_divideComplex, imaginary part

Percentage Accurate: 61.8% → 89.2%

Time: 18.8s

Alternatives: 14

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ 1.0 (hypot y.re y.im)) (* y.re (/ x.im (hypot y.re y.im))))
          (/ y.im (/ (pow (hypot y.re y.im) 2.0) x.re))))
        (t_1 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -1.35e+154)
     t_1
     (if (<= y.im -1.65e-106)
       t_0
       (if (<= y.im 3.1e-107)
         (/ (- x.im (* y.im (/ x.re y.re))) y.re)
         (if (<= y.im 1.35e+154) 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 = ((1.0 / hypot(y_46_re, y_46_im)) * (y_46_re * (x_46_im / hypot(y_46_re, y_46_im)))) - (y_46_im / (pow(hypot(y_46_re, y_46_im), 2.0) / x_46_re));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.35e+154) {
		tmp = t_1;
	} else if (y_46_im <= -1.65e-106) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-107) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((1.0 / Math.hypot(y_46_re, y_46_im)) * (y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im)))) - (y_46_im / (Math.pow(Math.hypot(y_46_re, y_46_im), 2.0) / x_46_re));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.35e+154) {
		tmp = t_1;
	} else if (y_46_im <= -1.65e-106) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-107) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.35e+154) {
		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 = ((1.0 / math.hypot(y_46_re, y_46_im)) * (y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im)))) - (y_46_im / (math.pow(math.hypot(y_46_re, y_46_im), 2.0) / x_46_re))
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.35e+154:
		tmp = t_1
	elif y_46_im <= -1.65e-106:
		tmp = t_0
	elif y_46_im <= 3.1e-107:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im)))) - Float64(y_46_im / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / x_46_re)))
	t_1 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.35e+154)
		tmp = t_1;
	elseif (y_46_im <= -1.65e-106)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-107)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.35e+154)
		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 = ((1.0 / hypot(y_46_re, y_46_im)) * (y_46_re * (x_46_im / hypot(y_46_re, y_46_im)))) - (y_46_im / ((hypot(y_46_re, y_46_im) ^ 2.0) / x_46_re));
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.35e+154)
		tmp = t_1;
	elseif (y_46_im <= -1.65e-106)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-107)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$46$im / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.35e+154], t$95$1, If[LessEqual[y$46$im, -1.65e-106], t$95$0, If[LessEqual[y$46$im, 3.1e-107], 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$46$im, 1.35e+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.35000000000000003e154 or 1.35000000000000003e154 < y.im

   1. Initial program 33.6%

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

   2. Taylor expanded in y.re around 0 83.3%

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

      1. fma-def 83.3%

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

      2. unpow2 83.3%

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

      3. associate-/l* 83.7%

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

   4. Simplified 83.7%

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

      1. expm1-log1p-u 37.5%

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

      2. expm1-udef 37.5%

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

      3. associate-/l* 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. expm1-def 39.0%

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

      2. expm1-log1p 85.3%

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

      3. associate-/r/ 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in x.re around 0 83.3%

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

       1. +-commutative 83.3%

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

       2. *-commutative 83.3%

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

       3. unpow2 83.3%

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

       4. times-frac 89.8%

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

       5. mul-1-neg 89.8%

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

       6. unsub-neg 89.8%

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

   11. Simplified 89.8%

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

   if -1.35000000000000003e154 < y.im < -1.65000000000000008e-106 or 3.10000000000000022e-107 < y.im < 1.35000000000000003e154

