_divideComplex, imaginary part

Percentage Accurate: 61.1% → 97.3%

Time: 12.7s

Alternatives: 12

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: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.0× speedup

Math

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

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))
  (* (/ y.im (hypot y.im y.re)) (/ (- x.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)), ((y_46_im / hypot(y_46_im, y_46_re)) * (-x_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(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(Float64(-x_46_re) / 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[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-x$46$re) / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

   1. div-sub 61.9%

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

   2. *-commutative 61.9%

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

   3. fma-define 61.9%

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

   4. add-sqr-sqrt 61.9%

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

   5. times-frac 63.5%

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

   6. fma-neg 63.5%

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

   7. fma-define 63.5%

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

   8. hypot-define 63.5%

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

   9. fma-define 63.5%

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

   10. hypot-define 78.0%

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

4. Applied egg-rr 78.0%

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

   1. *-commutative 78.0%

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

   2. add-sqr-sqrt 78.0%

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

   3. hypot-undefine 78.0%

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

   4. hypot-undefine 78.0%

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

   5. times-frac 96.6%

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

   6. hypot-undefine 80.1%

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

   7. +-commutative 80.1%

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

   8. hypot-define 96.6%

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

   9. hypot-undefine 80.1%

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

   10. +-commutative 80.1%

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

   11. hypot-define 96.6%

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

6. Applied egg-rr 96.6%

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

7. Final simplification 96.6%

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

Alternative 2: 84.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (* (/ y.im (hypot y.re y.im)) (/ (- 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);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (-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);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (y_46_im / Math.hypot(y_46_re, y_46_im)) * (-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)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (y_46_im / math.hypot(y_46_re, y_46_im)) * (-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(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))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(Float64(-x_46_re) / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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))) <= Inf)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (-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[(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], Infinity], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-x$46$re) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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))) < +inf.0

   1. Initial program 80.1%

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

      1. fma-define 80.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. associate-/r* 80.3%

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

      4. fma-define 80.3%

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

      5. hypot-define 80.3%

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

      6. fma-define 80.3%

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

      7. hypot-define 95.7%

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

   4. Applied egg-rr 95.7%

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

   if +inf.0 < (/.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 0.0%

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

      1. clear-num 0.0%

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

      2. associate-/r/ 0.0%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in x.im around 0 1.5%

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

      1. mul-1-neg 1.5%

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

      2. distribute-rgt-neg-in 1.5%

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

   7. Simplified 1.5%

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

      1. associate-*l/ 1.5%

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

      2. *-un-lft-identity 1.5%

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

      3. distribute-rgt-neg-out 1.5%

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

      4. remove-double-neg 1.5%

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

      5. distribute-rgt-neg-out 1.5%

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

      6. distribute-frac-neg 1.5%

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

      7. distribute-rgt-neg-out 1.5%

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

      8. remove-double-neg 1.5%

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

      9. +-commutative 1.5%

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

      10. add-sqr-sqrt 1.5%

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

      11. hypot-undefine 1.5%

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

      12. hypot-undefine 1.5%

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

      13. frac-times 56.4%

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

      14. *-commutative 56.4%

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

      15. distribute-rgt-neg-in 56.4%

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

      16. hypot-undefine 3.7%

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

      17. +-commutative 3.7%

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

      18. hypot-undefine 56.4%

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

   9. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(0 - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

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

Alternative 3: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -54000000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.im - y.im \cdot x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -54000000000.0)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 4.3e-116)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (if (<= y.im 3.1e+114)
       (/
        1.0
        (/ (+ (* y.re y.re) (* y.im y.im)) (- (* y.re x.im) (* y.im x.re))))
       (- (* (/ x.im y.im) (/ y.re y.im)) (* (/ x.re y.im) (/ y.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -54000000000.0) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 4.3e-116) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.1e+114) {
		tmp = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_im) - (y_46_im * x_46_re)));
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-54000000000.0d0)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 4.3d-116) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 3.1d+114) then
        tmp = 1.0d0 / (((y_46re * y_46re) + (y_46im * y_46im)) / ((y_46re * x_46im) - (y_46im * x_46re)))
    else
        tmp = ((x_46im / y_46im) * (y_46re / y_46im)) - ((x_46re / y_46im) * (y_46im / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -54000000000.0) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 4.3e-116) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.1e+114) {
		tmp = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_im) - (y_46_im * x_46_re)));
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -54000000000.0:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 4.3e-116:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 3.1e+114:
		tmp = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_im) - (y_46_im * x_46_re)))
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -54000000000.0)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 4.3e-116)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.1e+114)
		tmp = Float64(1.0 / Float64(Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)) / Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))));
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im)) - Float64(Float64(x_46_re / y_46_im) * Float64(y_46_im / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -54000000000.0)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 4.3e-116)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 3.1e+114)
		tmp = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_im) - (y_46_im * x_46_re)));
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -54000000000.0], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 4.3e-116], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+114], N[(1.0 / N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.4e10

