_divideComplex, imaginary part

Percentage Accurate: 61.2% → 85.5%

Time: 11.6s

Alternatives: 13

Speedup: 1.8×

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: 61.2% 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: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+239}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 1e+239)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (-
      (* t_0 (/ y.re (/ (hypot y.re y.im) x.im)))
      (* (/ x.re y.re) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+239) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+239) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (t_0 * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+239:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (t_0 * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+239)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(t_0 * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+239)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+239], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.99999999999999991e238

   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. Step-by-step derivation

      1. *-un-lft-identity 80.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. times-frac 80.1%

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

      4. hypot-def 80.1%

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

      5. hypot-def 96.2%

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

   3. Applied egg-rr 96.2%

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

   if 9.99999999999999991e238 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.6%

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

      1. div-sub 11.6%

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

      2. *-un-lft-identity 11.6%

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

      3. add-sqr-sqrt 11.6%

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

      4. times-frac 11.6%

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

      5. fma-neg 11.6%

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

      6. hypot-def 11.6%

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

      7. hypot-def 12.5%

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

      8. associate-/l* 19.7%

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

      9. add-sqr-sqrt 19.7%

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

      10. pow2 19.7%

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

      11. hypot-def 19.7%

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

   3. Applied egg-rr 19.7%

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

      1. fma-neg 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-/l* 59.9%

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

      4. associate-/r/ 58.3%

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

      5. *-commutative 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in y.im around 0 48.7%

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

      1. unpow2 48.7%

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

      2. times-frac 62.3%

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

   8. Simplified 62.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+239}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 2: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+262}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+262)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (* (/ -1.0 y.im) (- x.re (* x.im (/ y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+262) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (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 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+262) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (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 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+262:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+262)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im * Float64(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 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+262)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $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], 1e+262], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $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))) < 1e262

   1. Initial program 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. add-sqr-sqrt 80.3%

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

      3. times-frac 80.3%

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

      4. hypot-def 80.3%

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

      5. hypot-def 96.2%

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

   3. Applied egg-rr 96.2%

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

   if 1e262 < (/.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 10.8%

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

      1. *-un-lft-identity 10.8%

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

      2. add-sqr-sqrt 10.8%

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

      3. times-frac 10.8%

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

      4. hypot-def 10.8%

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

      5. hypot-def 20.0%

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

   3. Applied egg-rr 20.0%

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

   4. Taylor expanded in y.im around -inf 20.4%

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

      1. +-commutative 20.4%

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

      2. mul-1-neg 20.4%

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

      3. unsub-neg 20.4%

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

      4. associate-/l* 23.8%

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

      5. associate-/r/ 25.3%

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

   6. Simplified 25.3%

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

   7. Taylor expanded in y.im around -inf 57.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+262}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 3: 83.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.im \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{t_0} - \frac{x.re \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(t_1 - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))) (t_1 (* x.im (/ y.re y.im))))
   (if (<= y.im -9e+91)
     (/ (- x.re t_1) (hypot y.re y.im))
     (if (<= y.im -6e-124)
       (/ (- (* x.im y.re) (* x.re y.im)) t_0)
       (if (<= y.im 4.4e-128)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 3.2e+109)
           (- (/ (* x.im y.re) t_0) (/ (* x.re y.im) t_0))
           (* (/ 1.0 (hypot y.re y.im)) (- t_1 x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = x_46_im * (y_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -9e+91) {
		tmp = (x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -6e-124) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 4.4e-128) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.2e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_1 - x_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = x_46_im * (y_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -9e+91) {
		tmp = (x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -6e-124) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 4.4e-128) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.2e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_1 - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = x_46_im * (y_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -9e+91:
		tmp = (x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -6e-124:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0
	elif y_46_im <= 4.4e-128:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.2e+109:
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0)
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_1 - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(x_46_im * Float64(y_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -9e+91)
		tmp = Float64(Float64(x_46_re - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -6e-124)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / t_0);
	elseif (y_46_im <= 4.4e-128)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.2e+109)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) / t_0) - Float64(Float64(x_46_re * y_46_im) / t_0));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_1 - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = x_46_im * (y_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -9e+91)
		tmp = (x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -6e-124)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	elseif (y_46_im <= 4.4e-128)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.2e+109)
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_1 - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9e+91], N[(N[(x$46$re - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6e-124], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 4.4e-128], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.2e+109], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(x$46$re * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9e91

