_divideComplex, imaginary part

Percentage Accurate: 62.1% → 98.0%

Time: 8.9s

Alternatives: 10

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: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.0× speedup

Math

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

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)) (/ x.re (/ (hypot 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 (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)));
}

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)) - (x_46_re / (Math.hypot(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 (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - (x_46_re / (math.hypot(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(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(x_46_re / Float64(hypot(y_46_re, y_46_im) / y_46_im))))
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)) - (x_46_re / (hypot(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[(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[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

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

   1. *-un-lft-identity 67.4%

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

   2. add-sqr-sqrt 67.4%

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

   3. times-frac 67.4%

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

   4. hypot-def 67.4%

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

   5. hypot-def 81.9%

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

3. Applied egg-rr 81.9%

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

   1. div-sub 81.9%

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

5. Applied egg-rr 81.9%

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

   1. *-commutative 81.9%

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

   2. associate-/l* 90.3%

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

   3. associate-/l* 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 87.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+294}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \end{array} \]

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))) 1e+294)
     (* 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))) <= 1e+294) {
		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))) <= 1e+294) {
		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))) <= 1e+294:
		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))) <= 1e+294)
		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))) <= 1e+294)
		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], 1e+294], 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))) < 1.00000000000000007e294

   1. Initial program 80.1%

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

      1. *-un-lft-identity 80.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. times-frac 80.1%

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

      4. hypot-def 80.1%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 1.00000000000000007e294 < (/.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 14.9%

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

      1. *-un-lft-identity 14.9%

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

      2. add-sqr-sqrt 14.9%

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

      3. times-frac 14.9%

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

      4. hypot-def 14.9%

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

      5. hypot-def 24.3%

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

   3. Applied egg-rr 24.3%

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

      1. div-sub 24.3%

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

   5. Applied egg-rr 24.3%

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

      1. *-commutative 24.3%

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

      2. associate-/l* 62.9%

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

      3. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y.re around 0 84.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+294}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \]

Alternative 3: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.48 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -2.9e+46)
     (- (/ x.im y.re) (/ y.im (/ (* y.re y.re) x.re)))
     (if (<= y.re -4.2e-95)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.re 9.5e-137)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (if (<= y.re 1.48e+69)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
           (/ (- x.im (* y.im (/ x.re y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -2.9e+46) {
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	} else if (y_46_re <= -4.2e-95) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_re <= 9.5e-137) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.48e+69) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.9e+46)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_re)));
	elseif (y_46_re <= -4.2e-95)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_re <= 9.5e-137)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.48e+69)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e+46], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.2e-95], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-137], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.48e+69], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.9000000000000002e46

   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. Taylor expanded in y.re around inf 89.7%

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

      1. mul-1-neg 89.7%

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

      2. unsub-neg 89.7%

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

      3. unpow2 89.7%

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

      4. associate-/r* 87.8%

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

   4. Simplified 87.8%

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

   5. Taylor expanded in x.re around 0 89.7%

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

      1. unpow2 89.7%

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

      2. times-frac 87.9%

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

   7. Simplified 87.9%

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

      1. *-commutative 87.9%

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

      2. frac-2neg 87.9%

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

      3. frac-times 89.7%

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

      4. add-sqr-sqrt 40.5%

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

      5. sqrt-unprod 71.7%

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

      6. sqr-neg 71.7%

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

      7. sqrt-unprod 39.9%

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

      8. add-sqr-sqrt 80.4%

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

      9. frac-times 76.4%

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

      10. frac-2neg 76.4%

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

      11. times-frac 80.4%

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

      12. associate-/l* 80.4%

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

      13. add-sqr-sqrt 39.9%

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

      14. sqrt-unprod 71.8%

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

      15. sqr-neg 71.8%

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

      16. sqrt-unprod 40.5%

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

      17. add-sqr-sqrt 89.8%

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

   9. Applied egg-rr 89.8%

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

   if -2.9000000000000002e46 < y.re < -4.2e-95

   1. Initial program 81.6%

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

   if -4.2e-95 < y.re < 9.5000000000000007e-137

   1. Initial program 67.9%

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

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

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. unpow2 83.5%

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

      5. times-frac 91.8%

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

   4. Simplified 91.8%

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

   if 9.5000000000000007e-137 < y.re < 1.47999999999999999e69

   1. Initial program 89.6%

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

      1. sub-neg 89.6%

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

      2. +-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. distribute-lft-neg-in 89.6%