   1. Initial program 69.4%

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

      1. div-sub 69.4%

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

      2. *-un-lft-identity 69.4%

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

      3. add-sqr-sqrt 69.4%

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

      4. times-frac 69.5%

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

      5. fma-neg 69.5%

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

      6. hypot-def 69.5%

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

      7. hypot-def 74.7%

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

      8. associate-/l* 81.6%

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

      9. add-sqr-sqrt 81.6%

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

      10. pow2 81.6%

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

      11. hypot-def 81.6%

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

   3. Applied egg-rr 81.6%

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

      1. fma-neg 81.6%

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

      2. associate-/l* 94.9%

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

      3. associate-/r/ 94.8%

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

      4. associate-/l* 87.9%

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

      5. *-commutative 87.9%

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

      6. associate-/l* 93.9%

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

   5. Simplified 93.9%

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

   if -1.65000000000000008e-106 < y.im < 3.10000000000000022e-107

   1. Initial program 66.8%

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

   2. Taylor expanded in y.re around inf 82.8%

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

      1. +-commutative 82.8%

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

      2. mul-1-neg 82.8%

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

      3. unsub-neg 82.8%

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

      4. unpow2 82.8%

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

      5. associate-/l* 86.8%

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

      6. associate-/r/ 84.3%

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

   4. Simplified 84.3%

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

   5. Taylor expanded in x.im around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. mul-1-neg 82.8%

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

      3. unsub-neg 82.8%

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

      4. unpow2 82.8%

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

      5. times-frac 93.0%

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

   7. Simplified 93.0%

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

      1. associate-*r/ 93.6%

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

      2. sub-div 94.9%

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

   9. Applied egg-rr 94.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1.18 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{1}{\frac{y.im}{\frac{y.re}{y.im} \cdot x.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im)))))
   (if (<= y.re -1.15e+115)
     (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
     (if (<= y.re -1.18e-169)
       t_0
       (if (<= y.re 50000.0)
         (fma -1.0 (/ x.re y.im) (/ 1.0 (/ y.im (* (/ y.re y.im) x.im))))
         (if (<= y.re 1.5e+136)
           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 = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -1.15e+115) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1.18e-169) {
		tmp = t_0;
	} else if (y_46_re <= 50000.0) {
		tmp = fma(-1.0, (x_46_re / y_46_im), (1.0 / (y_46_im / ((y_46_re / y_46_im) * x_46_im))));
	} else if (y_46_re <= 1.5e+136) {
		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(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.15e+115)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -1.18e-169)
		tmp = t_0;
	elseif (y_46_re <= 50000.0)
		tmp = fma(-1.0, Float64(x_46_re / y_46_im), Float64(1.0 / Float64(y_46_im / Float64(Float64(y_46_re / y_46_im) * x_46_im))));
	elseif (y_46_re <= 1.5e+136)
		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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+115], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.18e-169], t$95$0, If[LessEqual[y$46$re, 50000.0], N[(-1.0 * N[(x$46$re / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+136], 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 < -1.15000000000000002e115

   1. Initial program 27.9%

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

   2. Taylor expanded in y.re around inf 88.1%

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

      1. +-commutative 88.1%

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

      2. mul-1-neg 88.1%

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

      3. unsub-neg 88.1%

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

      4. unpow2 88.1%

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

      5. associate-/l* 91.4%

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

      6. associate-/r/ 91.4%

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

   4. Simplified 91.4%

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

   if -1.15000000000000002e115 < y.re < -1.18e-169 or 5e4 < y.re < 1.49999999999999989e136

   1. Initial program 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. add-sqr-sqrt 79.1%

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

      3. times-frac 79.2%

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

      4. hypot-def 79.2%

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

      5. hypot-def 88.8%

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

   3. Applied egg-rr 88.8%

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

   if -1.18e-169 < y.re < 5e4

   1. Initial program 70.8%

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

   2. Taylor expanded in y.re around 0 93.2%

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

      1. fma-def 93.2%

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

      2. unpow2 93.2%

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

      3. associate-/l* 90.2%

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

   4. Simplified 90.2%

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

      1. clear-num 90.1%

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

      2. inv-pow 90.1%

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

      3. associate-/l* 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. unpow-1 91.3%

