_divideComplex, imaginary part

Percentage Accurate: 60.8% → 97.8%

Time: 10.5s

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: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (/ 1.0 (hypot y.re y.im))
  (- (* y.re (/ x.im (hypot y.re y.im))) (/ y.im (/ (hypot y.re y.im) x.re)))))

C

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - (y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - (y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)))

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re)));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. *-un-lft-identity 63.7%

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

   2. add-sqr-sqrt 63.7%

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

   3. times-frac 63.6%

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

   4. hypot-def 63.6%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

   1. div-sub 78.5%

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

   2. sub-neg 78.5%

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

   3. *-commutative 78.5%

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

   4. *-commutative 78.5%

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

5. Applied egg-rr 78.5%

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

   1. sub-neg 78.5%

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

   2. associate-*r/ 87.6%

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

   3. associate-/l* 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 2: 85.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+283)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (* (/ 1.0 y.re) (- x.im (/ x.re (/ y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+283) {
		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_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+283) {
		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_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+283:
		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_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+283)
		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_re) * Float64(x_46_im - Float64(x_46_re / 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 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+283)
		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_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+283], 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$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re / 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))) < 5.0000000000000004e283

   1. Initial program 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. add-sqr-sqrt 78.0%

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

      3. times-frac 77.9%

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

      4. hypot-def 77.9%

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

      5. hypot-def 95.7%

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

   3. Applied egg-rr 95.7%

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

   if 5.0000000000000004e283 < (/.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 11.4%

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

      1. *-un-lft-identity 11.4%

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

      2. add-sqr-sqrt 11.4%

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

      3. times-frac 11.4%

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

      4. hypot-def 11.4%

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

      5. hypot-def 15.3%

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

   3. Applied egg-rr 15.3%

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

   4. Taylor expanded in y.re around inf 22.9%

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

      1. mul-1-neg 22.9%

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

      2. unsub-neg 22.9%

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

      3. *-lft-identity 22.9%

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

      4. times-frac 26.4%

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

      5. /-rgt-identity 26.4%

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

   6. Simplified 26.4%

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

   7. Taylor expanded in y.re around inf 57.6%

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

      1. clear-num 57.6%

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

      2. un-div-inv 57.6%

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

   9. Applied egg-rr 57.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \end{array} \]

Alternative 3: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.2e+25)
   (* (/ 1.0 y.re) (- x.im (/ x.re (/ y.re y.im))))
   (if (<= y.re 1.85e-106)
     (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
     (if (<= y.re 3.5e+87)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (* (/ 1.0 (hypot y.re y.im)) (- x.im (* x.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 tmp;
	if (y_46_re <= -8.2e+25) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.85e-106) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 3.5e+87) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_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 tmp;
	if (y_46_re <= -8.2e+25) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.85e-106) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 3.5e+87) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_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):
	tmp = 0
	if y_46_re <= -8.2e+25:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	elif y_46_re <= 1.85e-106:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_re <= 3.5e+87:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_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)
	tmp = 0.0
	if (y_46_re <= -8.2e+25)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 1.85e-106)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_re <= 3.5e+87)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(x_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)
	tmp = 0.0;
	if (y_46_re <= -8.2e+25)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	elseif (y_46_re <= 1.85e-106)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_re <= 3.5e+87)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_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_] := If[LessEqual[y$46$re, -8.2e+25], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.85e-106], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+87], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.19999999999999933e25

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.4%

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

      4. hypot-def 55.4%

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

      5. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. unsub-neg 27.1%

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

      3. *-lft-identity 27.1%

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

      4. times-frac 27.1%

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

      5. /-rgt-identity 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in y.re around inf 86.4%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

   if -8.19999999999999933e25 < y.re < 1.8499999999999999e-106

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.7%

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

      4. hypot-def 67.7%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.im around -inf 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-*r/ 54.3%

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

      3. *-commutative 54.3%

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

      4. neg-mul-1 54.3%

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

      5. distribute-lft-neg-in 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in y.im around -inf 87.4%