   1. Initial program 36.8%

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

      1. *-un-lft-identity 36.8%

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

      2. add-sqr-sqrt 36.8%

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

      3. times-frac 36.9%

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

      4. hypot-def 36.9%

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

      5. hypot-def 60.9%

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

   3. Applied egg-rr 60.9%

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

   4. Taylor expanded in y.im around -inf 64.5%

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

      1. +-commutative 64.5%

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

      2. mul-1-neg 64.5%

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

      3. unsub-neg 64.5%

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

      4. associate-/l* 73.5%

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

      5. associate-/r/ 71.0%

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

   6. Simplified 71.0%

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

      1. expm1-log1p-u 70.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\right)\right)} \]

      2. expm1-udef 29.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\right)} - 1} \]

      3. associate-*l/ 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re - \frac{y.re}{y.im} \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      5. *-commutative 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.re - \color{blue}{x.im \cdot \frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   8. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 70.3%

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

      2. expm1-log1p 71.2%

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

   10. Simplified 71.2%

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

   if -9e91 < y.im < -6e-124

   1. Initial program 91.9%

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

   if -6e-124 < y.im < 4.40000000000000019e-128

   1. Initial program 62.4%

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

   2. Taylor expanded in y.re around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

      4. times-frac 87.3%

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

   4. Simplified 87.3%

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

      1. associate-*l/ 88.4%

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

      2. sub-div 89.9%

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

   6. Applied egg-rr 89.9%

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

   if 4.40000000000000019e-128 < y.im < 3.2000000000000001e109

   1. Initial program 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. add-sqr-sqrt 80.5%

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

      3. times-frac 80.6%

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

      4. hypot-def 80.6%

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

      5. hypot-def 89.0%

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

   3. Applied egg-rr 89.0%

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

   4. Taylor expanded in x.im around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. distribute-frac-neg 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. unpow2 80.5%

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

      6. unpow2 80.5%

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

      7. unpow2 80.5%

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

      8. unpow2 80.5%

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

   6. Simplified 80.5%

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

   if 3.2000000000000001e109 < y.im

   1. Initial program 36.5%

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

      1. *-un-lft-identity 36.5%

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

      2. add-sqr-sqrt 36.5%

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

      3. times-frac 36.5%

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

      4. hypot-def 36.5%

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

      5. hypot-def 55.7%

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

   3. Applied egg-rr 55.7%

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

   4. Taylor expanded in y.re around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. unsub-neg 83.5%

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

      3. associate-/l* 89.9%

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

      4. associate-/r/ 89.9%

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

   6. Simplified 89.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{y.re}{y.im} \cdot x.im - x.re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \end{array} \]

Alternative 4: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{t_0} - \frac{x.re \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -3.3e+92)
     (/ (- x.re (* x.im (/ y.re y.im))) (hypot y.re y.im))
     (if (<= y.im -3.6e-123)
       (/ (- (* x.im y.re) (* x.re y.im)) t_0)
       (if (<= y.im 3.5e-129)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 6.6e+109)
           (- (/ (* x.im y.re) t_0) (/ (* x.re y.im) t_0))
           (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -3.3e+92) {
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -3.6e-123) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 3.5e-129) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 6.6e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -3.3e+92) {
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -3.6e-123) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 3.5e-129) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 6.6e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -3.3e+92:
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -3.6e-123:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0
	elif y_46_im <= 3.5e-129:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 6.6e+109:
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0)
	else:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.3e+92)
		tmp = Float64(Float64(x_46_re - Float64(x_46_im * Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -3.6e-123)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / t_0);
	elseif (y_46_im <= 3.5e-129)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 6.6e+109)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) / t_0) - Float64(Float64(x_46_re * y_46_im) / t_0));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -3.3e+92)
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -3.6e-123)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	elseif (y_46_im <= 3.5e-129)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 6.6e+109)
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	else
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.3e+92], N[(N[(x$46$re - N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.6e-123], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-129], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.6e+109], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(x$46$re * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.29999999999999974e92