   1. Initial program 53.6%

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

   3. Taylor expanded in y.re around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. mul-1-neg 70.5%

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

      3. unsub-neg 70.5%

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

      4. unpow2 70.5%

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

      5. associate-/r* 73.6%

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

      6. div-sub 73.6%

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

      7. *-commutative 73.6%

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

      8. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -5.4e10 < y.im < 4.2999999999999997e-116

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. associate-/l* 90.6%

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

   5. Simplified 90.6%

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

      1. associate-*r/ 90.6%

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

      2. add-sqr-sqrt 35.8%

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

      3. sqrt-prod 71.6%

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

      4. sqr-neg 71.6%

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

      5. sqrt-unprod 40.3%

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

      6. add-sqr-sqrt 71.0%

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

      7. *-commutative 71.0%

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

      8. add-sqr-sqrt 40.3%

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

      9. sqrt-unprod 71.6%

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

      10. sqr-neg 71.6%

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

      11. sqrt-prod 35.8%

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

      12. add-sqr-sqrt 90.6%

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

   7. Applied egg-rr 90.6%

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

   if 4.2999999999999997e-116 < y.im < 3.1e114

   1. Initial program 83.2%

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

      1. clear-num 83.3%

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

   4. Applied egg-rr 83.3%

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

   if 3.1e114 < y.im

   1. Initial program 37.7%

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

   3. Taylor expanded in y.re around 0 37.7%

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

      1. unpow2 37.7%

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

   5. Simplified 37.7%

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

      1. div-sub 37.7%

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

      2. times-frac 45.0%

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

      3. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -54000000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.im - y.im \cdot x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -14600000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.18 \cdot 10^{+114}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -14600000000.0)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 1.7e-125)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (if (<= y.im 1.18e+114)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (- (* (/ x.im y.im) (/ y.re y.im)) (* (/ x.re y.im) (/ y.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -14600000000.0) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.7e-125) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.18e+114) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-14600000000.0d0)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 1.7d-125) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 1.18d+114) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = ((x_46im / y_46im) * (y_46re / y_46im)) - ((x_46re / y_46im) * (y_46im / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -14600000000.0) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.7e-125) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.18e+114) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -14600000000.0:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 1.7e-125:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 1.18e+114:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -14600000000.0)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 1.7e-125)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.18e+114)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im)) - Float64(Float64(x_46_re / y_46_im) * Float64(y_46_im / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -14600000000.0)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 1.7e-125)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.18e+114)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - ((x_46_re / y_46_im) * (y_46_im / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -14600000000.0], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.7e-125], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.18e+114], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.46e10

   1. Initial program 53.6%

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

   3. Taylor expanded in y.re around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. mul-1-neg 70.5%

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

      3. unsub-neg 70.5%

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

      4. unpow2 70.5%

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

      5. associate-/r* 73.6%

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

      6. div-sub 73.6%

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

      7. *-commutative 73.6%

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

      8. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -1.46e10 < y.im < 1.69999999999999988e-125