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

      2. associate-/l/ 92.4%

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

   8. Simplified 92.4%

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

   if 1.49999999999999989e136 < y.re

   1. Initial program 23.9%

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

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

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/l* 79.1%

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

      6. associate-/r/ 79.1%

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

   4. Simplified 79.1%

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

   5. Taylor expanded in x.im around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. times-frac 91.9%

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

   7. Simplified 91.9%

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

      1. associate-*r/ 92.0%

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

      2. sub-div 92.0%

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

   9. Applied egg-rr 92.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1.18 \cdot 10^{-169}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{1}{\frac{y.im}{\frac{y.re}{y.im} \cdot x.im}}\right)\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 3: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{1}{\frac{y.im}{\frac{y.re}{y.im} \cdot x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.5e+102)
   (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
   (if (<= y.re -1.4e-77)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 2.8e+51)
       (fma -1.0 (/ x.re y.im) (/ 1.0 (/ y.im (* (/ y.re y.im) x.im))))
       (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+102) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1.4e-77) {
		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 <= 2.8e+51) {
		tmp = fma(-1.0, (x_46_re / y_46_im), (1.0 / (y_46_im / ((y_46_re / y_46_im) * x_46_im))));
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.5e+102)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -1.4e-77)
		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 <= 2.8e+51)
		tmp = fma(-1.0, Float64(x_46_re / y_46_im), Float64(1.0 / Float64(y_46_im / Float64(Float64(y_46_re / y_46_im) * x_46_im))));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.5e+102], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.4e-77], 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, 2.8e+51], N[(-1.0 * N[(x$46$re / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.4999999999999996e102

   1. Initial program 31.2%

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

   2. Taylor expanded in y.re around inf 86.5%

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

      1. +-commutative 86.5%

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

      2. mul-1-neg 86.5%

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

      3. unsub-neg 86.5%

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

      4. unpow2 86.5%

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

      5. associate-/l* 89.5%

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

      6. associate-/r/ 89.4%

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

   4. Simplified 89.4%

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

   if -8.4999999999999996e102 < y.re < -1.4e-77

   1. Initial program 90.9%

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

   if -1.4e-77 < y.re < 2.80000000000000005e51

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around 0 86.5%

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

      1. fma-def 86.5%

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

      2. unpow2 86.5%

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

      3. associate-/l* 84.4%

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

   4. Simplified 84.4%

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

      1. clear-num 84.4%

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

      2. inv-pow 84.4%

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

      3. associate-/l* 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. unpow-1 85.4%

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

      2. associate-/l/ 86.9%

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

   8. Simplified 86.9%

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

   if 2.80000000000000005e51 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{1}{\frac{y.im}{\frac{y.re}{y.im} \cdot x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 4: 77.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -9.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{x.im}{y.im \cdot \frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.05e+103)
   (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
   (if (<= y.re -9.4e-77)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 3.7e+49)
       (fma -1.0 (/ x.re y.im) (/ x.im (* y.im (/ y.im y.re))))
       (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.05e+103) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -9.4e-77) {
		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 <= 3.7e+49) {
		tmp = fma(-1.0, (x_46_re / y_46_im), (x_46_im / (y_46_im * (y_46_im / y_46_re))));
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.05e+103)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -9.4e-77)
		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 <= 3.7e+49)
		tmp = fma(-1.0, Float64(x_46_re / y_46_im), Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.05e+103], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9.4e-77], 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, 3.7e+49], N[(-1.0 * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.0500000000000001e103

   1. Initial program 31.2%

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

   2. Taylor expanded in y.re around inf 86.5%

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

      1. +-commutative 86.5%

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

      2. mul-1-neg 86.5%

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

      3. unsub-neg 86.5%

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

      4. unpow2 86.5%

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

      5. associate-/l* 89.5%

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

      6. associate-/r/ 89.4%

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

   4. Simplified 89.4%

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

   if -1.0500000000000001e103 < y.re < -9.3999999999999998e-77

   1. Initial program 90.9%

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

   if -9.3999999999999998e-77 < y.re < 3.70000000000000018e49

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around 0 86.5%

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

      1. fma-def 86.5%

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

      2. unpow2 86.5%

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

      3. associate-/l* 84.4%

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

   4. Simplified 84.4%

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

      1. expm1-log1p-u 47.7%

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

      2. expm1-udef 42.5%

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

      3. associate-/l* 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. expm1-def 48.6%