   1. Initial program 36.8%

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

      1. *-un-lft-identity 36.8%

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

      2. add-sqr-sqrt 36.8%

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

      3. times-frac 36.9%

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

      4. hypot-def 36.9%

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

      5. hypot-def 60.9%

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

   3. Applied egg-rr 60.9%

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

   4. Taylor expanded in y.im around -inf 64.5%

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

      1. +-commutative 64.5%

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

      2. mul-1-neg 64.5%

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

      3. unsub-neg 64.5%

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

      4. associate-/l* 73.5%

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

      5. associate-/r/ 71.0%

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

   6. Simplified 71.0%

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

      1. expm1-log1p-u 70.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\right)\right)} \]

      2. expm1-udef 29.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\right)} - 1} \]

      3. associate-*l/ 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re - \frac{y.re}{y.im} \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      5. *-commutative 29.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.re - \color{blue}{x.im \cdot \frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   8. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 70.3%

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

      2. expm1-log1p 71.2%

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

   10. Simplified 71.2%

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

   if -3.29999999999999974e92 < y.im < -3.5999999999999997e-123

   1. Initial program 91.9%

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

   if -3.5999999999999997e-123 < y.im < 3.4999999999999997e-129

   1. Initial program 62.4%

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

   2. Taylor expanded in y.re around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

      4. times-frac 87.3%

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

   4. Simplified 87.3%

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

      1. associate-*l/ 88.4%

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

      2. sub-div 89.9%

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

   6. Applied egg-rr 89.9%

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

   if 3.4999999999999997e-129 < y.im < 6.5999999999999998e109

   1. Initial program 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. add-sqr-sqrt 80.5%

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

      3. times-frac 80.6%

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

      4. hypot-def 80.6%

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

      5. hypot-def 89.0%

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

   3. Applied egg-rr 89.0%

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

   4. Taylor expanded in x.im around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. distribute-frac-neg 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. unpow2 80.5%

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

      6. unpow2 80.5%

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

      7. unpow2 80.5%

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

      8. unpow2 80.5%

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

   6. Simplified 80.5%

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

   if 6.5999999999999998e109 < y.im

   1. Initial program 36.5%

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

   2. Taylor expanded in y.re around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. mul-1-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. unpow2 80.2%

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

      5. times-frac 89.6%

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

   4. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 1.58 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{t_0} - \frac{x.re \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -1e+73)
     t_1
     (if (<= y.im -8.5e-125)
       (/ (- (* x.im y.re) (* x.re y.im)) t_0)
       (if (<= y.im 1.58e-127)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 7e+109)
           (- (/ (* x.im y.re) t_0) (/ (* x.re y.im) t_0))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1e+73) {
		tmp = t_1;
	} else if (y_46_im <= -8.5e-125) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 1.58e-127) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-1d+73)) then
        tmp = t_1
    else if (y_46im <= (-8.5d-125)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / t_0
    else if (y_46im <= 1.58d-127) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 7d+109) then
        tmp = ((x_46im * y_46re) / t_0) - ((x_46re * y_46im) / t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1e+73) {
		tmp = t_1;
	} else if (y_46_im <= -8.5e-125) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	} else if (y_46_im <= 1.58e-127) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7e+109) {
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1e+73:
		tmp = t_1
	elif y_46_im <= -8.5e-125:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0
	elif y_46_im <= 1.58e-127:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 7e+109:
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1e+73)
		tmp = t_1;
	elseif (y_46_im <= -8.5e-125)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / t_0);
	elseif (y_46_im <= 1.58e-127)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 7e+109)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) / t_0) - Float64(Float64(x_46_re * y_46_im) / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1e+73)
		tmp = t_1;
	elseif (y_46_im <= -8.5e-125)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / t_0;
	elseif (y_46_im <= 1.58e-127)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 7e+109)
		tmp = ((x_46_im * y_46_re) / t_0) - ((x_46_re * y_46_im) / t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1e+73], t$95$1, If[LessEqual[y$46$im, -8.5e-125], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.58e-127], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7e+109], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(x$46$re * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.99999999999999983e72 or 6.99999999999999966e109 < y.im