   1. Initial program 72.9%

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

   3. Taylor expanded in y.re around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. associate-/l* 90.5%

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

   5. Simplified 90.5%

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

      1. associate-*r/ 90.6%

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

      2. add-sqr-sqrt 35.1%

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

      3. sqrt-prod 71.3%

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

      4. sqr-neg 71.3%

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

      5. sqrt-unprod 40.7%

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

      6. add-sqr-sqrt 70.7%

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

      7. *-commutative 70.7%

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

      8. add-sqr-sqrt 40.7%

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

      9. sqrt-unprod 71.3%

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

      10. sqr-neg 71.3%

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

      11. sqrt-prod 35.1%

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

      12. add-sqr-sqrt 90.6%

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

   7. Applied egg-rr 90.6%

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

   if 1.69999999999999988e-125 < y.im < 1.18000000000000005e114

   1. Initial program 83.5%

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

   if 1.18000000000000005e114 < y.im

   1. Initial program 37.7%

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

   3. Taylor expanded in y.re around 0 37.7%

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

      1. unpow2 37.7%

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

   5. Simplified 37.7%

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

      1. div-sub 37.7%

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

      2. times-frac 45.0%

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

      3. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -14600000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.18 \cdot 10^{+114}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im} \cdot \frac{y.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-28} \lor \neg \left(y.im \leq 2.9 \cdot 10^{-41}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -1.6e+89)
     t_0
     (if (<= y.im -8.5e+18)
       (/ (/ (* y.re x.im) y.im) y.im)
       (if (or (<= y.im -2.2e-28) (not (<= y.im 2.9e-41)))
         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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -1.6e+89) {
		tmp = t_0;
	} else if (y_46_im <= -8.5e+18) {
		tmp = ((y_46_re * x_46_im) / y_46_im) / y_46_im;
	} else if ((y_46_im <= -2.2e-28) || !(y_46_im <= 2.9e-41)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-1.6d+89)) then
        tmp = t_0
    else if (y_46im <= (-8.5d+18)) then
        tmp = ((y_46re * x_46im) / y_46im) / y_46im
    else if ((y_46im <= (-2.2d-28)) .or. (.not. (y_46im <= 2.9d-41))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -1.6e+89) {
		tmp = t_0;
	} else if (y_46_im <= -8.5e+18) {
		tmp = ((y_46_re * x_46_im) / y_46_im) / y_46_im;
	} else if ((y_46_im <= -2.2e-28) || !(y_46_im <= 2.9e-41)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -1.6e+89:
		tmp = t_0
	elif y_46_im <= -8.5e+18:
		tmp = ((y_46_re * x_46_im) / y_46_im) / y_46_im
	elif (y_46_im <= -2.2e-28) or not (y_46_im <= 2.9e-41):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.6e+89)
		tmp = t_0;
	elseif (y_46_im <= -8.5e+18)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) / y_46_im);
	elseif ((y_46_im <= -2.2e-28) || !(y_46_im <= 2.9e-41))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.6e+89)
		tmp = t_0;
	elseif (y_46_im <= -8.5e+18)
		tmp = ((y_46_re * x_46_im) / y_46_im) / y_46_im;
	elseif ((y_46_im <= -2.2e-28) || ~((y_46_im <= 2.9e-41)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+89], t$95$0, If[LessEqual[y$46$im, -8.5e+18], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$im, -2.2e-28], N[Not[LessEqual[y$46$im, 2.9e-41]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.59999999999999994e89 or -8.5e18 < y.im < -2.19999999999999996e-28 or 2.89999999999999977e-41 < y.im

   1. Initial program 53.8%

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

   3. Taylor expanded in y.re around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. distribute-neg-frac2 63.3%

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

   5. Simplified 63.3%

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

   if -1.59999999999999994e89 < y.im < -8.5e18

   1. Initial program 80.0%

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

   3. Taylor expanded in y.re around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

      4. unpow2 68.2%

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

      5. associate-/r* 68.4%

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

      6. div-sub 68.4%

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

      7. *-commutative 68.4%

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

      8. associate-/l* 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in y.re around inf 54.9%

