_divideComplex, imaginary part

Percentage Accurate: 61.5% → 98.1%

Time: 15.0s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.0× speedup

Math

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

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 (/ (hypot y.re y.im) y.im)) (/ -1.0 (hypot y.re y.im)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

   1. div-sub 57.9%

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

   2. sub-neg 57.9%

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

   3. *-commutative 57.9%

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

   4. add-sqr-sqrt 57.9%

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

   5. times-frac 60.3%

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

   6. fma-def 60.3%

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

   7. hypot-def 60.3%

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

   8. hypot-def 75.7%

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

   9. associate-/l* 78.7%

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

   10. add-sqr-sqrt 78.7%

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

   11. pow2 78.7%

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

   12. hypot-def 78.7%

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

4. Applied egg-rr 78.7%

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

   1. unpow2 78.7%

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

   2. *-un-lft-identity 78.7%

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

   3. times-frac 96.4%

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

6. Applied egg-rr 96.4%

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

   1. /-rgt-identity 96.4%

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

   2. *-un-lft-identity 96.4%

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

   3. times-frac 97.4%

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

8. Applied egg-rr 97.4%

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

9. Final simplification 97.4%

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

Alternative 2: 97.5% accurate, 0.0× speedup

Math

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

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 (hypot y.re y.im)) (/ (- y.im) (hypot y.re y.im)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

   1. div-sub 57.9%

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

   2. sub-neg 57.9%

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

   3. *-commutative 57.9%

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

   4. add-sqr-sqrt 57.9%

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

   5. times-frac 60.3%

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

   6. fma-def 60.3%

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

   7. hypot-def 60.3%

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

   8. hypot-def 75.7%

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

   9. associate-/l* 78.7%

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

   10. add-sqr-sqrt 78.7%

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

   11. pow2 78.7%

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

   12. hypot-def 78.7%

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

4. Applied egg-rr 78.7%

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

   1. unpow2 78.7%

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

   2. *-un-lft-identity 78.7%

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

   3. times-frac 96.4%

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

6. Applied egg-rr 96.4%

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

   1. /-rgt-identity 96.4%

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

   2. *-un-lft-identity 96.4%

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

   3. *-commutative 96.4%

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

   4. frac-times 97.1%

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

   5. clear-num 97.1%

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

8. Applied egg-rr 97.1%

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

9. Final simplification 97.1%

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

Alternative 3: 89.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+280)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (- x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+280) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+280)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(-x_46_re) / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000002e280

   1. Initial program 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. add-sqr-sqrt 76.2%

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

      3. times-frac 76.2%

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

      4. hypot-def 76.2%

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

      5. hypot-def 96.4%

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

   4. Applied egg-rr 96.4%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.6%

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

      1. div-sub 10.0%

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

      2. sub-neg 10.0%

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

      3. *-commutative 10.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. times-frac 17.5%

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

      6. fma-def 17.5%

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

      7. hypot-def 17.5%

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

      8. hypot-def 51.1%

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

      9. associate-/l* 61.8%

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

      10. add-sqr-sqrt 61.8%

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

      11. pow2 61.8%

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

      12. hypot-def 61.8%

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

   4. Applied egg-rr 61.8%

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

   5. Taylor expanded in y.re around 0 72.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 4 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 4e+263)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (/ x.im (* (hypot y.re y.im) (/ (hypot y.re y.im) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 4e+263) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 4e+263) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = x_46_im / (Math.hypot(y_46_re, y_46_im) * (Math.hypot(y_46_re, y_46_im) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 4e+263:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = x_46_im / (math.hypot(y_46_re, y_46_im) * (math.hypot(y_46_re, y_46_im) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 4e+263)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(y_46_re, y_46_im) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 4e+263)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+263], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.00000000000000006e263

