_divideComplex, imaginary part

Percentage Accurate: 61.7% → 97.4%

Time: 9.9s

Alternatives: 12

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 97.4% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (/ 1.0 (hypot y.re y.im))
  (- (/ y.re (/ (hypot y.re y.im) x.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 / (hypot(y_46_re, y_46_im) / x_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 / (Math.hypot(y_46_re, y_46_im) / x_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 / (math.hypot(y_46_re, y_46_im) / x_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(hypot(y_46_re, y_46_im) / x_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 / (hypot(y_46_re, y_46_im) / x_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[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $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 60.4%

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

   1. *-un-lft-identity 60.4%

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

   2. add-sqr-sqrt 60.4%

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

   3. times-frac 60.4%

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

   4. hypot-def 60.4%

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

   5. hypot-def 73.6%

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

3. Applied egg-rr 73.6%

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

   1. div-sub 73.6%

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

   2. *-commutative 73.6%

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

   3. *-commutative 73.6%

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

5. Applied egg-rr 73.6%

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

   1. associate-/l* 86.4%

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

   2. associate-/l* 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 87.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} + x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 2e+303)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (* t_0 (+ (/ y.re (/ (hypot y.re y.im) x.im)) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+303) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) + x_46_re);
	}
	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 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+303) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) + x_46_re);
	}
	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 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+303:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = t_0 * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) + x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+303)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(t_0 * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) + x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+303)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) + x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] + x$46$re), $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))) < 2e303

   1. Initial program 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. hypot-def 97.5%

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

   3. Applied egg-rr 97.5%

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

   if 2e303 < (/.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 7.5%

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

      1. *-un-lft-identity 7.5%

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

      2. add-sqr-sqrt 7.5%

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

      3. times-frac 7.5%

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

      4. hypot-def 7.5%

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

      5. hypot-def 12.4%

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

   3. Applied egg-rr 12.4%

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

      1. div-sub 12.4%

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

      2. *-commutative 12.4%

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

      3. *-commutative 12.4%

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

   5. Applied egg-rr 12.4%

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

      1. associate-/l* 56.7%

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

      2. associate-/l* 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y.im around -inf 70.8%

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

      1. mul-1-neg 70.8%

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

   10. Simplified 70.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} + x.re\right)\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im - t_0}{y.re}\\ \mathbf{elif}\;y.re \leq -1.42 \cdot 10^{-65}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{t_0}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (/ x.re y.re))))
   (if (<= y.re -3.5e+107)
     (/ (- x.im t_0) y.re)
     (if (<= y.re -1.42e-65)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 2.05e+55)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (- (/ x.im y.re) (/ t_0 y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+107) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= -1.42e-65) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.05e+55) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46im * (x_46re / y_46re)
    if (y_46re <= (-3.5d+107)) then
        tmp = (x_46im - t_0) / y_46re
    else if (y_46re <= (-1.42d-65)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 2.05d+55) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (t_0 / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+107) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= -1.42e-65) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.05e+55) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * (x_46_re / y_46_re)
	tmp = 0
	if y_46_re <= -3.5e+107:
		tmp = (x_46_im - t_0) / y_46_re
	elif y_46_re <= -1.42e-65:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 2.05e+55:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.5e+107)
		tmp = Float64(Float64(x_46_im - t_0) / y_46_re);
	elseif (y_46_re <= -1.42e-65)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.05e+55)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(t_0 / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * (x_46_re / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.5e+107)
		tmp = (x_46_im - t_0) / y_46_re;
	elseif (y_46_re <= -1.42e-65)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 2.05e+55)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+107], N[(N[(x$46$im - t$95$0), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.42e-65], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.05e+55], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.4999999999999997e107