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

   if 1.8499999999999999e-106 < y.re < 3.49999999999999986e87

   1. Initial program 91.2%

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

   if 3.49999999999999986e87 < y.re

   1. Initial program 47.5%

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

      1. *-un-lft-identity 47.5%

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

      2. add-sqr-sqrt 47.5%

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

      3. times-frac 47.5%

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

      4. hypot-def 47.5%

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

      5. hypot-def 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in y.re around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. *-lft-identity 89.6%

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

      4. times-frac 93.6%

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

      5. /-rgt-identity 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 4: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.re \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(t_1 - x.im\right)\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-107}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.im - t_1\right)\\ \end{array} \end{array} \]

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.re (/ y.im y.re))))
   (if (<= y.re -8.2e+25)
     (* t_0 (- t_1 x.im))
     (if (<= y.re 2.05e-107)
       (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
       (if (<= y.re 4.2e+81)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (* t_0 (- x.im t_1)))))))

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_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -8.2e+25) {
		tmp = t_0 * (t_1 - x_46_im);
	} else if (y_46_re <= 2.05e-107) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 4.2e+81) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0 * (x_46_im - t_1);
	}
	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_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -8.2e+25) {
		tmp = t_0 * (t_1 - x_46_im);
	} else if (y_46_re <= 2.05e-107) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 4.2e+81) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0 * (x_46_im - t_1);
	}
	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_re * (y_46_im / y_46_re)
	tmp = 0
	if y_46_re <= -8.2e+25:
		tmp = t_0 * (t_1 - x_46_im)
	elif y_46_re <= 2.05e-107:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_re <= 4.2e+81:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = t_0 * (x_46_im - t_1)
	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(x_46_re * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -8.2e+25)
		tmp = Float64(t_0 * Float64(t_1 - x_46_im));
	elseif (y_46_re <= 2.05e-107)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_re <= 4.2e+81)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(t_0 * Float64(x_46_im - 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 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = x_46_re * (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -8.2e+25)
		tmp = t_0 * (t_1 - x_46_im);
	elseif (y_46_re <= 2.05e-107)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_re <= 4.2e+81)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = t_0 * (x_46_im - 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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8.2e+25], N[(t$95$0 * N[(t$95$1 - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.05e-107], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+81], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$46$im - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.19999999999999933e25

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.4%

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

      4. hypot-def 55.4%

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

      5. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around -inf 86.5%

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

      1. +-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. unsub-neg 86.5%

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

      4. *-lft-identity 86.5%

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

      5. times-frac 87.9%

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

      6. /-rgt-identity 87.9%

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

   6. Simplified 87.9%

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

   if -8.19999999999999933e25 < y.re < 2.05e-107

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.7%

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

      4. hypot-def 67.7%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.im around -inf 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-*r/ 54.3%

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

      3. *-commutative 54.3%

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

      4. neg-mul-1 54.3%

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

      5. distribute-lft-neg-in 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in y.im around -inf 87.4%

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

   if 2.05e-107 < y.re < 4.1999999999999997e81

   1. Initial program 91.2%

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

   if 4.1999999999999997e81 < y.re

   1. Initial program 47.5%

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

      1. *-un-lft-identity 47.5%

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

      2. add-sqr-sqrt 47.5%

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

      3. times-frac 47.5%

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

      4. hypot-def 47.5%

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

      5. hypot-def 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in y.re around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. *-lft-identity 89.6%

