_divideComplex, imaginary part

Percentage Accurate: 61.0% → 98.0%

Time: 13.1s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.7%

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

   1. div-sub 57.3%

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

   2. *-commutative 57.3%

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

   3. add-sqr-sqrt 57.3%

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

   4. times-frac 58.8%

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

   5. fma-neg 58.8%

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

   6. hypot-define 58.8%

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

   7. hypot-define 72.9%

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

   8. associate-/l* 77.6%

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

   9. add-sqr-sqrt 77.6%

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

   10. pow2 77.6%

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

   11. hypot-define 77.6%

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

4. Applied egg-rr 77.6%

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

   1. *-un-lft-identity 77.6%

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

   2. unpow2 77.6%

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

   3. times-frac 94.4%

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

   4. hypot-undefine 77.6%

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

   5. +-commutative 77.6%

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

   6. hypot-define 94.4%

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

   7. hypot-undefine 77.6%

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

   8. +-commutative 77.6%

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

   9. hypot-define 94.4%

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

6. Applied egg-rr 94.4%

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

   1. associate-*l/ 94.4%

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

   2. *-un-lft-identity 94.4%

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

8. Applied egg-rr 94.4%

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

   1. associate-*r/ 97.6%

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

   2. clear-num 97.6%

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

   3. un-div-inv 97.6%

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

10. Applied egg-rr 97.6%

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

11. Final simplification 97.6%

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

Alternative 2: 91.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ t_1 := \mathsf{fma}\left(y.re \cdot \frac{1}{y.im}, \frac{x.im}{y.im}, \frac{x.re}{-y.im}\right)\\ \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -3.15 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 10^{-199}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (* x.re (/ y.im (- (pow (hypot y.re y.im) 2.0))))))
        (t_1 (fma (* y.re (/ 1.0 y.im)) (/ x.im y.im) (/ x.re (- y.im)))))
   (if (<= y.im -1.5e+146)
     t_1
     (if (<= y.im -3.15e-161)
       t_0
       (if (<= y.im 1e-199)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 4.2e+124) 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 = 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 / -pow(hypot(y_46_re, y_46_im), 2.0))));
	double t_1 = fma((y_46_re * (1.0 / y_46_im)), (x_46_im / y_46_im), (x_46_re / -y_46_im));
	double tmp;
	if (y_46_im <= -1.5e+146) {
		tmp = t_1;
	} else if (y_46_im <= -3.15e-161) {
		tmp = t_0;
	} else if (y_46_im <= 1e-199) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 4.2e+124) {
		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 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(y_46_im / Float64(-(hypot(y_46_re, y_46_im) ^ 2.0)))))
	t_1 = fma(Float64(y_46_re * Float64(1.0 / y_46_im)), Float64(x_46_im / y_46_im), Float64(x_46_re / Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.5e+146)
		tmp = t_1;
	elseif (y_46_im <= -3.15e-161)
		tmp = t_0;
	elseif (y_46_im <= 1e-199)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 4.2e+124)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(y$46$im / (-N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.5e+146], t$95$1, If[LessEqual[y$46$im, -3.15e-161], t$95$0, If[LessEqual[y$46$im, 1e-199], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+124], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.50000000000000001e146 or 4.20000000000000023e124 < y.im

   1. Initial program 30.2%

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

   3. Taylor expanded in y.re around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. associate-/l* 83.9%

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

   5. Simplified 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. pow2 83.9%

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

      3. times-frac 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. associate-*r* 90.6%

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

      2. fma-neg 90.7%

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

   9. Applied egg-rr 90.7%

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

   if -1.50000000000000001e146 < y.im < -3.1500000000000001e-161 or 9.99999999999999982e-200 < y.im < 4.20000000000000023e124