   1. Initial program 35.0%

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

   2. Taylor expanded in y.re around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. unpow2 78.2%

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

      4. times-frac 89.5%

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

   4. Simplified 89.5%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-*l/ 89.5%

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

      2. associate-*r/ 91.9%

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

      3. sub-div 91.9%

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

   8. Applied egg-rr 91.9%

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

   if -3.4999999999999997e107 < y.re < -1.41999999999999993e-65

   1. Initial program 91.9%

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

   if -1.41999999999999993e-65 < y.re < 2.04999999999999991e55

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

   if 2.04999999999999991e55 < y.re

   1. Initial program 51.3%

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

   2. Taylor expanded in y.re around inf 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

      3. unpow2 82.1%

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

      4. times-frac 88.1%

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

   4. Simplified 88.1%

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

      1. associate-*r/ 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.42 \cdot 10^{-65}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))
   (if (<= y.re -1.9e-63)
     t_1
     (if (<= y.re 9.2e-150)
       t_0
       (if (<= y.re 1e-123)
         (/ (* x.im (/ y.re y.im)) y.im)
         (if (<= y.re 8e+14) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.9e-63) {
		tmp = t_1;
	} else if (y_46_re <= 9.2e-150) {
		tmp = t_0;
	} else if (y_46_re <= 1e-123) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_re <= 8e+14) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    if (y_46re <= (-1.9d-63)) then
        tmp = t_1
    else if (y_46re <= 9.2d-150) then
        tmp = t_0
    else if (y_46re <= 1d-123) then
        tmp = (x_46im * (y_46re / y_46im)) / y_46im
    else if (y_46re <= 8d+14) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.9e-63) {
		tmp = t_1;
	} else if (y_46_re <= 9.2e-150) {
		tmp = t_0;
	} else if (y_46_re <= 1e-123) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_re <= 8e+14) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -1.9e-63:
		tmp = t_1
	elif y_46_re <= 9.2e-150:
		tmp = t_0
	elif y_46_re <= 1e-123:
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im
	elif y_46_re <= 8e+14:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.9e-63)
		tmp = t_1;
	elseif (y_46_re <= 9.2e-150)
		tmp = t_0;
	elseif (y_46_re <= 1e-123)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 8e+14)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	t_1 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.9e-63)
		tmp = t_1;
	elseif (y_46_re <= 9.2e-150)
		tmp = t_0;
	elseif (y_46_re <= 1e-123)
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	elseif (y_46_re <= 8e+14)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.9e-63], t$95$1, If[LessEqual[y$46$re, 9.2e-150], t$95$0, If[LessEqual[y$46$re, 1e-123], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8e+14], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.90000000000000009e-63 or 8e14 < y.re

   1. Initial program 56.2%

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

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

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

      3. unpow2 74.3%

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

      4. times-frac 80.3%

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

   4. Simplified 80.3%

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

      1. associate-*l/ 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*l/ 80.3%

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

      2. associate-*r/ 81.6%

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

      3. sub-div 81.6%

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

   8. Applied egg-rr 81.6%

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

   if -1.90000000000000009e-63 < y.re < 9.20000000000000011e-150 or 1.0000000000000001e-123 < y.re < 8e14

   1. Initial program 63.5%

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

   2. Taylor expanded in y.re around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. neg-mul-1 78.6%

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

   4. Simplified 78.6%

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

   if 9.20000000000000011e-150 < y.re < 1.0000000000000001e-123

   1. Initial program 89.5%

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

   2. Taylor expanded in x.im around inf 70.8%

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

      1. unpow2 70.8%

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

      2. unpow2 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in y.re around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. unpow2 70.7%

