_divideComplex, imaginary part

Percentage Accurate: 62.0% → 77.4%

Time: 8.4s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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: 77.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1650:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.5e+49)
   (/ x.re (- (- y.im) (* y.re (/ y.re y.im))))
   (if (<= y.im 1650.0)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (/ (- (* y.re (/ x.im y.im)) x.re) (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+49) {
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1650.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+49) {
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1650.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.5e+49:
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1650.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+49)
		tmp = Float64(x_46_re / Float64(Float64(-y_46_im) - Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1650.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.5e+49)
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1650.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.5e+49], N[(x$46$re / N[((-y$46$im) - N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1650.0], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.5000000000000002e49

   1. Initial program 49.3%

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

   2. Taylor expanded in x.im around 0 40.2%

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

      1. associate-*r/ 40.2%

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

      2. mul-1-neg 40.2%

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

      3. distribute-rgt-neg-out 40.2%

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

      4. associate-/l* 52.1%

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

      5. unpow2 52.1%

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

      6. fma-def 52.1%

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

      7. unpow2 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in y.re around 0 79.4%

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

      1. distribute-lft-out 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r/ 84.3%

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

   7. Simplified 84.3%

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

   if -2.5000000000000002e49 < y.im < 1650

   1. Initial program 72.4%

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

      1. *-un-lft-identity 72.4%

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

      2. add-sqr-sqrt 72.4%

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

      3. times-frac 72.2%

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

      4. hypot-def 72.2%

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

      5. hypot-def 81.6%

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

   3. Applied egg-rr 81.6%

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

   4. Taylor expanded in y.re around inf 81.7%

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

      1. metadata-eval 81.7%

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

      2. unpow2 81.7%

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

      3. cancel-sign-sub-inv 81.7%

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

      4. *-commutative 81.7%

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

      5. *-rgt-identity 81.7%

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

      6. associate-/r* 88.6%

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

      7. associate-*r/ 88.5%

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

      8. div-sub 90.1%

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

   6. Simplified 90.1%

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

   if 1650 < y.im

   1. Initial program 45.4%

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

      1. *-un-lft-identity 45.4%

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

      2. add-sqr-sqrt 45.4%

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

      3. times-frac 45.5%

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

      4. hypot-def 45.5%

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

      5. hypot-def 62.8%

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

   3. Applied egg-rr 62.8%

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

   4. Taylor expanded in y.re around 0 83.2%

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

      1. neg-mul-1 83.2%

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

      2. unsub-neg 83.2%

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

      3. *-lft-identity 83.2%

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

      4. times-frac 85.9%

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

      5. /-rgt-identity 85.9%

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

   6. Simplified 85.9%

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

      1. expm1-log1p-u 73.0%

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

      2. expm1-udef 33.3%

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

      3. associate-*l/ 33.3%

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

      4. *-un-lft-identity 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. expm1-def 73.2%

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

      2. expm1-log1p 86.2%

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

   10. Simplified 86.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1650:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+49} \lor \neg \left(y.im \leq 1.46 \cdot 10^{+26}\right):\\ \;\;\;\;\left(y.re \cdot \frac{x.im}{y.im} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.15e+49) (not (<= y.im 1.46e+26)))
   (* (- (* y.re (/ x.im y.im)) x.re) (/ 1.0 y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.15e+49) || !(y_46_im <= 1.46e+26)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.15d+49)) .or. (.not. (y_46im <= 1.46d+26))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) * (1.0d0 / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.15e+49) || !(y_46_im <= 1.46e+26)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.15e+49) or not (y_46_im <= 1.46e+26):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) * (1.0 / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.15e+49) || !(y_46_im <= 1.46e+26))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.15e+49) || ~((y_46_im <= 1.46e+26)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) * (1.0 / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.15e+49], N[Not[LessEqual[y$46$im, 1.46e+26]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.15e49 or 1.45999999999999992e26 < y.im

   1. Initial program 46.0%

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

      1. *-un-lft-identity 46.0%

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

      2. add-sqr-sqrt 46.0%

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

      3. times-frac 46.1%

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

      4. hypot-def 46.1%

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

      5. hypot-def 65.8%

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

   3. Applied egg-rr 65.8%

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

   4. Taylor expanded in y.re around 0 56.9%

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

      1. neg-mul-1 56.9%

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

      2. unsub-neg 56.9%

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

      3. *-lft-identity 56.9%

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

      4. times-frac 58.6%

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

      5. /-rgt-identity 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in y.re around 0 86.2%