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

   if -2.19999999999999996e-28 < y.im < 2.89999999999999977e-41

   1. Initial program 74.0%

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

   3. Taylor expanded in y.re around inf 68.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-28} \lor \neg \left(y.im \leq 2.9 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -2.75 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-24} \lor \neg \left(y.im \leq 5.8 \cdot 10^{-43}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -2.75e+53)
     t_0
     (if (<= y.im -6.5e+22)
       (/ (* x.im (/ y.re y.im)) y.im)
       (if (or (<= y.im -1.15e-24) (not (<= y.im 5.8e-43)))
         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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -2.75e+53) {
		tmp = t_0;
	} else if (y_46_im <= -6.5e+22) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if ((y_46_im <= -1.15e-24) || !(y_46_im <= 5.8e-43)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-2.75d+53)) then
        tmp = t_0
    else if (y_46im <= (-6.5d+22)) then
        tmp = (x_46im * (y_46re / y_46im)) / y_46im
    else if ((y_46im <= (-1.15d-24)) .or. (.not. (y_46im <= 5.8d-43))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -2.75e+53) {
		tmp = t_0;
	} else if (y_46_im <= -6.5e+22) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if ((y_46_im <= -1.15e-24) || !(y_46_im <= 5.8e-43)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -2.75e+53:
		tmp = t_0
	elif y_46_im <= -6.5e+22:
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im
	elif (y_46_im <= -1.15e-24) or not (y_46_im <= 5.8e-43):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.75e+53)
		tmp = t_0;
	elseif (y_46_im <= -6.5e+22)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im);
	elseif ((y_46_im <= -1.15e-24) || !(y_46_im <= 5.8e-43))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.75e+53)
		tmp = t_0;
	elseif (y_46_im <= -6.5e+22)
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	elseif ((y_46_im <= -1.15e-24) || ~((y_46_im <= 5.8e-43)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -2.75e+53], t$95$0, If[LessEqual[y$46$im, -6.5e+22], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$im, -1.15e-24], N[Not[LessEqual[y$46$im, 5.8e-43]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.74999999999999988e53 or -6.49999999999999979e22 < y.im < -1.1500000000000001e-24 or 5.8000000000000003e-43 < y.im

   1. Initial program 54.1%

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

   3. Taylor expanded in y.re around 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. distribute-neg-frac2 61.4%

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

   5. Simplified 61.4%

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

   if -2.74999999999999988e53 < y.im < -6.49999999999999979e22

   1. Initial program 99.4%

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

   3. Taylor expanded in y.re around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. unpow2 88.0%

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

      5. associate-/r* 88.4%

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

      6. div-sub 88.4%

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

      7. *-commutative 88.4%

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

      8. associate-/l* 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in y.re around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in y.re around inf 77.1%

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

       1. associate-/l* 71.1%

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

   11. Simplified 71.1%

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

   if -1.1500000000000001e-24 < y.im < 5.8000000000000003e-43

   1. Initial program 74.0%

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

   3. Taylor expanded in y.re around inf 68.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.75 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-24} \lor \neg \left(y.im \leq 5.8 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-25} \lor \neg \left(y.im \leq 3.2 \cdot 10^{-41}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -1.75e+55)
     t_0
     (if (<= y.im -2.4e+18)
       (* (/ x.im y.im) (/ y.re y.im))
       (if (or (<= y.im -7.4e-25) (not (<= y.im 3.2e-41)))
         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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -1.75e+55) {
		tmp = t_0;
	} else if (y_46_im <= -2.4e+18) {
		tmp = (x_46_im / y_46_im) * (y_46_re / y_46_im);
	} else if ((y_46_im <= -7.4e-25) || !(y_46_im <= 3.2e-41)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-1.75d+55)) then
        tmp = t_0
    else if (y_46im <= (-2.4d+18)) then
        tmp = (x_46im / y_46im) * (y_46re / y_46im)
    else if ((y_46im <= (-7.4d-25)) .or. (.not. (y_46im <= 3.2d-41))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -1.75e+55) {
		tmp = t_0;
	} else if (y_46_im <= -2.4e+18) {
		tmp = (x_46_im / y_46_im) * (y_46_re / y_46_im);
	} else if ((y_46_im <= -7.4e-25) || !(y_46_im <= 3.2e-41)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -1.75e+55:
		tmp = t_0
	elif y_46_im <= -2.4e+18:
		tmp = (x_46_im / y_46_im) * (y_46_re / y_46_im)
	elif (y_46_im <= -7.4e-25) or not (y_46_im <= 3.2e-41):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.75e+55)
		tmp = t_0;
	elseif (y_46_im <= -2.4e+18)
		tmp = Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im));
	elseif ((y_46_im <= -7.4e-25) || !(y_46_im <= 3.2e-41))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.75e+55)
		tmp = t_0;
	elseif (y_46_im <= -2.4e+18)
		tmp = (x_46_im / y_46_im) * (y_46_re / y_46_im);
	elseif ((y_46_im <= -7.4e-25) || ~((y_46_im <= 3.2e-41)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -1.75e+55], t$95$0, If[LessEqual[y$46$im, -2.4e+18], N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$im, -7.4e-25], N[Not[LessEqual[y$46$im, 3.2e-41]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.75000000000000005e55 or -2.4e18 < y.im < -7.40000000000000017e-25 or 3.20000000000000012e-41 < y.im

