-commutative 40.9%

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

      4. fma-define 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in y.re around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. neg-mul-1 79.8%

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

   7. Simplified 79.8%

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

      1. neg-sub0 79.8%

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

      2. sub-neg 79.8%

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

      3. add-sqr-sqrt 31.7%

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

      4. sqrt-unprod 50.5%

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

      5. sqr-neg 50.5%

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

      6. sqrt-unprod 18.8%

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

      7. add-sqr-sqrt 31.8%

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

   9. Applied egg-rr 31.8%

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

       1. +-lft-identity 31.8%

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

   11. Simplified 31.8%

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

Alternative 11: 43.3% 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 58.3%

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

   1. fma-neg 58.3%

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

   2. distribute-rgt-neg-out 58.3%

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

   3. +-commutative 58.3%

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

   4. fma-define 58.3%

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

3. Simplified 58.3%

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

5. Taylor expanded in y.re around inf 38.3%

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

Reproduce

herbie shell --seed 2024141 
(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))))