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

      2. expm1-log1p 85.4%

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

      3. associate-/r/ 85.3%

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

   8. Simplified 85.3%

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

   if 3.70000000000000018e49 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -9.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{x.im}{y.im \cdot \frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 5: 77.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{x.im}{\frac{y.im}{\frac{y.re}{y.im}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8e+102)
   (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
   (if (<= y.re -2.7e-77)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.02e+49)
       (fma -1.0 (/ x.re y.im) (/ x.im (/ y.im (/ y.re y.im))))
       (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e+102) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -2.7e-77) {
		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 <= 1.02e+49) {
		tmp = fma(-1.0, (x_46_re / y_46_im), (x_46_im / (y_46_im / (y_46_re / y_46_im))));
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8e+102)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -2.7e-77)
		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 <= 1.02e+49)
		tmp = fma(-1.0, Float64(x_46_re / y_46_im), Float64(x_46_im / Float64(y_46_im / Float64(y_46_re / y_46_im))));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8e+102], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.7e-77], 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, 1.02e+49], N[(-1.0 * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / N[(y$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.99999999999999982e102

   1. Initial program 31.2%

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

   2. Taylor expanded in y.re around inf 86.5%

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

      1. +-commutative 86.5%

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

      2. mul-1-neg 86.5%

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

      3. unsub-neg 86.5%

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

      4. unpow2 86.5%

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

      5. associate-/l* 89.5%

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

      6. associate-/r/ 89.4%

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

   4. Simplified 89.4%

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

   if -7.99999999999999982e102 < y.re < -2.7e-77

   1. Initial program 90.9%

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

   if -2.7e-77 < y.re < 1.02e49

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around 0 86.5%

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

      1. fma-def 86.5%

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

      2. unpow2 86.5%

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

      3. associate-/l* 84.4%

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

   4. Simplified 84.4%

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

      1. expm1-log1p-u 47.7%

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

      2. expm1-udef 42.5%

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

      3. associate-/l* 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. expm1-def 48.6%

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

      2. expm1-log1p 85.4%

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

      3. associate-/r/ 85.3%

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

   8. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. clear-num 85.4%

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

      3. un-div-inv 85.4%

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

   10. Applied egg-rr 85.4%

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

   if 1.02e49 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{x.im}{\frac{y.im}{\frac{y.re}{y.im}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 6: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+54}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.8e+104)
   (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
   (if (<= y.re -5.4e-78)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.75e+54)
       (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
       (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e+104) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -5.4e-78) {
		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 <= 1.75e+54) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.8d+104)) then
        tmp = (x_46im / y_46re) - (y_46im * (x_46re / (y_46re * y_46re)))
    else if (y_46re <= (-5.4d-78)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1.75d+54) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e+104) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= -5.4e-78) {
		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 <= 1.75e+54) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.8e+104:
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)))
	elif y_46_re <= -5.4e-78:
		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 <= 1.75e+54:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.8e+104)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -5.4e-78)
		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 <= 1.75e+54)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.8e+104)
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	elseif (y_46_re <= -5.4e-78)
		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 <= 1.75e+54)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.8e+104], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.4e-78], 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, 1.75e+54], 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[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.8e104

   1. Initial program 31.2%

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

   2. Taylor expanded in y.re around inf 86.5%

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

      1. +-commutative 86.5%

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

      2. mul-1-neg 86.5%

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

      3. unsub-neg 86.5%

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

      4. unpow2 86.5%

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

      5. associate-/l* 89.5%

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

      6. associate-/r/ 89.4%

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

   4. Simplified 89.4%

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

   if -2.8e104 < y.re < -5.39999999999999987e-78

   1. Initial program 90.9%

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

   if -5.39999999999999987e-78 < y.re < 1.7500000000000001e54

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around 0 86.5%

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

      1. fma-def 86.5%

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

      2. unpow2 86.5%

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

      3. associate-/l* 84.4%

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

   4. Simplified 84.4%

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

      1. expm1-log1p-u 47.7%

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

      2. expm1-udef 42.5%

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

      3. associate-/l* 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. expm1-def 48.6%

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

      2. expm1-log1p 85.4%

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

      3. associate-/r/ 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in x.re around 0 86.5%