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

      3. times-frac 52.1%

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

   7. Simplified 52.1%

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

      1. associate-*l/ 70.7%

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

   9. Applied egg-rr 70.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im - t_0}{y.re}\\ \mathbf{elif}\;y.re \leq -5.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{t_0}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (/ x.re y.re))))
   (if (<= y.re -2.45e+76)
     (/ (- x.im t_0) y.re)
     (if (<= y.re -5.3e-61)
       (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.85e+55)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (- (/ x.im y.re) (/ t_0 y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -2.45e+76) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= -5.3e-61) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.85e+55) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46im * (x_46re / y_46re)
    if (y_46re <= (-2.45d+76)) then
        tmp = (x_46im - t_0) / y_46re
    else if (y_46re <= (-5.3d-61)) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1.85d+55) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (t_0 / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -2.45e+76) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= -5.3e-61) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.85e+55) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * (x_46_re / y_46_re)
	tmp = 0
	if y_46_re <= -2.45e+76:
		tmp = (x_46_im - t_0) / y_46_re
	elif y_46_re <= -5.3e-61:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.85e+55:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.45e+76)
		tmp = Float64(Float64(x_46_im - t_0) / y_46_re);
	elseif (y_46_re <= -5.3e-61)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.85e+55)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(t_0 / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * (x_46_re / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.45e+76)
		tmp = (x_46_im - t_0) / y_46_re;
	elseif (y_46_re <= -5.3e-61)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.85e+55)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.45e+76], N[(N[(x$46$im - t$95$0), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.3e-61], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+55], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.45000000000000013e76

   1. Initial program 38.8%

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

   2. Taylor expanded in y.re around inf 78.0%

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

      1. mul-1-neg 78.0%

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

      2. unsub-neg 78.0%

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

      3. unpow2 78.0%

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

      4. times-frac 88.3%

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

   4. Simplified 88.3%

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

      1. associate-*l/ 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. associate-*l/ 88.3%

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

      2. associate-*r/ 90.4%

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

      3. sub-div 90.4%

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

   8. Applied egg-rr 90.4%

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

   if -2.45000000000000013e76 < y.re < -5.3e-61

   1. Initial program 94.8%

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

   2. Taylor expanded in x.im around inf 85.5%

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

      1. unpow2 85.5%

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

      2. unpow2 85.5%

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

   4. Simplified 85.5%

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

   if -5.3e-61 < y.re < 1.8500000000000001e55

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

   if 1.8500000000000001e55 < y.re

   1. Initial program 51.3%

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

   2. Taylor expanded in y.re around inf 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

      3. unpow2 82.1%

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

      4. times-frac 88.1%

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

   4. Simplified 88.1%

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

      1. associate-*r/ 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -5.3 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.9 \cdot 10^{-62} \lor \neg \left(y.re \leq 2.3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.9e-62) (not (<= y.re 2.3e+53)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5.9e-62) || !(y_46_re <= 2.3e+53)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_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 <= (-5.9d-62)) .or. (.not. (y_46re <= 2.3d+53))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = ((y_46re / y_46im) * (x_46im / y_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 <= -5.9e-62) || !(y_46_re <= 2.3e+53)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -5.9e-62) or not (y_46_re <= 2.3e+53):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -5.9e-62) || !(y_46_re <= 2.3e+53))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -5.9e-62) || ~((y_46_re <= 2.3e+53)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -5.9e-62], N[Not[LessEqual[y$46$re, 2.3e+53]], $MachinePrecision]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.9000000000000004e-62 or 2.3000000000000002e53 < y.re

   1. Initial program 54.4%

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

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

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

      1. mul-1-neg 76.3%

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

      2. unsub-neg 76.3%

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

      3. unpow2 76.3%

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

      4. times-frac 82.6%

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

   4. Simplified 82.6%

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

      1. associate-*l/ 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. associate-*l/ 82.6%