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

      5. fma-def 89.7%

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

   3. Applied egg-rr 89.7%

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

   if 1.47999999999999999e69 < y.re

   1. Initial program 52.1%

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

   2. Taylor expanded in y.re around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. unsub-neg 81.5%

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

      3. unpow2 81.5%

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

      4. associate-/r* 83.5%

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

   4. Simplified 83.5%

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

   5. Taylor expanded in x.re around 0 81.5%

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

      1. unpow2 81.5%

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

      2. times-frac 82.0%

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

   7. Simplified 82.0%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 85.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.48 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.9e+46)
     (- (/ x.im y.re) (/ y.im (/ (* y.re y.re) x.re)))
     (if (<= y.re -1.1e-98)
       t_0
       (if (<= y.re 8.4e-137)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (if (<= y.re 4.9e+69)
           t_0
           (/ (- x.im (* y.im (/ x.re y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.9e+46) {
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	} else if (y_46_re <= -1.1e-98) {
		tmp = t_0;
	} else if (y_46_re <= 8.4e-137) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 4.9e+69) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.9d+46)) then
        tmp = (x_46im / y_46re) - (y_46im / ((y_46re * y_46re) / x_46re))
    else if (y_46re <= (-1.1d-98)) then
        tmp = t_0
    else if (y_46re <= 8.4d-137) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= 4.9d+69) then
        tmp = t_0
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.9e+46) {
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	} else if (y_46_re <= -1.1e-98) {
		tmp = t_0;
	} else if (y_46_re <= 8.4e-137) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 4.9e+69) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.9e+46:
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re))
	elif y_46_re <= -1.1e-98:
		tmp = t_0
	elif y_46_re <= 8.4e-137:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= 4.9e+69:
		tmp = t_0
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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)))
	tmp = 0.0
	if (y_46_re <= -2.9e+46)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_re)));
	elseif (y_46_re <= -1.1e-98)
		tmp = t_0;
	elseif (y_46_re <= 8.4e-137)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 4.9e+69)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.9e+46)
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	elseif (y_46_re <= -1.1e-98)
		tmp = t_0;
	elseif (y_46_re <= 8.4e-137)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 4.9e+69)
		tmp = t_0;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = 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, -2.9e+46], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.1e-98], t$95$0, If[LessEqual[y$46$re, 8.4e-137], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.9e+69], t$95$0, N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.9000000000000002e46

   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. Taylor expanded in y.re around inf 89.7%

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

      1. mul-1-neg 89.7%

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

      2. unsub-neg 89.7%

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

      3. unpow2 89.7%

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

      4. associate-/r* 87.8%

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

   4. Simplified 87.8%

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

   5. Taylor expanded in x.re around 0 89.7%

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

      1. unpow2 89.7%

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

      2. times-frac 87.9%

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

   7. Simplified 87.9%

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

      1. *-commutative 87.9%

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

      2. frac-2neg 87.9%

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

      3. frac-times 89.7%

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

      4. add-sqr-sqrt 40.5%

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

      5. sqrt-unprod 71.7%

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

      6. sqr-neg 71.7%

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

      7. sqrt-unprod 39.9%

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

      8. add-sqr-sqrt 80.4%

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

      9. frac-times 76.4%

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

      10. frac-2neg 76.4%

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

      11. times-frac 80.4%

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

      12. associate-/l* 80.4%

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

      13. add-sqr-sqrt 39.9%

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

      14. sqrt-unprod 71.8%

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

      15. sqr-neg 71.8%

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

      16. sqrt-unprod 40.5%

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

      17. add-sqr-sqrt 89.8%

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

   9. Applied egg-rr 89.8%

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

   if -2.9000000000000002e46 < y.re < -1.09999999999999998e-98 or 8.39999999999999967e-137 < y.re < 4.9e69