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 80.0%

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

   4. Simplified 80.0%

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

   if -9.99999999999999983e72 < y.im < -8.5000000000000002e-125

   1. Initial program 93.4%

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

   if -8.5000000000000002e-125 < y.im < 1.58e-127

   1. Initial program 62.4%

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

   2. Taylor expanded in y.re around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

      4. times-frac 87.3%

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

   4. Simplified 87.3%

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

      1. associate-*l/ 88.4%

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

      2. sub-div 89.9%

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

   6. Applied egg-rr 89.9%

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

   if 1.58e-127 < y.im < 6.99999999999999966e109

   1. Initial program 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. add-sqr-sqrt 80.5%

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

      3. times-frac 80.6%

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

      4. hypot-def 80.6%

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

      5. hypot-def 89.0%

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

   3. Applied egg-rr 89.0%

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

   4. Taylor expanded in x.im around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. distribute-frac-neg 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. unpow2 80.5%

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

      6. unpow2 80.5%

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

      7. unpow2 80.5%

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

      8. unpow2 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.58 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -2.25e+73)
     t_1
     (if (<= y.im -2.4e-123)
       t_0
       (if (<= y.im 2.7e-128)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 7.2e+109) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.25e+73) {
		tmp = t_1;
	} else if (y_46_im <= -2.4e-123) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e-128) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.2e+109) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-2.25d+73)) then
        tmp = t_1
    else if (y_46im <= (-2.4d-123)) then
        tmp = t_0
    else if (y_46im <= 2.7d-128) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 7.2d+109) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.25e+73) {
		tmp = t_1;
	} else if (y_46_im <= -2.4e-123) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e-128) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.2e+109) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.25e+73:
		tmp = t_1
	elif y_46_im <= -2.4e-123:
		tmp = t_0
	elif y_46_im <= 2.7e-128:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 7.2e+109:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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)))
	t_1 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.25e+73)
		tmp = t_1;
	elseif (y_46_im <= -2.4e-123)
		tmp = t_0;
	elseif (y_46_im <= 2.7e-128)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 7.2e+109)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.25e+73)
		tmp = t_1;
	elseif (y_46_im <= -2.4e-123)
		tmp = t_0;
	elseif (y_46_im <= 2.7e-128)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 7.2e+109)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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]}, Block[{t$95$1 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.25e+73], t$95$1, If[LessEqual[y$46$im, -2.4e-123], t$95$0, If[LessEqual[y$46$im, 2.7e-128], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+109], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.24999999999999992e73 or 7.2e109 < y.im

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 80.0%

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

   4. Simplified 80.0%

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

   if -2.24999999999999992e73 < y.im < -2.4e-123 or 2.70000000000000006e-128 < y.im < 7.2e109

   1. Initial program 87.1%

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

   if -2.4e-123 < y.im < 2.70000000000000006e-128

   1. Initial program 62.4%

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

   2. Taylor expanded in y.re around inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

      4. times-frac 87.3%

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

   4. Simplified 87.3%

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

      1. associate-*l/ 88.4%

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

      2. sub-div 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.02 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.8e-46) (not (<= y.im 1.02e-54)))
   (* (/ -1.0 y.im) (- x.re (* x.im (/ y.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 <= -2.8e-46) || !(y_46_im <= 1.02e-54)) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (y_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 <= (-2.8d-46)) .or. (.not. (y_46im <= 1.02d-54))) then
        tmp = ((-1.0d0) / y_46im) * (x_46re - (x_46im * (y_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 <= -2.8e-46) || !(y_46_im <= 1.02e-54)) {
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (y_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 <= -2.8e-46) or not (y_46_im <= 1.02e-54):
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (y_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 <= -2.8e-46) || !(y_46_im <= 1.02e-54))
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(x_46_im * Float64(y_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.8e-46) || ~((y_46_im <= 1.02e-54)))
		tmp = (-1.0 / y_46_im) * (x_46_re - (x_46_im * (y_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, -2.8e-46], N[Not[LessEqual[y$46$im, 1.02e-54]], $MachinePrecision]], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.7999999999999998e-46 or 1.01999999999999999e-54 < y.im