   1. Initial program 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. add-sqr-sqrt 76.1%

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

      3. times-frac 76.0%

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

      4. hypot-def 76.1%

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

      5. hypot-def 96.4%

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

   4. Applied egg-rr 96.4%

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

   if 4.00000000000000006e263 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 14.9%

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

   3. Taylor expanded in x.im around inf 14.3%

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

      1. associate-/l* 22.0%

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

      2. unpow2 22.0%

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

      3. fma-udef 22.0%

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

   5. Simplified 22.0%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-div 5.9%

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

      3. fma-udef 5.9%

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

      4. +-commutative 5.9%

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

      5. pow2 5.9%

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

      6. hypot-udef 5.9%

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

      7. sqrt-div 5.9%

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

      8. fma-udef 5.9%

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

      9. +-commutative 5.9%

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

      10. pow2 5.9%

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

      11. hypot-udef 19.2%

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

      12. times-frac 5.9%

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

      13. add-sqr-sqrt 22.0%

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

      14. *-un-lft-identity 22.0%

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

      15. times-frac 55.0%

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

   7. Applied egg-rr 55.0%

      \[\leadsto \frac{x.im}{\color{blue}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{1} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 4 \cdot 10^{+263}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5000:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e+51)
   (/ x.im (* (hypot y.re y.im) (/ (hypot y.re y.im) y.re)))
   (if (<= y.re -7.2e-133)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 5000.0)
       (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
       (- (/ x.im y.re) (* y.im (/ x.re (pow y.re 2.0))))))))

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 <= -5.6e+51) {
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_re));
	} else if (y_46_re <= -7.2e-133) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5000.0) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / pow(y_46_re, 2.0)));
	}
	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 <= -5.6e+51) {
		tmp = x_46_im / (Math.hypot(y_46_re, y_46_im) * (Math.hypot(y_46_re, y_46_im) / y_46_re));
	} else if (y_46_re <= -7.2e-133) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5000.0) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / Math.pow(y_46_re, 2.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5.6e+51:
		tmp = x_46_im / (math.hypot(y_46_re, y_46_im) * (math.hypot(y_46_re, y_46_im) / y_46_re))
	elif y_46_re <= -7.2e-133:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 5000.0:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / math.pow(y_46_re, 2.0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.6e+51)
		tmp = Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(y_46_re, y_46_im) / y_46_re)));
	elseif (y_46_re <= -7.2e-133)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 5000.0)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / (y_46_re ^ 2.0))));
	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 <= -5.6e+51)
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_re));
	elseif (y_46_re <= -7.2e-133)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 5000.0)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.6e+51], N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.2e-133], 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, 5000.0], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.60000000000000009e51