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

      2. associate-*r/ 84.0%

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

      3. sub-div 84.0%

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

   8. Applied egg-rr 84.0%

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

   if -5.9000000000000004e-62 < y.re < 2.3000000000000002e53

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.9 \cdot 10^{-62} \lor \neg \left(y.re \leq 2.3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im - t_0}{y.re}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+56}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{t_0}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (/ x.re y.re))))
   (if (<= y.re -1.1e-60)
     (/ (- x.im t_0) y.re)
     (if (<= y.re 8e+56)
       (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
       (- (/ x.im y.re) (/ t_0 y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -1.1e-60) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= 8e+56) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46im * (x_46re / y_46re)
    if (y_46re <= (-1.1d-60)) then
        tmp = (x_46im - t_0) / y_46re
    else if (y_46re <= 8d+56) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (t_0 / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * (x_46_re / y_46_re);
	double tmp;
	if (y_46_re <= -1.1e-60) {
		tmp = (x_46_im - t_0) / y_46_re;
	} else if (y_46_re <= 8e+56) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * (x_46_re / y_46_re)
	tmp = 0
	if y_46_re <= -1.1e-60:
		tmp = (x_46_im - t_0) / y_46_re
	elif y_46_re <= 8e+56:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.1e-60)
		tmp = Float64(Float64(x_46_im - t_0) / y_46_re);
	elseif (y_46_re <= 8e+56)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(t_0 / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * (x_46_re / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.1e-60)
		tmp = (x_46_im - t_0) / y_46_re;
	elseif (y_46_re <= 8e+56)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (t_0 / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.1e-60], N[(N[(x$46$im - t$95$0), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 8e+56], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.0999999999999999e-60

   1. Initial program 57.8%

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

   2. Taylor expanded in y.re around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. unsub-neg 69.9%

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

      3. unpow2 69.9%

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

      4. times-frac 76.7%

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

   4. Simplified 76.7%

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

      1. associate-*l/ 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. associate-*l/ 76.7%

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

      2. associate-*r/ 78.1%

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

      3. sub-div 78.1%

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

   8. Applied egg-rr 78.1%

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

   if -1.0999999999999999e-60 < y.re < 8.00000000000000074e56

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

   if 8.00000000000000074e56 < y.re

   1. Initial program 51.3%

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

   2. Taylor expanded in y.re around inf 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

      3. unpow2 82.1%

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

      4. times-frac 88.1%

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

   4. Simplified 88.1%

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

      1. associate-*r/ 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+56}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.re -3.6e-64)
     (/ x.im y.re)
     (if (<= y.re 9.2e-150)
       t_0
       (if (<= y.re 4.4e-123)
         (/ (* x.im (/ y.re y.im)) y.im)
         (if (<= y.re 4.2e+54) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_re <= -3.6e-64) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9.2e-150) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e-123) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_re <= 4.2e+54) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46re <= (-3.6d-64)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 9.2d-150) then
        tmp = t_0
    else if (y_46re <= 4.4d-123) then
        tmp = (x_46im * (y_46re / y_46im)) / y_46im
    else if (y_46re <= 4.2d+54) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_re <= -3.6e-64) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9.2e-150) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e-123) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_re <= 4.2e+54) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_re <= -3.6e-64:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 9.2e-150:
		tmp = t_0
	elif y_46_re <= 4.4e-123:
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im
	elif y_46_re <= 4.2e+54:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.6e-64)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 9.2e-150)
		tmp = t_0;
	elseif (y_46_re <= 4.4e-123)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 4.2e+54)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -3.6e-64)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 9.2e-150)
		tmp = t_0;
	elseif (y_46_re <= 4.4e-123)
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	elseif (y_46_re <= 4.2e+54)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e-64], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 9.2e-150], t$95$0, If[LessEqual[y$46$re, 4.4e-123], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+54], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.5999999999999998e-64 or 4.19999999999999972e54 < y.re

   1. Initial program 54.4%

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

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

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

   if -3.5999999999999998e-64 < y.re < 9.20000000000000011e-150 or 4.40000000000000013e-123 < y.re < 4.19999999999999972e54

   1. Initial program 65.2%

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

   2. Taylor expanded in y.re around 0 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

   4. Simplified 74.9%

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

   if 9.20000000000000011e-150 < y.re < 4.40000000000000013e-123

   1. Initial program 89.5%

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

   2. Taylor expanded in x.im around inf 70.8%

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

      1. unpow2 70.8%

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

      2. unpow2 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in y.re around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. unpow2 70.7%