   1. Initial program 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. add-sqr-sqrt 56.0%

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

      3. times-frac 56.0%

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

      4. hypot-def 56.0%

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

      5. hypot-def 70.5%

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

   3. Applied egg-rr 70.5%

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

   4. Taylor expanded in y.im around -inf 39.3%

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

      1. +-commutative 39.3%

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

      2. mul-1-neg 39.3%

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

      3. unsub-neg 39.3%

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

      4. associate-/l* 41.6%

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

      5. associate-/r/ 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in y.im around -inf 72.4%

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

   if -2.7999999999999998e-46 < y.im < 1.01999999999999999e-54

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. unpow2 79.2%

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

      4. times-frac 83.9%

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

   4. Simplified 83.9%

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

      1. associate-*l/ 84.6%

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

      2. sub-div 85.7%

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.02 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.04 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3e-46) (not (<= y.im 1.04e-54)))
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3e-46) || !(y_46_im <= 1.04e-54)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3d-46)) .or. (.not. (y_46im <= 1.04d-54))) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3e-46) || !(y_46_im <= 1.04e-54)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3e-46) or not (y_46_im <= 1.04e-54):
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3e-46) || !(y_46_im <= 1.04e-54))
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3e-46) || ~((y_46_im <= 1.04e-54)))
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3e-46], N[Not[LessEqual[y$46$im, 1.04e-54]], $MachinePrecision]], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.99999999999999987e-46 or 1.04e-54 < y.im

   1. Initial program 56.0%

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

   2. Taylor expanded in y.re around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. times-frac 73.5%

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

   4. Simplified 73.5%

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

   if -2.99999999999999987e-46 < y.im < 1.04e-54

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. unpow2 79.2%

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

      4. times-frac 83.9%

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

   4. Simplified 83.9%

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

      1. associate-*l/ 84.6%

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

      2. sub-div 85.7%

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.04 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.28 \cdot 10^{+54}\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3e-46) (not (<= y.im 1.28e+54)))
   (- (/ 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 <= -3e-46) || !(y_46_im <= 1.28e+54)) {
		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 <= (-3d-46)) .or. (.not. (y_46im <= 1.28d+54))) 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 <= -3e-46) || !(y_46_im <= 1.28e+54)) {
		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 <= -3e-46) or not (y_46_im <= 1.28e+54):
		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 <= -3e-46) || !(y_46_im <= 1.28e+54))
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3e-46) || ~((y_46_im <= 1.28e+54)))
		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, -3e-46], N[Not[LessEqual[y$46$im, 1.28e+54]], $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 < -2.99999999999999987e-46 or 1.28e54 < y.im