   1. Initial program 39.1%

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

   3. Taylor expanded in x.im around inf 37.8%

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

      1. associate-/l* 46.1%

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

      2. unpow2 46.1%

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

      3. fma-udef 46.1%

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

   5. Simplified 46.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-div 0.0%

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

      3. fma-udef 0.0%

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

      4. +-commutative 0.0%

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

      5. pow2 0.0%

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

      6. hypot-udef 0.0%

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

      7. sqrt-div 0.0%

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

      8. fma-udef 0.0%

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

      9. +-commutative 0.0%

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

      10. pow2 0.0%

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

      11. hypot-udef 0.0%

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

      12. times-frac 0.0%

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

      13. add-sqr-sqrt 46.1%

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

      14. *-un-lft-identity 46.1%

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

      15. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

   if -5.60000000000000009e51 < y.re < -7.2000000000000008e-133

   1. Initial program 74.5%

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

   if -7.2000000000000008e-133 < y.re < 5e3

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. mul-1-neg 81.0%

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

      3. unsub-neg 81.0%

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

      4. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   if 5e3 < y.re

   1. Initial program 56.4%

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

   3. Taylor expanded in y.re around inf 79.5%

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

      1. +-commutative 79.5%

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

      2. mul-1-neg 79.5%

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

      3. unsub-neg 79.5%

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

      4. associate-/l* 75.5%

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

      5. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5000:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-224}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (* y.im (/ x.re (pow y.re 2.0))))))
   (if (<= y.re -1.2e+48)
     t_1
     (if (<= y.re -1.2e-132)
       t_0
       (if (<= y.re 1.75e-224)
         (/ (- x.re) y.im)
         (if (<= y.re 1.7e+72) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -1.2e+48) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-132) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e-224) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.7e+72) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (y_46im * (x_46re / (y_46re ** 2.0d0)))
    if (y_46re <= (-1.2d+48)) then
        tmp = t_1
    else if (y_46re <= (-1.2d-132)) then
        tmp = t_0
    else if (y_46re <= 1.75d-224) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 1.7d+72) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / Math.pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -1.2e+48) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-132) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e-224) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.7e+72) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / math.pow(y_46_re, 2.0)))
	tmp = 0
	if y_46_re <= -1.2e+48:
		tmp = t_1
	elif y_46_re <= -1.2e-132:
		tmp = t_0
	elif y_46_re <= 1.75e-224:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 1.7e+72:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / (y_46_re ^ 2.0))))
	tmp = 0.0
	if (y_46_re <= -1.2e+48)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-132)
		tmp = t_0;
	elseif (y_46_re <= 1.75e-224)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.7e+72)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -1.2e+48)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-132)
		tmp = t_0;
	elseif (y_46_re <= 1.75e-224)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 1.7e+72)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.2e+48], t$95$1, If[LessEqual[y$46$re, -1.2e-132], t$95$0, If[LessEqual[y$46$re, 1.75e-224], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+72], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.2000000000000001e48 or 1.6999999999999999e72 < y.re

   1. Initial program 42.0%

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

   3. Taylor expanded in y.re around inf 75.9%

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

      1. +-commutative 75.9%

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

      2. mul-1-neg 75.9%

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

      3. unsub-neg 75.9%

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

      4. associate-/l* 77.1%

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

      5. associate-/r/ 78.0%

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

   5. Simplified 78.0%

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

   if -1.2000000000000001e48 < y.re < -1.20000000000000008e-132 or 1.75000000000000009e-224 < y.re < 1.6999999999999999e72

   1. Initial program 77.2%

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

   if -1.20000000000000008e-132 < y.re < 1.75000000000000009e-224

   1. Initial program 65.5%

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

   3. Taylor expanded in y.re around 0 79.3%

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

      1. associate-*r/ 79.3%

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

      2. neg-mul-1 79.3%

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

   5. Simplified 79.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-224}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.45 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -3.6e+130)
     (/ x.re (hypot y.re y.im))
     (if (<= y.im -9e-245)
       t_0
       (if (<= y.im 4.5e-26)
         (/ x.im y.re)
         (if (<= y.im 3.45e+146) t_0 (/ x.re (- (hypot y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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_im <= -3.6e+130) {
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -9e-245) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.45e+146) {
		tmp = t_0;
	} else {
		tmp = x_46_re / -hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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_im <= -3.6e+130) {
		tmp = x_46_re / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -9e-245) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.45e+146) {
		tmp = t_0;
	} else {
		tmp = x_46_re / -Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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_im <= -3.6e+130:
		tmp = x_46_re / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -9e-245:
		tmp = t_0
	elif y_46_im <= 4.5e-26:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.45e+146:
		tmp = t_0
	else:
		tmp = x_46_re / -math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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_im <= -3.6e+130)
		tmp = Float64(x_46_re / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -9e-245)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-26)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.45e+146)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / Float64(-hypot(y_46_re, y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3.6e+130)
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -9e-245)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-26)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.45e+146)
		tmp = t_0;
	else
		tmp = x_46_re / -hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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$46$im, -3.6e+130], N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9e-245], t$95$0, If[LessEqual[y$46$im, 4.5e-26], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.45e+146], t$95$0, N[(x$46$re / (-N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.6000000000000001e130