   1. Initial program 54.1%

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

   3. Taylor expanded in y.re around 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. distribute-neg-frac2 61.4%

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

   5. Simplified 61.4%

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

   if -1.75000000000000005e55 < y.im < -2.4e18

   1. Initial program 99.4%

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

   3. Taylor expanded in y.re around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. unpow2 88.0%

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

      5. associate-/r* 88.4%

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

      6. div-sub 88.4%

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

      7. *-commutative 88.4%

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

      8. associate-/l* 88.2%

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

   5. Simplified 88.2%

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

   6. Taylor expanded in y.re around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

   8. Simplified 88.2%

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

   9. Taylor expanded in y.re around inf 76.7%

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

       1. unpow2 76.7%

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

       2. times-frac 70.5%

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

   11. Simplified 70.5%

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

   if -7.40000000000000017e-25 < y.im < 3.20000000000000012e-41

   1. Initial program 74.0%

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

   3. Taylor expanded in y.re around inf 68.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-25} \lor \neg \left(y.im \leq 3.2 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -41000000000 \lor \neg \left(y.im \leq 5.5 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -41000000000.0) (not (<= y.im 5.5e+42)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ (* y.im x.re) y.re)) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -41000000000.0) || !(y_46_im <= 5.5e+42)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-41000000000.0d0)) .or. (.not. (y_46im <= 5.5d+42))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -41000000000.0) || !(y_46_im <= 5.5e+42)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -41000000000.0) or not (y_46_im <= 5.5e+42):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -41000000000.0) || !(y_46_im <= 5.5e+42))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -41000000000.0) || ~((y_46_im <= 5.5e+42)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -41000000000.0], N[Not[LessEqual[y$46$im, 5.5e+42]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1e10 or 5.50000000000000001e42 < y.im

   1. Initial program 52.7%

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

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

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

      1. +-commutative 72.6%

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

      2. mul-1-neg 72.6%

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

      3. unsub-neg 72.6%

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

      4. unpow2 72.6%

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

      5. associate-/r* 75.1%

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

      6. div-sub 75.1%

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

      7. *-commutative 75.1%

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

      8. associate-/l* 80.9%

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

   5. Simplified 80.9%

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

   if -4.1e10 < y.im < 5.50000000000000001e42

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. associate-/l* 83.3%

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

   5. Simplified 83.3%

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

      1. associate-*r/ 83.4%

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

      2. add-sqr-sqrt 42.9%

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

      3. sqrt-prod 69.3%

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

      4. sqr-neg 69.3%

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

      5. sqrt-unprod 29.7%

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

      6. add-sqr-sqrt 64.9%

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

      7. *-commutative 64.9%

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

      8. add-sqr-sqrt 29.7%

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

      9. sqrt-unprod 69.3%

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

      10. sqr-neg 69.3%

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

      11. sqrt-prod 42.9%

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

      12. add-sqr-sqrt 83.4%

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

   7. Applied egg-rr 83.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -41000000000 \lor \neg \left(y.im \leq 5.5 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+112} \lor \neg \left(y.im \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.4e+112) (not (<= y.im 7e+41)))
   (/ 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 <= -5.4e+112) || !(y_46_im <= 7e+41)) {
		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 <= (-5.4d+112)) .or. (.not. (y_46im <= 7d+41))) 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 <= -5.4e+112) || !(y_46_im <= 7e+41)) {
		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 <= -5.4e+112) or not (y_46_im <= 7e+41):
		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 <= -5.4e+112) || !(y_46_im <= 7e+41))
		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 <= -5.4e+112) || ~((y_46_im <= 7e+41)))
		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, -5.4e+112], N[Not[LessEqual[y$46$im, 7e+41]], $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 < -5.4000000000000002e112 or 6.9999999999999998e41 < y.im

   1. Initial program 47.6%

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

   3. Taylor expanded in y.re around 0 66.4%

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

      1. mul-1-neg 66.4%

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

      2. distribute-neg-frac2 66.4%

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

   5. Simplified 66.4%

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

   if -5.4000000000000002e112 < y.im < 6.9999999999999998e41

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

      3. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+112} \lor \neg \left(y.im \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+94} \lor \neg \left(y.im \leq 1.15 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.5e+94) (not (<= y.im 1.15e-41)))
   (/ 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 <= -4.5e+94) || !(y_46_im <= 1.15e-41)) {
		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 <= (-4.5d+94)) .or. (.not. (y_46im <= 1.15d-41))) 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 <= -4.5e+94) || !(y_46_im <= 1.15e-41)) {
		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 <= -4.5e+94) or not (y_46_im <= 1.15e-41):
		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 <= -4.5e+94) || !(y_46_im <= 1.15e-41))
		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 <= -4.5e+94) || ~((y_46_im <= 1.15e-41)))
		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, -4.5e+94], N[Not[LessEqual[y$46$im, 1.15e-41]], $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 < -4.49999999999999972e94 or 1.15000000000000005e-41 < y.im