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

      4. times-frac 93.6%

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

      5. /-rgt-identity 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot \frac{y.im}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-107}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 5: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9e+25)
   (* (/ 1.0 y.re) (- x.im (/ x.re (/ y.re y.im))))
   (if (<= y.re 9.4e-107)
     (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
     (if (<= y.re 2.5e+86)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (* (/ 1.0 y.re) (- x.im (* x.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 tmp;
	if (y_46_re <= -9e+25) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 9.4e-107) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 2.5e+86) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-9d+25)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re / (y_46re / y_46im)))
    else if (y_46re <= 9.4d-107) then
        tmp = ((-1.0d0) / y_46im) * (x_46re - ((y_46re * x_46im) / y_46im))
    else if (y_46re <= 2.5d+86) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9e+25) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 9.4e-107) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 2.5e+86) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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):
	tmp = 0
	if y_46_re <= -9e+25:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	elif y_46_re <= 9.4e-107:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_re <= 2.5e+86:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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)
	tmp = 0.0
	if (y_46_re <= -9e+25)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 9.4e-107)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_re <= 2.5e+86)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_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)
	tmp = 0.0;
	if (y_46_re <= -9e+25)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	elseif (y_46_re <= 9.4e-107)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_re <= 2.5e+86)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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_] := If[LessEqual[y$46$re, -9e+25], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.4e-107], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+86], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.0000000000000006e25

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.4%

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

      4. hypot-def 55.4%

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

      5. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. unsub-neg 27.1%

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

      3. *-lft-identity 27.1%

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

      4. times-frac 27.1%

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

      5. /-rgt-identity 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in y.re around inf 86.4%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

   if -9.0000000000000006e25 < y.re < 9.39999999999999995e-107

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.7%

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

      4. hypot-def 67.7%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.im around -inf 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-*r/ 54.3%

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

      3. *-commutative 54.3%

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

      4. neg-mul-1 54.3%

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

      5. distribute-lft-neg-in 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in y.im around -inf 87.4%

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

   if 9.39999999999999995e-107 < y.re < 2.4999999999999999e86

   1. Initial program 91.2%

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

   if 2.4999999999999999e86 < y.re

   1. Initial program 47.5%

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

      1. *-un-lft-identity 47.5%

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

      2. add-sqr-sqrt 47.5%

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

      3. times-frac 47.5%

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

      4. hypot-def 47.5%

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

      5. hypot-def 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in y.re around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. *-lft-identity 89.6%

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

      4. times-frac 93.6%

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

      5. /-rgt-identity 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in y.re around inf 93.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 6: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25} \lor \neg \left(y.re \leq 4.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8.2e+25) (not (<= y.re 4.5e-14)))
   (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
   (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.2e+25) || !(y_46_re <= 4.5e-14)) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-8.2d+25)) .or. (.not. (y_46re <= 4.5d-14))) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.2e+25) || !(y_46_re <= 4.5e-14)) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -8.2e+25) or not (y_46_re <= 4.5e-14):
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -8.2e+25) || !(y_46_re <= 4.5e-14))
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -8.2e+25) || ~((y_46_re <= 4.5e-14)))
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -8.2e+25], N[Not[LessEqual[y$46$re, 4.5e-14]], $MachinePrecision]], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.19999999999999933e25 or 4.4999999999999998e-14 < y.re

   1. Initial program 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. add-sqr-sqrt 57.1%

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

      3. times-frac 57.0%

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

      4. hypot-def 57.0%

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

      5. hypot-def 74.0%

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

   3. Applied egg-rr 74.0%

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

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

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

      3. *-lft-identity 54.0%

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

      4. times-frac 54.8%

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

      5. /-rgt-identity 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in y.re around inf 85.6%

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

   if -8.19999999999999933e25 < y.re < 4.4999999999999998e-14

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. add-sqr-sqrt 71.1%

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

      3. times-frac 71.1%

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

      4. hypot-def 71.1%

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

      5. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in y.im around -inf 51.1%

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

      1. +-commutative 51.1%

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

      2. associate-*r/ 51.1%

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

      3. *-commutative 51.1%

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

      4. neg-mul-1 51.1%

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

      5. distribute-lft-neg-in 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in y.im around -inf 84.6%

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

      1. associate-*l/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. associate-*r/ 82.6%