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

      3. times-frac 52.1%

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

   7. Simplified 52.1%

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

      1. associate-*l/ 70.7%

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

   9. Applied egg-rr 70.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9: 63.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.38 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.38e-64)
   (/ x.im y.re)
   (if (<= y.re 2.1e+55) (/ (- x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.38e-64) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.1e+55) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.38d-64)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.1d+55) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.38e-64) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.1e+55) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.38e-64:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.1e+55:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.38e-64)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.1e+55)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.38e-64)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.1e+55)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.38e-64], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.1e+55], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.37999999999999998e-64 or 2.1000000000000001e55 < y.re

   1. Initial program 54.4%

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

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

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

   if -1.37999999999999998e-64 < y.re < 2.1000000000000001e55

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   4. Simplified 70.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.38 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 46.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+235}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.5e+235)
   (/ x.re y.im)
   (if (<= y.im 7.2e+149) (/ 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 <= -6.5e+235) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 7.2e+149) {
		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 <= (-6.5d+235)) then
        tmp = x_46re / y_46im
    else if (y_46im <= 7.2d+149) 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 <= -6.5e+235) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 7.2e+149) {
		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 <= -6.5e+235:
		tmp = x_46_re / y_46_im
	elif y_46_im <= 7.2e+149:
		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 <= -6.5e+235)
		tmp = Float64(x_46_re / y_46_im);
	elseif (y_46_im <= 7.2e+149)
		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 <= -6.5e+235)
		tmp = x_46_re / y_46_im;
	elseif (y_46_im <= 7.2e+149)
		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, -6.5e+235], N[(x$46$re / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+149], 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 < -6.5000000000000001e235 or 7.1999999999999999e149 < y.im

   1. Initial program 42.8%

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

      1. *-un-lft-identity 42.8%

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

      2. add-sqr-sqrt 42.8%

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

      3. times-frac 42.8%

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

      4. hypot-def 42.8%

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

      5. hypot-def 62.9%

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

   3. Applied egg-rr 62.9%

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

   4. Taylor expanded in y.re around 0 81.9%

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

      1. neg-mul-1 81.9%

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

      2. unsub-neg 81.9%

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

      3. associate-/l* 84.5%

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

   6. Simplified 84.5%

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

   7. Taylor expanded in y.im around -inf 42.1%

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

   if -6.5000000000000001e235 < y.im < 7.1999999999999999e149

   1. Initial program 64.7%

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

   2. Taylor expanded in y.re around inf 54.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+235}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11: 43.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 7.2e+149) (/ x.im y.re) (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 7.2e+149) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 7.2d+149) then
        tmp = x_46im / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 7.2e+149) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 7.2e+149:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 7.2e+149)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 7.2e+149)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 7.2e+149], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 7.1999999999999999e149

   1. Initial program 63.8%

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

   2. Taylor expanded in y.re around inf 52.3%

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

   if 7.1999999999999999e149 < y.im

   1. Initial program 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. times-frac 41.0%

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

      4. hypot-def 41.0%

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

      5. hypot-def 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in y.re around 0 92.3%

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

      1. neg-mul-1 92.3%

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

      2. unsub-neg 92.3%

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

      3. associate-/l* 95.8%

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

   6. Simplified 95.8%

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

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

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 9.4% 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 60.4%

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

   1. *-un-lft-identity 60.4%

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

   2. add-sqr-sqrt 60.4%

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

   3. times-frac 60.4%

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

   4. hypot-def 60.4%

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

   5. hypot-def 73.6%

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

3. Applied egg-rr 73.6%

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

4. Taylor expanded in y.re around 0 32.9%

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

   1. neg-mul-1 32.9%

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

   2. unsub-neg 32.9%

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

   3. associate-/l* 33.4%

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

6. Simplified 33.4%

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

7. Taylor expanded in y.re around inf 8.8%

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

8. Final simplification 8.8%

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

Reproduce

herbie shell --seed 2023240 
(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))))