   1. Initial program 70.2%

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

      1. div-sub 69.4%

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

      2. *-commutative 69.4%

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

      3. add-sqr-sqrt 69.4%

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

      4. times-frac 68.5%

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

      5. fma-neg 68.5%

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

      6. hypot-define 68.5%

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

      7. hypot-define 82.4%

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

      8. associate-/l* 89.6%

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

      9. add-sqr-sqrt 89.6%

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

      10. pow2 89.6%

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

      11. hypot-define 89.6%

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

   4. Applied egg-rr 89.6%

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

   if -3.1500000000000001e-161 < y.im < 9.99999999999999982e-200

   1. Initial program 71.9%

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

   3. Taylor expanded in y.re around inf 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. *-commutative 88.0%

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

      5. associate-/l* 84.8%

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

   5. Simplified 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. unpow2 84.8%

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

      3. times-frac 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. associate-*l/ 86.6%

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

      2. *-lft-identity 86.6%

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

   9. Simplified 86.6%

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

       1. *-un-lft-identity 86.6%

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

       2. associate-*r/ 93.4%

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

       3. sub-div 93.4%

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

       4. clear-num 93.4%

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

       5. un-div-inv 93.4%

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

   11. Applied egg-rr 93.4%

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

       1. *-lft-identity 93.4%

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

       2. associate-/r/ 95.0%

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

   13. Simplified 95.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(y.re \cdot \frac{1}{y.im}, \frac{x.im}{y.im}, \frac{x.re}{-y.im}\right)\\ \mathbf{elif}\;y.im \leq -3.15 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 10^{-199}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re \cdot \frac{1}{y.im}, \frac{x.im}{y.im}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.7%

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

   1. div-sub 57.3%

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

   2. *-commutative 57.3%

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

   3. add-sqr-sqrt 57.3%

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

   4. times-frac 58.8%

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

   5. fma-neg 58.8%

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

   6. hypot-define 58.8%

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

   7. hypot-define 72.9%

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

   8. associate-/l* 77.6%

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

   9. add-sqr-sqrt 77.6%

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

   10. pow2 77.6%

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

   11. hypot-define 77.6%

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

4. Applied egg-rr 77.6%

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

   1. *-un-lft-identity 77.6%

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

   2. unpow2 77.6%

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

   3. times-frac 94.4%

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

   4. hypot-undefine 77.6%

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

   5. +-commutative 77.6%

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

   6. hypot-define 94.4%

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

   7. hypot-undefine 77.6%

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

   8. +-commutative 77.6%

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

   9. hypot-define 94.4%

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

6. Applied egg-rr 94.4%

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

   1. associate-*l/ 94.4%

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

   2. *-un-lft-identity 94.4%

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

8. Applied egg-rr 94.4%

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

9. Final simplification 94.4%

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

Alternative 4: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-21}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.46 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re \cdot \frac{1}{y.im}, \frac{x.im}{y.im}, \frac{x.re}{-y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.im -1.2e-21)
     (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
     (if (<= y.im 1e-93)
       t_0
       (if (<= y.im 1.46e+51)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 5.8e+67)
           t_0
           (fma (* y.re (/ 1.0 y.im)) (/ x.im 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 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_im <= -1.2e-21) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 1e-93) {
		tmp = t_0;
	} else if (y_46_im <= 1.46e+51) {
		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_im <= 5.8e+67) {
		tmp = t_0;
	} else {
		tmp = fma((y_46_re * (1.0 / y_46_im)), (x_46_im / 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(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_im <= -1.2e-21)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 1e-93)
		tmp = t_0;
	elseif (y_46_im <= 1.46e+51)
		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_im <= 5.8e+67)
		tmp = t_0;
	else
		tmp = fma(Float64(y_46_re * Float64(1.0 / y_46_im)), Float64(x_46_im / y_46_im), Float64(x_46_re / Float64(-y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e-21], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1e-93], t$95$0, If[LessEqual[y$46$im, 1.46e+51], 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, 5.8e+67], t$95$0, N[(N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.2e-21

   1. Initial program 47.1%

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

   3. Taylor expanded in y.re around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. *-commutative 76.6%

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

      5. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. pow2 76.8%

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

      3. times-frac 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. associate-*l/ 79.5%

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

      2. *-un-lft-identity 79.5%

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

   9. Applied egg-rr 79.5%

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

   if -1.2e-21 < y.im < 9.999999999999999e-94 or 1.4600000000000001e51 < y.im < 5.80000000000000047e67