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

       1. +-commutative 86.5%

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

       2. *-commutative 86.5%

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

       3. unpow2 86.5%

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

       4. times-frac 84.6%

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

       5. mul-1-neg 84.6%

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

       6. unsub-neg 84.6%

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

   11. Simplified 84.6%

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

   if 1.7500000000000001e54 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+54}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 7: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-47} \lor \neg \left(y.re \leq 2.35 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{x.im}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.1e-47) (not (<= y.re 2.35e+51)))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (- (* y.re (/ x.im (* y.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 <= -1.1e-47) || !(y_46_re <= 2.35e+51)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (y_46_re * (x_46_im / (y_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 <= (-1.1d-47)) .or. (.not. (y_46re <= 2.35d+51))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = (y_46re * (x_46im / (y_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 <= -1.1e-47) || !(y_46_re <= 2.35e+51)) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = (y_46_re * (x_46_im / (y_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 <= -1.1e-47) or not (y_46_re <= 2.35e+51):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = (y_46_re * (x_46_im / (y_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 <= -1.1e-47) || !(y_46_re <= 2.35e+51))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(y_46_re * Float64(x_46_im / Float64(y_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 <= -1.1e-47) || ~((y_46_re <= 2.35e+51)))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = (y_46_re * (x_46_im / (y_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, -1.1e-47], N[Not[LessEqual[y$46$re, 2.35e+51]], $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[(y$46$re * N[(x$46$im / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.10000000000000009e-47 or 2.3500000000000001e51 < y.re

   1. Initial program 48.9%

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

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

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 77.3%

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

      6. associate-/r/ 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in x.im around 0 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 81.6%

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

   7. Simplified 81.6%

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

      1. associate-*r/ 81.9%

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

      2. sub-div 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.10000000000000009e-47 < y.re < 2.3500000000000001e51

   1. Initial program 70.6%

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

   2. Taylor expanded in y.re around 0 85.4%

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

      1. +-commutative 85.4%

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

      2. mul-1-neg 85.4%

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

      3. unsub-neg 85.4%

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

      4. unpow2 85.4%

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

      5. associate-/l* 83.4%

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

      6. associate-/r/ 82.5%

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

   4. Simplified 82.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-47} \lor \neg \left(y.re \leq 2.35 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{x.im}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.18 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.16e-47) (not (<= y.re 1.18e+51)))
   (/ (- 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 <= -1.16e-47) || !(y_46_re <= 1.18e+51)) {
		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 <= (-1.16d-47)) .or. (.not. (y_46re <= 1.18d+51))) 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 <= -1.16e-47) || !(y_46_re <= 1.18e+51)) {
		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 <= -1.16e-47) or not (y_46_re <= 1.18e+51):
		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 <= -1.16e-47) || !(y_46_re <= 1.18e+51))
		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 <= -1.16e-47) || ~((y_46_re <= 1.18e+51)))
		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, -1.16e-47], N[Not[LessEqual[y$46$re, 1.18e+51]], $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 < -1.1600000000000001e-47 or 1.18e51 < y.re

   1. Initial program 48.9%

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

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

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 77.3%

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

      6. associate-/r/ 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in x.im around 0 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 81.6%

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

   7. Simplified 81.6%

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

      1. associate-*r/ 81.9%

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

      2. sub-div 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.1600000000000001e-47 < y.re < 1.18e51

   1. Initial program 70.6%

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

   2. Taylor expanded in y.re around 0 85.4%

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

      1. fma-def 85.4%

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

      2. unpow2 85.4%

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

      3. associate-/l* 83.4%

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

   4. Simplified 83.4%

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

      1. expm1-log1p-u 46.2%

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

      2. expm1-udef 41.2%

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

      3. associate-/l* 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 84.3%

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

      3. associate-/r/ 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in x.re around 0 85.4%

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

       1. +-commutative 85.4%

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

       2. *-commutative 85.4%

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

       3. unpow2 85.4%

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

       4. times-frac 83.6%

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

       5. mul-1-neg 83.6%

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

       6. unsub-neg 83.6%

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

   11. Simplified 83.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.18 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 9: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.16e-47)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 7.2e+48)
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
     (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e-47) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 7.2e+48) {
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.16d-47)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 7.2d+48) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e-47) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 7.2e+48) {
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.16e-47:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 7.2e+48:
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.16e-47)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 7.2e+48)
		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(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.16e-47)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 7.2e+48)
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.16e-47], 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, 7.2e+48], 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[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1600000000000001e-47