   1. Initial program 86.4%

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

   if -1.09999999999999998e-98 < y.re < 8.39999999999999967e-137

   1. Initial program 67.9%

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

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

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. unpow2 83.5%

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

      5. times-frac 91.8%

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

   4. Simplified 91.8%

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

   if 4.9e69 < y.re

   1. Initial program 52.1%

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

   2. Taylor expanded in y.re around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. unsub-neg 81.5%

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

      3. unpow2 81.5%

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

      4. associate-/r* 83.5%

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

   4. Simplified 83.5%

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

   5. Taylor expanded in x.re around 0 81.5%

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

      1. unpow2 81.5%

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

      2. times-frac 82.0%

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

   7. Simplified 82.0%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 85.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -65000 \lor \neg \left(y.re \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -65000.0) (not (<= y.re 6e-23)))
   (/ (- 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 <= -65000.0) || !(y_46_re <= 6e-23)) {
		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 <= (-65000.0d0)) .or. (.not. (y_46re <= 6d-23))) 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 <= -65000.0) || !(y_46_re <= 6e-23)) {
		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 <= -65000.0) or not (y_46_re <= 6e-23):
		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 <= -65000.0) || !(y_46_re <= 6e-23))
		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 <= -65000.0) || ~((y_46_re <= 6e-23)))
		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, -65000.0], N[Not[LessEqual[y$46$re, 6e-23]], $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 < -65000 or 6.00000000000000006e-23 < y.re

   1. Initial program 61.0%

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

   2. Taylor expanded in y.re around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. unsub-neg 81.4%

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

      3. unpow2 81.4%

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

      4. associate-/r* 81.5%

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

   4. Simplified 81.5%

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

   5. Taylor expanded in x.re around 0 81.4%

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

      1. unpow2 81.4%

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

      2. times-frac 80.9%

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

   7. Simplified 80.9%

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

      1. associate-*r/ 82.4%

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

      2. sub-div 82.4%

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

   9. Applied egg-rr 82.4%

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

   if -65000 < y.re < 6.00000000000000006e-23

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -65000 \lor \neg \left(y.re \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -86000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -86000.0)
   (- (/ x.im y.re) (/ y.im (/ (* y.re y.re) x.re)))
   (if (<= y.re 2.1e-21)
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
     (/ (- x.im (* y.im (/ x.re y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -86000.0) {
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	} else if (y_46_re <= 2.1e-21) {
		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_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-86000.0d0)) then
        tmp = (x_46im / y_46re) - (y_46im / ((y_46re * y_46re) / x_46re))
    else if (y_46re <= 2.1d-21) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -86000.0) {
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	} else if (y_46_re <= 2.1e-21) {
		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_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -86000.0:
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re))
	elif y_46_re <= 2.1e-21:
		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_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -86000.0)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_re)));
	elseif (y_46_re <= 2.1e-21)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -86000.0)
		tmp = (x_46_im / y_46_re) - (y_46_im / ((y_46_re * y_46_re) / x_46_re));
	elseif (y_46_re <= 2.1e-21)
		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_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -86000.0], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-21], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -86000

   1. Initial program 62.3%

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

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

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

      1. mul-1-neg 86.4%

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

      2. unsub-neg 86.4%

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

      3. unpow2 86.4%

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

      4. associate-/r* 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in x.re around 0 86.4%