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

   if -2.15e49 < y.im < 1.45999999999999992e26

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.6%

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

      4. hypot-def 71.6%

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

      5. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

   4. Taylor expanded in y.re around inf 79.9%

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

      1. metadata-eval 79.9%

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

      2. unpow2 79.9%

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

      3. cancel-sign-sub-inv 79.9%

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

      4. *-commutative 79.9%

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

      5. *-rgt-identity 79.9%

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

      6. associate-/r* 86.4%

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

      7. associate-*r/ 86.4%

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

      8. div-sub 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+49} \lor \neg \left(y.im \leq 1.46 \cdot 10^{+26}\right):\\ \;\;\;\;\left(y.re \cdot \frac{x.im}{y.im} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 3: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+53} \lor \neg \left(y.im \leq 1.46 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7e+53) (not (<= y.im 1.46e+26)))
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e+53) || !(y_46_im <= 1.46e+26)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7d+53)) .or. (.not. (y_46im <= 1.46d+26))) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e+53) || !(y_46_im <= 1.46e+26)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7e+53) or not (y_46_im <= 1.46e+26):
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7e+53) || !(y_46_im <= 1.46e+26))
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7e+53) || ~((y_46_im <= 1.46e+26)))
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7e+53], N[Not[LessEqual[y$46$im, 1.46e+26]], $MachinePrecision]], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.00000000000000038e53 or 1.45999999999999992e26 < y.im

   1. Initial program 46.0%

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

   2. Taylor expanded in y.re around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. mul-1-neg 81.0%

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

      3. unsub-neg 81.0%

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

      4. unpow2 81.0%

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

      5. times-frac 86.5%

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

   4. Simplified 86.5%

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

   if -7.00000000000000038e53 < y.im < 1.45999999999999992e26

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.6%

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

      4. hypot-def 71.6%

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

      5. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

   4. Taylor expanded in y.re around inf 79.9%

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

      1. metadata-eval 79.9%

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

      2. unpow2 79.9%

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

      3. cancel-sign-sub-inv 79.9%

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

      4. *-commutative 79.9%

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

      5. *-rgt-identity 79.9%

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

      6. associate-/r* 86.4%

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

      7. associate-*r/ 86.4%

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

      8. div-sub 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+53} \lor \neg \left(y.im \leq 1.46 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re}{\frac{y.im \cdot y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.1e+50)
   (- (/ y.re (/ (* y.im y.im) x.im)) (/ x.re y.im))
   (if (<= y.im 1.3e+26)
     (/ (- x.im (* x.re (/ y.im 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_im <= -3.1e+50) {
		tmp = (y_46_re / ((y_46_im * y_46_im) / x_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 1.3e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_46im <= (-3.1d+50)) then
        tmp = (y_46re / ((y_46im * y_46im) / x_46im)) - (x_46re / y_46im)
    else if (y_46im <= 1.3d+26) then
        tmp = (x_46im - (x_46re * (y_46im / 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_im <= -3.1e+50) {
		tmp = (y_46_re / ((y_46_im * y_46_im) / x_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 1.3e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_im <= -3.1e+50:
		tmp = (y_46_re / ((y_46_im * y_46_im) / x_46_im)) - (x_46_re / y_46_im)
	elif y_46_im <= 1.3e+26:
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_im <= -3.1e+50)
		tmp = Float64(Float64(y_46_re / Float64(Float64(y_46_im * y_46_im) / x_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 1.3e+26)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / 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_im <= -3.1e+50)
		tmp = (y_46_re / ((y_46_im * y_46_im) / x_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_im <= 1.3e+26)
		tmp = (x_46_im - (x_46_re * (y_46_im / 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[LessEqual[y$46$im, -3.1e+50], N[(N[(y$46$re / N[(N[(y$46$im * y$46$im), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+26], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / 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 3 regimes