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

      4. add-sqr-sqrt 37.0%

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

      5. sqrt-unprod 70.1%

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

      6. sqr-neg 70.1%

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

      7. sqrt-unprod 35.4%

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

      8. add-sqr-sqrt 61.6%

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

      9. associate-*l* 61.6%

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

      10. neg-mul-1 61.6%

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

      11. associate-*r/ 62.4%

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

      12. neg-mul-1 62.4%

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

      13. sub-neg 62.4%

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

      14. associate-/l* 61.6%

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

      15. add-sqr-sqrt 26.1%

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

      16. sqrt-unprod 67.4%

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

      17. sqr-neg 67.4%

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

      18. sqrt-unprod 46.0%

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

      19. add-sqr-sqrt 83.1%

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

   9. Applied egg-rr 83.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+25} \lor \neg \left(y.re \leq 4.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e+26)
   (* (/ 1.0 y.re) (- x.im (/ x.re (/ y.re y.im))))
   (if (<= y.re 1.85e-10)
     (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)
     (* (/ 1.0 y.re) (- x.im (* x.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 tmp;
	if (y_46_re <= -2.4e+26) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.85e-10) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.4d+26)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re / (y_46re / y_46im)))
    else if (y_46re <= 1.85d-10) then
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    else
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e+26) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.85e-10) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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):
	tmp = 0
	if y_46_re <= -2.4e+26:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	elif y_46_re <= 1.85e-10:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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)
	tmp = 0.0
	if (y_46_re <= -2.4e+26)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 1.85e-10)
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_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)
	tmp = 0.0;
	if (y_46_re <= -2.4e+26)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	elseif (y_46_re <= 1.85e-10)
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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_] := If[LessEqual[y$46$re, -2.4e+26], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.85e-10], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.40000000000000005e26

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.4%

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

      4. hypot-def 55.4%

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

      5. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. unsub-neg 27.1%

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

      3. *-lft-identity 27.1%

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

      4. times-frac 27.1%

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

      5. /-rgt-identity 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in y.re around inf 86.4%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

   if -2.40000000000000005e26 < y.re < 1.85000000000000007e-10

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. add-sqr-sqrt 71.1%

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

      3. times-frac 71.1%

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

      4. hypot-def 71.1%

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

      5. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in y.im around -inf 51.1%

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

      1. +-commutative 51.1%

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

      2. associate-*r/ 51.1%

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

      3. *-commutative 51.1%

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

      4. neg-mul-1 51.1%

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

      5. distribute-lft-neg-in 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in y.im around -inf 84.6%