   1. Initial program 72.4%

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

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

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

      1. +-commutative 84.0%

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

      2. mul-1-neg 84.0%

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

      3. unsub-neg 84.0%

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

      4. *-commutative 84.0%

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

      5. associate-/l* 82.3%

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

   5. Simplified 82.3%

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

      1. *-un-lft-identity 82.3%

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

      2. unpow2 82.3%

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

      3. times-frac 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. associate-*l/ 84.1%

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

      2. *-lft-identity 84.1%

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

   9. Simplified 84.1%

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

       1. *-un-lft-identity 84.1%

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

       2. associate-*r/ 88.6%

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

       3. sub-div 90.4%

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

       4. clear-num 90.4%

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

       5. un-div-inv 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. *-lft-identity 90.4%

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

       2. associate-/r/ 91.3%

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

   13. Simplified 91.3%

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

   if 9.999999999999999e-94 < y.im < 1.4600000000000001e51

   1. Initial program 82.8%

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

   if 5.80000000000000047e67 < y.im

   1. Initial program 35.0%

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

   3. Taylor expanded in y.re around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. mul-1-neg 75.4%

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

      3. unsub-neg 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-/l* 80.3%

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

   5. Simplified 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. pow2 80.3%

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

      3. times-frac 80.3%

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

   7. Applied egg-rr 80.3%

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

      1. associate-*r* 86.2%

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

      2. fma-neg 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-21}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{-93}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.46 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re \cdot \frac{1}{y.im}, \frac{x.im}{y.im}, \frac{x.re}{-y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ t_1 := y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* x.re (/ y.im y.re))) y.re))
        (t_1 (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))))
   (if (<= y.im -1.65e-22)
     t_1
     (if (<= y.im 1.1e-93)
       t_0
       (if (<= y.im 1.6e+51)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 5.1e+67) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double t_1 = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.65e-22) {
		tmp = t_1;
	} else if (y_46_im <= 1.1e-93) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+51) {
		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_im <= 5.1e+67) {
		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 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    t_1 = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-1.65d-22)) then
        tmp = t_1
    else if (y_46im <= 1.1d-93) then
        tmp = t_0
    else if (y_46im <= 1.6d+51) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 5.1d+67) 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 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double t_1 = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.65e-22) {
		tmp = t_1;
	} else if (y_46_im <= 1.1e-93) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+51) {
		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_im <= 5.1e+67) {
		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 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	t_1 = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.65e-22:
		tmp = t_1
	elif y_46_im <= 1.1e-93:
		tmp = t_0
	elif y_46_im <= 1.6e+51:
		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_im <= 5.1e+67:
		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(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	t_1 = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.65e-22)
		tmp = t_1;
	elseif (y_46_im <= 1.1e-93)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+51)
		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_im <= 5.1e+67)
		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 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	t_1 = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.65e-22)
		tmp = t_1;
	elseif (y_46_im <= 1.1e-93)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+51)
		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_im <= 5.1e+67)
		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[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e-22], t$95$1, If[LessEqual[y$46$im, 1.1e-93], t$95$0, If[LessEqual[y$46$im, 1.6e+51], 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, 5.1e+67], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65e-22 or 5.1000000000000002e67 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. *-commutative 76.1%

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

      5. associate-/l* 78.2%

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

   5. Simplified 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. pow2 78.2%

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

      3. times-frac 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. associate-*l/ 79.8%

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

      2. *-un-lft-identity 79.8%

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

   9. Applied egg-rr 79.8%

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

   if -1.65e-22 < y.im < 1.09999999999999998e-93 or 1.6000000000000001e51 < y.im < 5.1000000000000002e67

   1. Initial program 72.4%

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

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

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

      1. +-commutative 84.0%

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

      2. mul-1-neg 84.0%

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

      3. unsub-neg 84.0%

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

      4. *-commutative 84.0%

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

      5. associate-/l* 82.3%

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

   5. Simplified 82.3%

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

      1. *-un-lft-identity 82.3%

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

      2. unpow2 82.3%

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

      3. times-frac 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. associate-*l/ 84.1%

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

      2. *-lft-identity 84.1%

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

   9. Simplified 84.1%

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

       1. *-un-lft-identity 84.1%

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

       2. associate-*r/ 88.6%

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

       3. sub-div 90.4%

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

       4. clear-num 90.4%

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

       5. un-div-inv 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. *-lft-identity 90.4%