   1. Initial program 52.3%

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

   2. Taylor expanded in y.re around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   4. Simplified 67.3%

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

   if -2.99999999999999987e-46 < y.im < 1.28e54

   1. Initial program 72.3%

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

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

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

      1. mul-1-neg 72.7%

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

      2. unsub-neg 72.7%

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

      3. unpow2 72.7%

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

      4. times-frac 78.0%

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

   4. Simplified 78.0%

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

      1. associate-*l/ 78.7%

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

      2. sub-div 79.6%

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

   6. Applied egg-rr 79.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-46} \lor \neg \left(y.im \leq 1.28 \cdot 10^{+54}\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 10: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.re}{y.re} \cdot \frac{-y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.re y.im))))
   (if (<= y.im -1.1e-46)
     t_0
     (if (<= y.im -5.2e-112)
       (* (/ x.re y.re) (/ (- y.im) y.re))
       (if (<= y.im 9.8e+53) (/ x.im y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -(x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.1e-46) {
		tmp = t_0;
	} else if (y_46_im <= -5.2e-112) {
		tmp = (x_46_re / y_46_re) * (-y_46_im / y_46_re);
	} else if (y_46_im <= 9.8e+53) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x_46re / y_46im)
    if (y_46im <= (-1.1d-46)) then
        tmp = t_0
    else if (y_46im <= (-5.2d-112)) then
        tmp = (x_46re / y_46re) * (-y_46im / y_46re)
    else if (y_46im <= 9.8d+53) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -(x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.1e-46) {
		tmp = t_0;
	} else if (y_46_im <= -5.2e-112) {
		tmp = (x_46_re / y_46_re) * (-y_46_im / y_46_re);
	} else if (y_46_im <= 9.8e+53) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -(x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.1e-46:
		tmp = t_0
	elif y_46_im <= -5.2e-112:
		tmp = (x_46_re / y_46_re) * (-y_46_im / y_46_re)
	elif y_46_im <= 9.8e+53:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(-Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.1e-46)
		tmp = t_0;
	elseif (y_46_im <= -5.2e-112)
		tmp = Float64(Float64(x_46_re / y_46_re) * Float64(Float64(-y_46_im) / y_46_re));
	elseif (y_46_im <= 9.8e+53)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -(x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.1e-46)
		tmp = t_0;
	elseif (y_46_im <= -5.2e-112)
		tmp = (x_46_re / y_46_re) * (-y_46_im / y_46_re);
	elseif (y_46_im <= 9.8e+53)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = (-N[(x$46$re / y$46$im), $MachinePrecision])}, If[LessEqual[y$46$im, -1.1e-46], t$95$0, If[LessEqual[y$46$im, -5.2e-112], N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[((-y$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.8e+53], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.1e-46 or 9.80000000000000036e53 < y.im

   1. Initial program 52.3%

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

   2. Taylor expanded in y.re around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   4. Simplified 67.3%

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

   if -1.1e-46 < y.im < -5.19999999999999983e-112

   1. Initial program 94.6%

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

   2. Taylor expanded in y.re around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unpow2 74.8%

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

      4. times-frac 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in x.im around 0 64.2%

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

      1. mul-1-neg 64.2%

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

      2. *-commutative 64.2%

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

      3. unpow2 64.2%

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

      4. times-frac 69.0%

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

      5. distribute-rgt-neg-in 69.0%

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

      6. distribute-frac-neg 69.0%

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

   7. Simplified 69.0%

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

   if -5.19999999999999983e-112 < y.im < 9.80000000000000036e53

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around inf 65.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{-46}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.re}{y.re} \cdot \frac{-y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11: 64.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-46} \lor \neg \left(y.im \leq 8.8 \cdot 10^{+53}\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.5e-46) (not (<= y.im 8.8e+53)))
   (- (/ 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-46) || !(y_46_im <= 8.8e+53)) {
		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-46)) .or. (.not. (y_46im <= 8.8d+53))) 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-46) || !(y_46_im <= 8.8e+53)) {
		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-46) or not (y_46_im <= 8.8e+53):
		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-46) || !(y_46_im <= 8.8e+53))
		tmp = Float64(-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-46) || ~((y_46_im <= 8.8e+53)))
		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-46], N[Not[LessEqual[y$46$im, 8.8e+53]], $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 < -2.49999999999999996e-46 or 8.79999999999999994e53 < y.im

   1. Initial program 52.3%

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

   2. Taylor expanded in y.re around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   4. Simplified 67.3%

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

   if -2.49999999999999996e-46 < y.im < 8.79999999999999994e53

   1. Initial program 72.3%

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

   2. Taylor expanded in y.re around inf 62.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-46} \lor \neg \left(y.im \leq 8.8 \cdot 10^{+53}\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12: 10.2% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_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_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 63.2%

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

   1. *-un-lft-identity 63.2%

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

   2. add-sqr-sqrt 63.2%

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

   3. times-frac 63.2%

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

   4. hypot-def 63.2%

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

   5. hypot-def 77.4%

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

3. Applied egg-rr 77.4%

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

4. Taylor expanded in y.im around -inf 28.2%

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

   1. +-commutative 28.2%

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

   2. mul-1-neg 28.2%

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

   3. unsub-neg 28.2%

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

   4. associate-/l* 29.0%

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

   5. associate-/r/ 29.4%

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

6. Simplified 29.4%

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

7. Taylor expanded in y.re around -inf 10.1%

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

8. Final simplification 10.1%

   \[\leadsto \frac{x.im}{y.im} \]

Alternative 13: 42.1% 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 63.2%

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

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

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

3. Final simplification 43.1%

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

Reproduce

herbie shell --seed 2023195 
(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))))