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

      1. associate-*l/ 84.8%

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

      2. distribute-lft-in 84.8%

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

      3. associate-*r/ 82.6%

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

      4. add-sqr-sqrt 37.0%

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

      5. sqrt-unprod 70.1%

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

      6. sqr-neg 70.1%

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

      7. sqrt-unprod 35.4%

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

      8. add-sqr-sqrt 61.6%

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

      9. associate-*l* 61.6%

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

      10. neg-mul-1 61.6%

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

      11. associate-*r/ 62.4%

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

      12. neg-mul-1 62.4%

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

      13. sub-neg 62.4%

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

      14. associate-/l* 61.6%

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

      15. add-sqr-sqrt 26.1%

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

      16. sqrt-unprod 67.4%

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

      17. sqr-neg 67.4%

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

      18. sqrt-unprod 46.0%

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

      19. add-sqr-sqrt 83.1%

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

   9. Applied egg-rr 83.1%

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

   if 1.85000000000000007e-10 < y.re

   1. Initial program 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. add-sqr-sqrt 58.8%

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

      3. times-frac 58.7%

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

      4. hypot-def 58.7%

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

      5. hypot-def 75.8%

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

   3. Applied egg-rr 75.8%

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

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

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. *-lft-identity 83.4%

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

      4. times-frac 85.0%

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

      5. /-rgt-identity 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in y.re around inf 84.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 8: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+26)
   (* (/ 1.0 y.re) (- x.im (/ x.re (/ y.re y.im))))
   (if (<= y.re 2.95e-12)
     (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
     (* (/ 1.0 y.re) (- x.im (* x.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 tmp;
	if (y_46_re <= -1.15e+26) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 2.95e-12) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.15d+26)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re / (y_46re / y_46im)))
    else if (y_46re <= 2.95d-12) then
        tmp = ((-1.0d0) / y_46im) * (x_46re - ((y_46re * x_46im) / y_46im))
    else
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+26) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	} else if (y_46_re <= 2.95e-12) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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):
	tmp = 0
	if y_46_re <= -1.15e+26:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	elif y_46_re <= 2.95e-12:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	else:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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)
	tmp = 0.0
	if (y_46_re <= -1.15e+26)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 2.95e-12)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_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)
	tmp = 0.0;
	if (y_46_re <= -1.15e+26)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	elseif (y_46_re <= 2.95e-12)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	else
		tmp = (1.0 / y_46_re) * (x_46_im - (x_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_] := If[LessEqual[y$46$re, -1.15e+26], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.95e-12], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15e26

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 55.4%

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

      4. hypot-def 55.4%

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

      5. hypot-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   4. Taylor expanded in y.re around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. unsub-neg 27.1%

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

      3. *-lft-identity 27.1%

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

      4. times-frac 27.1%

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

      5. /-rgt-identity 27.1%

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

   6. Simplified 27.1%

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

   7. Taylor expanded in y.re around inf 86.4%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.4%

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

   9. Applied egg-rr 86.4%

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

   if -1.15e26 < y.re < 2.95e-12

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. add-sqr-sqrt 71.1%

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

      3. times-frac 71.1%

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

      4. hypot-def 71.1%

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

      5. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in y.im around -inf 51.1%

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

      1. +-commutative 51.1%

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

      2. associate-*r/ 51.1%

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

      3. *-commutative 51.1%

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

      4. neg-mul-1 51.1%

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

      5. distribute-lft-neg-in 51.1%

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

   6. Simplified 51.1%

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

   7. Taylor expanded in y.im around -inf 84.6%

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

   if 2.95e-12 < y.re

   1. Initial program 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. add-sqr-sqrt 58.8%

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

      3. times-frac 58.7%

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

      4. hypot-def 58.7%

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

      5. hypot-def 75.8%

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

   3. Applied egg-rr 75.8%

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

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

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. *-lft-identity 83.4%

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

      4. times-frac 85.0%

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

      5. /-rgt-identity 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in y.re around inf 84.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 9: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+46} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.6e+46) (not (<= y.re 1.15e+27)))
   (/ x.im y.re)
   (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.6e+46) || !(y_46_re <= 1.15e+27)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.6d+46)) .or. (.not. (y_46re <= 1.15d+27))) then
        tmp = x_46im / y_46re
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.6e+46) || !(y_46_re <= 1.15e+27)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.6e+46) or not (y_46_re <= 1.15e+27):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.6e+46) || !(y_46_re <= 1.15e+27))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.6e+46) || ~((y_46_re <= 1.15e+27)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.6e+46], N[Not[LessEqual[y$46$re, 1.15e+27]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.6000000000000001e46 or 1.15e27 < y.re

   1. Initial program 53.7%

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

   2. Taylor expanded in y.re around inf 80.9%

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

   if -4.6000000000000001e46 < y.re < 1.15e27

   1. Initial program 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. add-sqr-sqrt 73.0%

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

      3. times-frac 72.9%

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

      4. hypot-def 72.9%

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

      5. hypot-def 84.2%

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

   3. Applied egg-rr 84.2%

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

   4. Taylor expanded in y.im around -inf 48.2%

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

      1. +-commutative 48.2%

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

      2. associate-*r/ 48.2%

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

      3. *-commutative 48.2%

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

      4. neg-mul-1 48.2%

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

      5. distribute-lft-neg-in 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in y.im around -inf 81.0%