   1. Initial program 32.7%

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

   3. Taylor expanded in x.im around 0 28.5%

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

      1. associate-*r* 28.5%

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

      2. neg-mul-1 28.5%

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

      3. *-commutative 28.5%

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

   5. Simplified 28.5%

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

      1. fma-def 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. fma-def 28.5%

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

      4. hypot-udef 28.5%

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

      5. fma-def 28.5%

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

      6. hypot-udef 28.5%

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

      7. times-frac 82.0%

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

      8. add-sqr-sqrt 31.2%

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

      9. sqrt-unprod 38.0%

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

      10. sqr-neg 38.0%

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

      11. sqrt-unprod 11.3%

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

      12. add-sqr-sqrt 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. clear-num 22.0%

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

      2. frac-2neg 22.0%

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

      3. frac-times 22.0%

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

      4. *-un-lft-identity 22.0%

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

      5. add-sqr-sqrt 10.7%

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

      6. sqrt-unprod 47.4%

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

      7. sqr-neg 47.4%

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

      8. sqrt-unprod 48.0%

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

      9. add-sqr-sqrt 79.7%

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

   9. Applied egg-rr 79.7%

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

   10. Taylor expanded in y.im around -inf 77.6%

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

   if -3.6000000000000001e130 < y.im < -8.99999999999999937e-245 or 4.4999999999999999e-26 < y.im < 3.44999999999999978e146