   1. Initial program 58.8%

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

   2. Taylor expanded in y.re around inf 79.3%

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

      1. +-commutative 79.3%

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

      2. mul-1-neg 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-/l* 79.5%

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

      6. associate-/r/ 79.4%

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

   4. Simplified 79.4%

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

   5. Taylor expanded in x.im around 0 79.3%

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

      1. +-commutative 79.3%

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

      2. mul-1-neg 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. times-frac 77.4%

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

   7. Simplified 77.4%

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

      1. associate-*r/ 77.9%

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

      2. sub-div 77.9%

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

   9. Applied egg-rr 77.9%

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

   if -1.1600000000000001e-47 < y.re < 7.19999999999999967e48

   1. Initial program 70.6%

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

   2. Taylor expanded in y.re around 0 85.4%

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

      1. fma-def 85.4%

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

      2. unpow2 85.4%

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

      3. associate-/l* 83.4%

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

   4. Simplified 83.4%

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

      1. expm1-log1p-u 46.2%

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

      2. expm1-udef 41.2%

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

      3. associate-/l* 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 84.3%

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

      3. associate-/r/ 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in x.re around 0 85.4%

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

       1. +-commutative 85.4%

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

       2. *-commutative 85.4%

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

       3. unpow2 85.4%

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

       4. times-frac 83.6%

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

       5. mul-1-neg 83.6%

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

       6. unsub-neg 83.6%

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

   11. Simplified 83.6%

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

   if 7.19999999999999967e48 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 10: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.16e-47)
   (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
   (if (<= y.re 4.4e+52)
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
     (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e-47) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= 4.4e+52) {
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.16d-47)) then
        tmp = (x_46im / y_46re) - (y_46im * (x_46re / (y_46re * y_46re)))
    else if (y_46re <= 4.4d+52) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - ((x_46re / y_46re) * (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e-47) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= 4.4e+52) {
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.16e-47:
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)))
	elif y_46_re <= 4.4e+52:
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.16e-47)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 4.4e+52)
		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(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.16e-47)
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	elseif (y_46_re <= 4.4e+52)
		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) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.16e-47], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.4e+52], 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[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1600000000000001e-47

   1. Initial program 58.8%

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

   2. Taylor expanded in y.re around inf 79.3%

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

      1. +-commutative 79.3%

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

      2. mul-1-neg 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-/l* 79.5%

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

      6. associate-/r/ 79.4%

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

   4. Simplified 79.4%

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

   if -1.1600000000000001e-47 < y.re < 4.4e52

   1. Initial program 70.6%

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

   2. Taylor expanded in y.re around 0 85.4%

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

      1. fma-def 85.4%

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

      2. unpow2 85.4%

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

      3. associate-/l* 83.4%

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

   4. Simplified 83.4%

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

      1. expm1-log1p-u 46.2%

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

      2. expm1-udef 41.2%

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

      3. associate-/l* 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 84.3%

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

      3. associate-/r/ 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in x.re around 0 85.4%

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

       1. +-commutative 85.4%

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

       2. *-commutative 85.4%

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

       3. unpow2 85.4%

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

       4. times-frac 83.6%

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

       5. mul-1-neg 83.6%

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

       6. unsub-neg 83.6%

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

   11. Simplified 83.6%

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

   if 4.4e52 < y.re

   1. Initial program 39.0%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/l* 75.2%

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

      6. associate-/r/ 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x.im around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 11: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+17} \lor \neg \left(y.im \leq 4.6 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5e+17) (not (<= y.im 4.6e-17)))
   (/ (- x.re) (+ y.im (* y.re (/ y.re y.im))))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5e+17) || !(y_46_im <= 4.6e-17)) {
		tmp = -x_46_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5d+17)) .or. (.not. (y_46im <= 4.6d-17))) then
        tmp = -x_46re / (y_46im + (y_46re * (y_46re / y_46im)))
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5e+17) || !(y_46_im <= 4.6e-17)) {
		tmp = -x_46_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5e+17) or not (y_46_im <= 4.6e-17):
		tmp = -x_46_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)))
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5e+17) || !(y_46_im <= 4.6e-17))
		tmp = Float64(Float64(-x_46_re) / Float64(y_46_im + Float64(y_46_re * Float64(y_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_im <= -5e+17) || ~((y_46_im <= 4.6e-17)))
		tmp = -x_46_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5e+17], N[Not[LessEqual[y$46$im, 4.6e-17]], $MachinePrecision]], N[((-x$46$re) / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if y.im < -5e17 or 4.60000000000000018e-17 < y.im