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

      1. associate-*l/ 81.1%

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

      2. distribute-lft-in 81.1%

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

      3. associate-*r/ 79.1%

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

      4. add-sqr-sqrt 36.0%

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

      5. sqrt-unprod 67.7%

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

      6. sqr-neg 67.7%

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

      7. sqrt-unprod 33.8%

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

      8. add-sqr-sqrt 59.1%

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

      9. associate-*l* 59.1%

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

      10. neg-mul-1 59.1%

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

      11. associate-*r/ 59.9%

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

      12. neg-mul-1 59.9%

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

      13. sub-neg 59.9%

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

      14. associate-/l* 59.1%

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

      15. add-sqr-sqrt 25.3%

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

      16. sqrt-unprod 64.5%

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

      17. sqr-neg 64.5%

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

      18. sqrt-unprod 43.5%

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

      19. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+46} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 10: 65.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+26} \lor \neg \left(y.re \leq 4.8 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.3e+26) (not (<= y.re 4.8e-31)))
   (/ x.im y.re)
   (- (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.3e+26) || !(y_46_re <= 4.8e-31)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.3d+26)) .or. (.not. (y_46re <= 4.8d-31))) then
        tmp = x_46im / y_46re
    else
        tmp = -(x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.3e+26) || !(y_46_re <= 4.8e-31)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.3e+26) or not (y_46_re <= 4.8e-31):
		tmp = x_46_im / y_46_re
	else:
		tmp = -(x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.3e+26) || !(y_46_re <= 4.8e-31))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(-Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.3e+26) || ~((y_46_re <= 4.8e-31)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -(x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.3e+26], N[Not[LessEqual[y$46$re, 4.8e-31]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], (-N[(x$46$re / y$46$im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.3000000000000001e26 or 4.8e-31 < y.re

   1. Initial program 58.0%

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

   2. Taylor expanded in y.re around inf 75.5%

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

   if -2.3000000000000001e26 < y.re < 4.8e-31

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+26} \lor \neg \left(y.re \leq 4.8 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 2.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 2.4e+211) (/ x.im y.re) (/ x.re y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.4e+211) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 2.4d+211) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.4e+211) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 2.4e+211:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 2.4e+211)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 2.4e+211)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 2.4e+211], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 2.40000000000000018e211

   1. Initial program 64.2%

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

   2. Taylor expanded in y.re around inf 50.4%

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

   if 2.40000000000000018e211 < y.im

   1. Initial program 56.5%

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

      1. *-un-lft-identity 56.5%

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

      2. add-sqr-sqrt 56.5%

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

      3. times-frac 56.5%

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

      4. hypot-def 56.5%

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

      5. hypot-def 67.5%

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

   3. Applied egg-rr 67.5%

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

   4. Taylor expanded in y.im around -inf 50.0%

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

      1. +-commutative 50.0%

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

      2. associate-*r/ 50.0%

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

      3. *-commutative 50.0%

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

      4. neg-mul-1 50.0%

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

      5. distribute-lft-neg-in 50.0%

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

   6. Simplified 50.0%

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

   7. Taylor expanded in y.re around 0 52.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 2.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 12: 9.6% 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 63.7%

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

   1. *-un-lft-identity 63.7%

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

   2. add-sqr-sqrt 63.7%

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

   3. times-frac 63.6%

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

   4. hypot-def 63.6%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

4. Taylor expanded in y.im around -inf 29.7%

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

   1. +-commutative 29.7%

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

   2. associate-*r/ 29.7%

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

   3. *-commutative 29.7%

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

   4. neg-mul-1 29.7%

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

   5. distribute-lft-neg-in 29.7%

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

6. Simplified 29.7%

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

7. Taylor expanded in y.re around -inf 10.4%

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

8. Final simplification 10.4%

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

Alternative 13: 42.2% 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 63.7%

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

2. Taylor expanded in y.re around inf 48.3%

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

3. Final simplification 48.3%

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

Reproduce

herbie shell --seed 2023319 
(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))))