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

       2. associate-/r/ 91.3%

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

   13. Simplified 91.3%

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

   if 1.09999999999999998e-93 < y.im < 1.6000000000000001e51

   1. Initial program 82.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-22}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-22} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7e-22) (not (<= y.im 1.8e+67)))
   (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e-22) || !(y_46_im <= 1.8e+67)) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7d-22)) .or. (.not. (y_46im <= 1.8d+67))) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e-22) || !(y_46_im <= 1.8e+67)) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7e-22) or not (y_46_im <= 1.8e+67):
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7e-22) || !(y_46_im <= 1.8e+67))
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7e-22) || ~((y_46_im <= 1.8e+67)))
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7e-22], N[Not[LessEqual[y$46$im, 1.8e+67]], $MachinePrecision]], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.00000000000000011e-22 or 1.7999999999999999e67 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. *-commutative 76.1%

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

      5. associate-/l* 78.2%

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

   5. Simplified 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. pow2 78.2%

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

      3. times-frac 79.8%

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

   7. Applied egg-rr 79.8%

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

      1. associate-*l/ 79.8%

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

      2. *-un-lft-identity 79.8%

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

   9. Applied egg-rr 79.8%

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

   if -7.00000000000000011e-22 < y.im < 1.7999999999999999e67

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. unpow2 77.4%

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

      3. times-frac 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

   9. Simplified 80.1%

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

       1. *-un-lft-identity 80.1%

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

       2. associate-*r/ 84.0%

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

       3. sub-div 85.5%

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

       4. clear-num 85.5%

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

       5. un-div-inv 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. *-lft-identity 85.5%

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

       2. associate-/r/ 86.2%

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

   13. Simplified 86.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-22} \lor \neg \left(y.im \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-21} \lor \neg \left(y.im \leq 2.4 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.9e-21) (not (<= y.im 2.4e+68)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.9e-21) || !(y_46_im <= 2.4e+68)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.9d-21)) .or. (.not. (y_46im <= 2.4d+68))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.9e-21) || !(y_46_im <= 2.4e+68)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.9e-21) or not (y_46_im <= 2.4e+68):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.9e-21) || !(y_46_im <= 2.4e+68))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.9e-21) || ~((y_46_im <= 2.4e+68)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.9e-21], N[Not[LessEqual[y$46$im, 2.4e+68]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8999999999999999e-21 or 2.40000000000000008e68 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -1.8999999999999999e-21 < y.im < 2.40000000000000008e68

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. unpow2 77.4%

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

      3. times-frac 80.1%

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

   7. Applied egg-rr 80.1%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

   9. Simplified 80.1%

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

       1. *-un-lft-identity 80.1%

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

       2. associate-*r/ 84.0%

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

       3. sub-div 85.5%

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

       4. clear-num 85.5%

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

       5. un-div-inv 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. *-lft-identity 85.5%

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

       2. associate-/r/ 86.2%

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

   13. Simplified 86.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-21} \lor \neg \left(y.im \leq 2.4 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{-21} \lor \neg \left(y.im \leq 2.35 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.4e-21) (not (<= y.im 2.35e+67)))
   (/ x.re (- y.im))
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e-21) || !(y_46_im <= 2.35e+67)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.4d-21)) .or. (.not. (y_46im <= 2.35d+67))) then
        tmp = x_46re / -y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e-21) || !(y_46_im <= 2.35e+67)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.4e-21) or not (y_46_im <= 2.35e+67):
		tmp = x_46_re / -y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.4e-21) || !(y_46_im <= 2.35e+67))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.4e-21) || ~((y_46_im <= 2.35e+67)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.4e-21], N[Not[LessEqual[y$46$im, 2.35e+67]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.40000000000000002e-21 or 2.35000000000000009e67 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if -1.40000000000000002e-21 < y.im < 2.35000000000000009e67

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf 67.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{-21} \lor \neg \left(y.im \leq 2.35 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.7%

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

3. Taylor expanded in y.re around inf 44.4%

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

4. Final simplification 44.4%

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

Reproduce

herbie shell --seed 2024039 
(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))))