   1. Initial program 78.7%

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

   if -8.99999999999999937e-245 < y.im < 4.4999999999999999e-26

   1. Initial program 69.5%

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

   3. Taylor expanded in y.re around inf 81.1%

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

   if 3.44999999999999978e146 < y.im

   1. Initial program 13.5%

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

   3. Taylor expanded in x.im around 0 14.8%

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

      1. associate-*r* 14.8%

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

      2. neg-mul-1 14.8%

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

      3. *-commutative 14.8%

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

   5. Simplified 14.8%

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

      1. fma-def 14.8%

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

      2. add-sqr-sqrt 14.8%

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

      3. fma-def 14.8%

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

      4. hypot-udef 14.8%

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

      5. fma-def 14.8%

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

      6. hypot-udef 14.8%

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

      7. times-frac 68.5%

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

      8. add-sqr-sqrt 30.9%

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

      9. sqrt-unprod 34.0%

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

      10. sqr-neg 34.0%

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

      11. sqrt-unprod 9.6%

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

      12. add-sqr-sqrt 15.1%

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

   7. Applied egg-rr 15.1%

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

      1. clear-num 15.1%

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

      2. frac-2neg 15.1%

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

      3. frac-times 15.1%

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

      4. *-un-lft-identity 15.1%

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

      5. add-sqr-sqrt 5.5%

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

      6. sqrt-unprod 30.0%

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

      7. sqr-neg 30.0%

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

      8. sqrt-unprod 37.4%

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

      9. add-sqr-sqrt 68.5%

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

   9. Applied egg-rr 68.5%

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

   10. Taylor expanded in y.re around 0 65.6%

       \[\leadsto \frac{x.re}{\color{blue}{1} \cdot \left(-\mathsf{hypot}\left(y.re, y.im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-245}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.45 \cdot 10^{+146}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 0.0058:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (* y.im (/ x.re (pow y.re 2.0))))))
   (if (<= y.re -6.5e+51)
     t_0
     (if (<= y.re -1.1e-132)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 0.0058)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ 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_re) - (y_46_im * (x_46_re / pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -6.5e+51) {
		tmp = t_0;
	} else if (y_46_re <= -1.1e-132) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 0.0058) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - (y_46im * (x_46re / (y_46re ** 2.0d0)))
    if (y_46re <= (-6.5d+51)) then
        tmp = t_0
    else if (y_46re <= (-1.1d-132)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 0.0058d0) then
        tmp = (x_46im / ((y_46im ** 2.0d0) / y_46re)) - (x_46re / y_46im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / Math.pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -6.5e+51) {
		tmp = t_0;
	} else if (y_46_re <= -1.1e-132) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 0.0058) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / math.pow(y_46_re, 2.0)))
	tmp = 0
	if y_46_re <= -6.5e+51:
		tmp = t_0
	elif y_46_re <= -1.1e-132:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 0.0058:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (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(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / (y_46_re ^ 2.0))))
	tmp = 0.0
	if (y_46_re <= -6.5e+51)
		tmp = t_0;
	elseif (y_46_re <= -1.1e-132)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 0.0058)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -6.5e+51)
		tmp = t_0;
	elseif (y_46_re <= -1.1e-132)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 0.0058)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e+51], t$95$0, If[LessEqual[y$46$re, -1.1e-132], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.0058], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.5e51 or 0.0058 < y.re

   1. Initial program 46.3%

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

   3. Taylor expanded in y.re around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. unsub-neg 76.3%

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

      4. associate-/l* 76.5%

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

      5. associate-/r/ 78.2%

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

   5. Simplified 78.2%

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

   if -6.5e51 < y.re < -1.09999999999999995e-132

   1. Initial program 74.5%

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

   if -1.09999999999999995e-132 < y.re < 0.0058

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. mul-1-neg 81.0%

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

      3. unsub-neg 81.0%

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

      4. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 0.0058:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-243}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

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.im -5.5e+130)
     (/ x.re (hypot y.re y.im))
     (if (<= y.im -2.2e-243)
       t_0
       (if (<= y.im 4.5e-26)
         (/ x.im y.re)
         (if (<= y.im 5.3e+150) t_0 (/ (- x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -5.5e+130) {
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -2.2e-243) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 5.3e+150) {
		tmp = t_0;
	} else {
		tmp = -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 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_im <= -5.5e+130) {
		tmp = x_46_re / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -2.2e-243) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 5.3e+150) {
		tmp = t_0;
	} 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):
	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_im <= -5.5e+130:
		tmp = x_46_re / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -2.2e-243:
		tmp = t_0
	elif y_46_im <= 4.5e-26:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 5.3e+150:
		tmp = t_0
	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)
	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_im <= -5.5e+130)
		tmp = Float64(x_46_re / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -2.2e-243)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-26)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 5.3e+150)
		tmp = t_0;
	else
		tmp = Float64(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)
	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_im <= -5.5e+130)
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -2.2e-243)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-26)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 5.3e+150)
		tmp = t_0;
	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_] := 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$46$im, -5.5e+130], N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.2e-243], t$95$0, If[LessEqual[y$46$im, 4.5e-26], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.3e+150], t$95$0, N[((-x$46$re) / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.4999999999999997e130

   1. Initial program 32.7%

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

   3. Taylor expanded in x.im around 0 28.5%

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

      1. associate-*r* 28.5%

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

      2. neg-mul-1 28.5%

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

      3. *-commutative 28.5%

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

   5. Simplified 28.5%

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

      1. fma-def 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. fma-def 28.5%

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

      4. hypot-udef 28.5%

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

      5. fma-def 28.5%

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

      6. hypot-udef 28.5%

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

      7. times-frac 82.0%

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

      8. add-sqr-sqrt 31.2%

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

      9. sqrt-unprod 38.0%

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

      10. sqr-neg 38.0%

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

      11. sqrt-unprod 11.3%

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

      12. add-sqr-sqrt 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. clear-num 22.0%

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

      2. frac-2neg 22.0%

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

      3. frac-times 22.0%

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

      4. *-un-lft-identity 22.0%

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

      5. add-sqr-sqrt 10.7%

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

      6. sqrt-unprod 47.4%

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

      7. sqr-neg 47.4%

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

      8. sqrt-unprod 48.0%

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

      9. add-sqr-sqrt 79.7%

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

   9. Applied egg-rr 79.7%

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

   10. Taylor expanded in y.im around -inf 77.6%

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

   if -5.4999999999999997e130 < y.im < -2.1999999999999999e-243 or 4.4999999999999999e-26 < y.im < 5.30000000000000013e150