   1. Initial program 52.0%

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

   3. Taylor expanded in y.re around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. distribute-neg-frac2 64.0%

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

   5. Simplified 64.0%

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

   if -4.49999999999999972e94 < y.im < 1.15000000000000005e-41

   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

   3. Taylor expanded in y.re around inf 62.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+94} \lor \neg \left(y.im \leq 1.15 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+212} \lor \neg \left(y.im \leq 1.32 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.5e+212) (not (<= y.im 1.32e+88)))
   (/ 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 <= -2.5e+212) || !(y_46_im <= 1.32e+88)) {
		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 <= (-2.5d+212)) .or. (.not. (y_46im <= 1.32d+88))) 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 <= -2.5e+212) || !(y_46_im <= 1.32e+88)) {
		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 <= -2.5e+212) or not (y_46_im <= 1.32e+88):
		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 <= -2.5e+212) || !(y_46_im <= 1.32e+88))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.5e+212) || ~((y_46_im <= 1.32e+88)))
		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, -2.5e+212], N[Not[LessEqual[y$46$im, 1.32e+88]], $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 < -2.49999999999999996e212 or 1.3200000000000001e88 < y.im

   1. Initial program 46.0%

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

   3. Taylor expanded in y.re around 0 44.5%

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

      1. unpow2 44.5%

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

   5. Simplified 44.5%

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

   6. Taylor expanded in x.im around 0 39.0%

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

      1. mul-1-neg 40.4%

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

      2. distribute-rgt-neg-in 40.4%

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

   8. Simplified 39.0%

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

      1. associate-/l* 41.7%

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

      2. clear-num 41.7%

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

      3. add-sqr-sqrt 29.0%

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

      4. sqrt-prod 41.7%

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

      5. sqr-neg 41.7%

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

      6. sqrt-unprod 12.6%

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

      7. add-sqr-sqrt 37.2%

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

      8. distribute-lft-neg-in 37.2%

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

      9. neg-sub0 37.2%

         \[\leadsto x.re \cdot \frac{1}{\frac{\color{blue}{0 - y.im \cdot y.im}}{-y.im}} \]

      10. metadata-eval 37.2%

          \[\leadsto x.re \cdot \frac{1}{\frac{\color{blue}{0 \cdot 0} - y.im \cdot y.im}{-y.im}} \]

      11. neg-sub0 37.2%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{\color{blue}{0 - y.im}}} \]

      12. sub-neg 37.2%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{\color{blue}{0 + \left(-y.im\right)}}} \]

      13. add-sqr-sqrt 12.6%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{0 + \color{blue}{\sqrt{-y.im} \cdot \sqrt{-y.im}}}} \]

      14. sqrt-unprod 10.3%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{0 + \color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}}}} \]

      15. sqr-neg 10.3%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{0 + \sqrt{\color{blue}{y.im \cdot y.im}}}} \]

      16. sqrt-prod 29.0%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{0 + \color{blue}{\sqrt{y.im} \cdot \sqrt{y.im}}}} \]

      17. add-sqr-sqrt 41.7%

          \[\leadsto x.re \cdot \frac{1}{\frac{0 \cdot 0 - y.im \cdot y.im}{0 + \color{blue}{y.im}}} \]

      18. flip-- 71.7%

          \[\leadsto x.re \cdot \frac{1}{\color{blue}{0 - y.im}} \]

      19. neg-sub0 71.7%

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

      20. add-sqr-sqrt 20.5%

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

      21. sqrt-unprod 37.2%

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

      22. sqr-neg 37.2%

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

      23. sqrt-prod 24.2%

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

      24. add-sqr-sqrt 36.6%

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

   10. Applied egg-rr 36.6%

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

   if -2.49999999999999996e212 < y.im < 1.3200000000000001e88

   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 inf 53.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+212} \lor \neg \left(y.im \leq 1.32 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.6% 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 64.8%

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

3. Taylor expanded in y.re around inf 44.3%

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

Reproduce

herbie shell --seed 2024097 
(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))))