   1. Initial program 51.2%

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

   2. Taylor expanded in x.im around 0 43.6%

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

      1. mul-1-neg 43.6%

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

      2. *-commutative 43.6%

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

      3. distribute-rgt-neg-in 43.6%

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

   4. Simplified 43.6%

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

   5. Taylor expanded in x.re around 0 43.6%

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

      1. mul-1-neg 43.6%

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

      2. associate-/l* 50.5%

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

      3. unpow2 50.5%

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

      4. unpow2 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in y.im around 0 71.5%

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

      1. unpow2 71.5%

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

      2. associate-*r/ 76.2%

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

   10. Simplified 76.2%

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

   if -5e17 < y.im < 4.60000000000000018e-17

   1. Initial program 69.4%

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

   2. Taylor expanded in y.re around inf 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. unsub-neg 78.4%

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

      4. unpow2 78.4%

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

      5. associate-/l* 80.2%

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

      6. associate-/r/ 79.4%

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

   4. Simplified 79.4%

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

   5. Taylor expanded in x.im around 0 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. unsub-neg 78.4%

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

      4. unpow2 78.4%

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

      5. times-frac 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*r/ 86.1%

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

      2. sub-div 87.0%

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

   9. Applied egg-rr 87.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+17} \lor \neg \left(y.im \leq 4.6 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 12: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-50} \lor \neg \left(y.re \leq 1.95 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.25e-50) (not (<= y.re 1.95e+49)))
   (/ (- 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 <= -1.25e-50) || !(y_46_re <= 1.95e+49)) {
		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 <= (-1.25d-50)) .or. (.not. (y_46re <= 1.95d+49))) 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 <= -1.25e-50) || !(y_46_re <= 1.95e+49)) {
		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 <= -1.25e-50) or not (y_46_re <= 1.95e+49):
		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 <= -1.25e-50) || !(y_46_re <= 1.95e+49))
		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 <= -1.25e-50) || ~((y_46_re <= 1.95e+49)))
		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, -1.25e-50], N[Not[LessEqual[y$46$re, 1.95e+49]], $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 < -1.24999999999999992e-50 or 1.95e49 < y.re

   1. Initial program 48.9%

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

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

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 77.3%

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

      6. associate-/r/ 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in x.im around 0 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 81.6%

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

   7. Simplified 81.6%

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

      1. associate-*r/ 81.9%

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

      2. sub-div 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.24999999999999992e-50 < y.re < 1.95e49

   1. Initial program 70.6%

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

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

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   4. Simplified 68.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-50} \lor \neg \left(y.re \leq 1.95 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 13: 63.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-14} \lor \neg \left(y.im \leq 4.7 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.2e-14) (not (<= y.im 4.7e-17)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e-14) || !(y_46_im <= 4.7e-17)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.2d-14)) .or. (.not. (y_46im <= 4.7d-17))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e-14) || !(y_46_im <= 4.7e-17)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.2e-14) or not (y_46_im <= 4.7e-17):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.2e-14) || !(y_46_im <= 4.7e-17))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.2e-14) || ~((y_46_im <= 4.7e-17)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.2e-14], N[Not[LessEqual[y$46$im, 4.7e-17]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1999999999999998e-14 or 4.7e-17 < y.im

   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 0 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   4. Simplified 66.7%

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

   if -4.1999999999999998e-14 < y.im < 4.7e-17

   1. Initial program 70.0%

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

   2. Taylor expanded in y.re around inf 73.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-14} \lor \neg \left(y.im \leq 4.7 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 14: 42.8% 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 59.6%

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

2. Taylor expanded in y.re around inf 45.4%

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

3. Final simplification 45.4%

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

Reproduce

herbie shell --seed 2023283 
(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))))