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

      1. unpow2 86.4%

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

      2. times-frac 84.9%

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

   7. Simplified 84.9%

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

      1. *-commutative 84.9%

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

      2. frac-2neg 84.9%

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

      3. frac-times 86.4%

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

      4. add-sqr-sqrt 37.9%

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

      5. sqrt-unprod 67.8%

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

      6. sqr-neg 67.8%

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

      7. sqrt-unprod 37.3%

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

      8. add-sqr-sqrt 74.8%

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

      9. frac-times 71.5%

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

      10. frac-2neg 71.5%

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

      11. times-frac 74.8%

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

      12. associate-/l* 74.9%

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

      13. add-sqr-sqrt 37.3%

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

      14. sqrt-unprod 67.8%

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

      15. sqr-neg 67.8%

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

      16. sqrt-unprod 37.9%

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

      17. add-sqr-sqrt 86.5%

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

   9. Applied egg-rr 86.5%

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

   if -86000 < y.re < 2.10000000000000013e-21

   1. Initial program 73.2%

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

   2. Taylor expanded in y.re around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 81.5%

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

   4. Simplified 81.5%

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

   if 2.10000000000000013e-21 < y.re

   1. Initial program 60.0%

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

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

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

      1. mul-1-neg 77.1%

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

      2. unsub-neg 77.1%

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

      3. unpow2 77.1%

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

      4. associate-/r* 78.7%

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

   4. Simplified 78.7%

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

   5. Taylor expanded in x.re around 0 77.1%

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

      1. unpow2 77.1%

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

      2. times-frac 77.5%

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

   7. Simplified 77.5%

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

      1. associate-*r/ 80.2%

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

      2. sub-div 80.2%

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -86000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re \cdot y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-22} \lor \neg \left(y.re \leq 6 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.5e-22) (not (<= y.re 6e-69)))
   (/ (- x.im (* y.im (/ x.re y.re))) 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 <= -4.5e-22) || !(y_46_re <= 6e-69)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / 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 <= (-4.5d-22)) .or. (.not. (y_46re <= 6d-69))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / 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 <= -4.5e-22) || !(y_46_re <= 6e-69)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / 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 <= -4.5e-22) or not (y_46_re <= 6e-69):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / 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 <= -4.5e-22) || !(y_46_re <= 6e-69))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / 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 <= -4.5e-22) || ~((y_46_re <= 6e-69)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / 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, -4.5e-22], N[Not[LessEqual[y$46$re, 6e-69]], $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[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.49999999999999987e-22 or 5.99999999999999978e-69 < y.re

   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 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

      3. unpow2 77.4%

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

      4. associate-/r* 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in x.re around 0 77.4%

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

      1. unpow2 77.4%

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

      2. times-frac 77.0%

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

   7. Simplified 77.0%

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

      1. associate-*r/ 78.3%

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

      2. sub-div 78.3%

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

   9. Applied egg-rr 78.3%

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

   if -4.49999999999999987e-22 < y.re < 5.99999999999999978e-69

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-22} \lor \neg \left(y.re \leq 6 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 8: 62.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.6e-14)
   (/ x.im y.re)
   (if (<= y.re 5.4e-68) (/ (- 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 <= -7.6e-14) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.4e-68) {
		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 <= (-7.6d-14)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 5.4d-68) 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 <= -7.6e-14) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 5.4e-68) {
		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 <= -7.6e-14:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 5.4e-68:
		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 <= -7.6e-14)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 5.4e-68)
		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 <= -7.6e-14)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 5.4e-68)
		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, -7.6e-14], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.4e-68], 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 < -7.6000000000000004e-14 or 5.4000000000000003e-68 < y.re

   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 65.8%

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

   if -7.6000000000000004e-14 < y.re < 5.4000000000000003e-68

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9: 46.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+191}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.2e+90)
   (/ x.re y.im)
   (if (<= y.im 2.85e+191) (/ 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 <= -4.2e+90) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 2.85e+191) {
		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 <= (-4.2d+90)) then
        tmp = x_46re / y_46im
    else if (y_46im <= 2.85d+191) 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 <= -4.2e+90) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 2.85e+191) {
		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 <= -4.2e+90:
		tmp = x_46_re / y_46_im
	elif y_46_im <= 2.85e+191:
		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 <= -4.2e+90)
		tmp = Float64(x_46_re / y_46_im);
	elseif (y_46_im <= 2.85e+191)
		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 <= -4.2e+90)
		tmp = x_46_re / y_46_im;
	elseif (y_46_im <= 2.85e+191)
		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, -4.2e+90], N[(x$46$re / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.85e+191], 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 < -4.19999999999999961e90 or 2.85000000000000001e191 < y.im

   1. Initial program 46.8%

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

      1. *-un-lft-identity 46.8%

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

      2. add-sqr-sqrt 46.8%

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

      3. times-frac 46.8%

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

      4. hypot-def 46.8%

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

      5. hypot-def 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. Step-by-step derivation

      1. div-sub 70.1%

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

   5. Applied egg-rr 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-/l* 76.8%

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

      3. associate-/l* 99.7%

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

   7. Simplified 99.7%

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

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

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

   9. Taylor expanded in y.im around -inf 42.4%

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

   if -4.19999999999999961e90 < y.im < 2.85000000000000001e191

   1. Initial program 73.5%

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

   2. Taylor expanded in y.re around inf 50.7%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+191}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 10: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

2. Taylor expanded in y.re around inf 42.8%

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

3. Final simplification 42.8%

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

Reproduce

herbie shell --seed 2023178 
(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))))