   1. Initial program 78.7%

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

   if -2.1999999999999999e-243 < y.im < 4.4999999999999999e-26

   1. Initial program 69.5%

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

   3. Taylor expanded in y.re around inf 81.1%

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

   if 5.30000000000000013e150 < y.im

   1. Initial program 13.5%

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

   3. Taylor expanded in y.re around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   5. Simplified 65.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-243}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -3.9e+130)
     t_0
     (if (<= y.im -6e-239)
       t_1
       (if (<= y.im 4.5e-26) (/ x.im y.re) (if (<= y.im 1e+150) t_1 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_re / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3.9e+130) {
		tmp = t_0;
	} else if (y_46_im <= -6e-239) {
		tmp = t_1;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1e+150) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-3.9d+130)) then
        tmp = t_0
    else if (y_46im <= (-6d-239)) then
        tmp = t_1
    else if (y_46im <= 4.5d-26) then
        tmp = x_46im / y_46re
    else if (y_46im <= 1d+150) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3.9e+130) {
		tmp = t_0;
	} else if (y_46_im <= -6e-239) {
		tmp = t_1;
	} else if (y_46_im <= 4.5e-26) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1e+150) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -3.9e+130:
		tmp = t_0
	elif y_46_im <= -6e-239:
		tmp = t_1
	elif y_46_im <= 4.5e-26:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1e+150:
		tmp = t_1
	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(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -3.9e+130)
		tmp = t_0;
	elseif (y_46_im <= -6e-239)
		tmp = t_1;
	elseif (y_46_im <= 4.5e-26)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1e+150)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3.9e+130)
		tmp = t_0;
	elseif (y_46_im <= -6e-239)
		tmp = t_1;
	elseif (y_46_im <= 4.5e-26)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1e+150)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.9e+130], t$95$0, If[LessEqual[y$46$im, -6e-239], t$95$1, If[LessEqual[y$46$im, 4.5e-26], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1e+150], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.9000000000000002e130 or 9.99999999999999981e149 < y.im

   1. Initial program 24.7%

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

   3. Taylor expanded in y.re around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   if -3.9000000000000002e130 < y.im < -5.9999999999999996e-239 or 4.4999999999999999e-26 < y.im < 9.99999999999999981e149

   1. Initial program 78.7%

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

   if -5.9999999999999996e-239 < y.im < 4.4999999999999999e-26

   1. Initial program 69.5%

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

   3. Taylor expanded in y.re around inf 81.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-239}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+150}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{y.re \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -1.22e+48)
     (/ x.im y.re)
     (if (<= y.re 6.2e-29)
       (/ (- x.re) y.im)
       (if (<= y.re 2.05e+17)
         (/ (* y.re x.im) t_0)
         (if (<= y.re 1.5e+42) (/ (* y.im (- x.re)) t_0) (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -1.22e+48) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 6.2e-29) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.05e+17) {
		tmp = (y_46_re * x_46_im) / t_0;
	} else if (y_46_re <= 1.5e+42) {
		tmp = (y_46_im * -x_46_re) / t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46re <= (-1.22d+48)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 6.2d-29) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 2.05d+17) then
        tmp = (y_46re * x_46im) / t_0
    else if (y_46re <= 1.5d+42) then
        tmp = (y_46im * -x_46re) / t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -1.22e+48) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 6.2e-29) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.05e+17) {
		tmp = (y_46_re * x_46_im) / t_0;
	} else if (y_46_re <= 1.5e+42) {
		tmp = (y_46_im * -x_46_re) / t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_re <= -1.22e+48:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 6.2e-29:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 2.05e+17:
		tmp = (y_46_re * x_46_im) / t_0
	elif y_46_re <= 1.5e+42:
		tmp = (y_46_im * -x_46_re) / t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.22e+48)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 6.2e-29)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 2.05e+17)
		tmp = Float64(Float64(y_46_re * x_46_im) / t_0);
	elseif (y_46_re <= 1.5e+42)
		tmp = Float64(Float64(y_46_im * Float64(-x_46_re)) / t_0);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.22e+48)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 6.2e-29)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 2.05e+17)
		tmp = (y_46_re * x_46_im) / t_0;
	elseif (y_46_re <= 1.5e+42)
		tmp = (y_46_im * -x_46_re) / t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.22e+48], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-29], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.05e+17], N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+42], N[(N[(y$46$im * (-x$46$re)), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.22000000000000004e48 or 1.50000000000000014e42 < y.re

   1. Initial program 42.8%

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

   3. Taylor expanded in y.re around inf 74.2%

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

   if -1.22000000000000004e48 < y.re < 6.20000000000000052e-29

   1. Initial program 71.1%

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

   3. Taylor expanded in y.re around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   5. Simplified 65.4%

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

   if 6.20000000000000052e-29 < y.re < 2.05e17

   1. Initial program 99.5%

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

   3. Taylor expanded in x.im around inf 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 2.05e17 < y.re < 1.50000000000000014e42

   1. Initial program 99.5%

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

   3. Taylor expanded in x.im around 0 88.5%

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

      1. associate-*r* 88.5%

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

      2. neg-mul-1 88.5%

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

      3. *-commutative 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9.2e+49)
   (/ x.im y.re)
   (if (<= y.re 5.2e-26)
     (/ (- x.re) y.im)
     (if (<= y.re 1.05e+71)
       (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9.2e+49) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.2e-26) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.05e+71) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-9.2d+49)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 5.2d-26) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 1.05d+71) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9.2e+49) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.2e-26) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.05e+71) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -9.2e+49:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 5.2e-26:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 1.05e+71:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -9.2e+49)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 5.2e-26)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.05e+71)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -9.2e+49)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 5.2e-26)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 1.05e+71)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -9.2e+49], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.2e-26], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+71], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.20000000000000008e49 or 1.04999999999999995e71 < y.re

   1. Initial program 42.0%

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

   3. Taylor expanded in y.re around inf 74.3%

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

   if -9.20000000000000008e49 < y.re < 5.2000000000000002e-26

   1. Initial program 71.1%

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

   3. Taylor expanded in y.re around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   5. Simplified 65.4%

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

   if 5.2000000000000002e-26 < y.re < 1.04999999999999995e71

   1. Initial program 93.3%

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

   3. Taylor expanded in x.im around inf 60.8%

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

      1. *-commutative 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+44} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e+44) (not (<= y.re 1.4e-26)))
   (/ 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 <= -6.5e+44) || !(y_46_re <= 1.4e-26)) {
		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 <= (-6.5d+44)) .or. (.not. (y_46re <= 1.4d-26))) 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 <= -6.5e+44) || !(y_46_re <= 1.4e-26)) {
		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 <= -6.5e+44) or not (y_46_re <= 1.4e-26):
		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 <= -6.5e+44) || !(y_46_re <= 1.4e-26))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.5e+44) || ~((y_46_re <= 1.4e-26)))
		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, -6.5e+44], N[Not[LessEqual[y$46$re, 1.4e-26]], $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 < -6.50000000000000018e44 or 1.4000000000000001e-26 < y.re

   1. Initial program 48.6%

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

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

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

   if -6.50000000000000018e44 < y.re < 1.4000000000000001e-26

   1. Initial program 71.1%

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

   3. Taylor expanded in y.re around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   5. Simplified 65.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+44} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in y.re around inf 43.7%

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

4. Final simplification 43.7%

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

Reproduce

herbie shell --seed 2024027 
(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))))