2. if y.im < -3.10000000000000003e50

   1. Initial program 49.3%

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

   2. Taylor expanded in y.re around 0 79.6%

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

      1. +-commutative 79.6%

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

      2. mul-1-neg 79.6%

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

      3. unsub-neg 79.6%

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

      4. associate-/l* 82.8%

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

      5. unpow2 82.8%

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

   4. Simplified 82.8%

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

   if -3.10000000000000003e50 < y.im < 1.30000000000000001e26

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.6%

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

      4. hypot-def 71.6%

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

      5. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

   4. Taylor expanded in y.re around inf 79.9%

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

      1. metadata-eval 79.9%

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

      2. unpow2 79.9%

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

      3. cancel-sign-sub-inv 79.9%

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

      4. *-commutative 79.9%

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

      5. *-rgt-identity 79.9%

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

      6. associate-/r* 86.4%

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

      7. associate-*r/ 86.4%

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

      8. div-sub 87.8%

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

   6. Simplified 87.8%

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

   if 1.30000000000000001e26 < y.im

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. mul-1-neg 82.0%

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

      3. unsub-neg 82.0%

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

      4. unpow2 82.0%

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

      5. times-frac 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re}{\frac{y.im \cdot y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9.5e+49)
   (/ x.re (- (- y.im) (* y.re (/ y.re y.im))))
   (if (<= y.im 1.25e+26)
     (/ (- x.im (* x.re (/ y.im 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_im <= -9.5e+49) {
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.25e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_46im <= (-9.5d+49)) then
        tmp = x_46re / (-y_46im - (y_46re * (y_46re / y_46im)))
    else if (y_46im <= 1.25d+26) then
        tmp = (x_46im - (x_46re * (y_46im / 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_im <= -9.5e+49) {
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.25e+26) {
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_im <= -9.5e+49:
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1.25e+26:
		tmp = (x_46_im - (x_46_re * (y_46_im / 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_im <= -9.5e+49)
		tmp = Float64(x_46_re / Float64(Float64(-y_46_im) - Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1.25e+26)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / 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_im <= -9.5e+49)
		tmp = x_46_re / (-y_46_im - (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1.25e+26)
		tmp = (x_46_im - (x_46_re * (y_46_im / 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[LessEqual[y$46$im, -9.5e+49], N[(x$46$re / N[((-y$46$im) - N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.25e+26], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / 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 3 regimes

2. if y.im < -9.49999999999999969e49

   1. Initial program 49.3%

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

   2. Taylor expanded in x.im around 0 40.2%

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

      1. associate-*r/ 40.2%

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

      2. mul-1-neg 40.2%

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

      3. distribute-rgt-neg-out 40.2%

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

      4. associate-/l* 52.1%

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

      5. unpow2 52.1%

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

      6. fma-def 52.1%

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

      7. unpow2 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in y.re around 0 79.4%

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

      1. distribute-lft-out 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r/ 84.3%

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

   7. Simplified 84.3%

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

   if -9.49999999999999969e49 < y.im < 1.25e26

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.6%

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

      4. hypot-def 71.6%

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

      5. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

   4. Taylor expanded in y.re around inf 79.9%

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

      1. metadata-eval 79.9%

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

      2. unpow2 79.9%

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

      3. cancel-sign-sub-inv 79.9%

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

      4. *-commutative 79.9%

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

      5. *-rgt-identity 79.9%

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

      6. associate-/r* 86.4%

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

      7. associate-*r/ 86.4%

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

      8. div-sub 87.8%

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

   6. Simplified 87.8%

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

   if 1.25e26 < y.im

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. mul-1-neg 82.0%

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

      3. unsub-neg 82.0%

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

      4. unpow2 82.0%

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

      5. times-frac 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+87} \lor \neg \left(y.im \leq 3.4 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.5e+87) (not (<= y.im 3.4e+21)))
   (/ (- x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.5e+87) || !(y_46_im <= 3.4e+21)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.5d+87)) .or. (.not. (y_46im <= 3.4d+21))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.5e+87) || !(y_46_im <= 3.4e+21)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.5e+87) or not (y_46_im <= 3.4e+21):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.5e+87) || !(y_46_im <= 3.4e+21))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.5e+87) || ~((y_46_im <= 3.4e+21)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.5e+87], N[Not[LessEqual[y$46$im, 3.4e+21]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.5000000000000003e87 or 3.4e21 < y.im

   1. Initial program 45.6%

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

   2. Taylor expanded in y.re around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. neg-mul-1 80.3%

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

   4. Simplified 80.3%

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

   if -4.5000000000000003e87 < y.im < 3.4e21

   1. Initial program 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. add-sqr-sqrt 71.0%

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

      3. times-frac 70.8%

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

      4. hypot-def 70.8%

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

      5. hypot-def 79.3%

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

   3. Applied egg-rr 79.3%

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

   4. Taylor expanded in y.re around inf 77.4%

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

      1. metadata-eval 77.4%

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

      2. unpow2 77.4%

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

      3. cancel-sign-sub-inv 77.4%

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

      4. *-commutative 77.4%

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

      5. *-rgt-identity 77.4%

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

      6. associate-/r* 83.6%

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

      7. associate-*r/ 84.9%

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

      8. div-sub 86.3%

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

   6. Simplified 86.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+87} \lor \neg \left(y.im \leq 3.4 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+48} \lor \neg \left(y.im \leq 1.65 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.8e+48) (not (<= y.im 1.65e+20)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.8e+48) || !(y_46_im <= 1.65e+20)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.8d+48)) .or. (.not. (y_46im <= 1.65d+20))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.8e+48) || !(y_46_im <= 1.65e+20)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.8e+48) or not (y_46_im <= 1.65e+20):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.8e+48) || !(y_46_im <= 1.65e+20))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.8e+48) || ~((y_46_im <= 1.65e+20)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.8e+48], N[Not[LessEqual[y$46$im, 1.65e+20]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.80000000000000012e48 or 1.65e20 < y.im

   1. Initial program 46.8%

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

   2. Taylor expanded in y.re around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

   4. Simplified 77.8%

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

   if -2.80000000000000012e48 < y.im < 1.65e20

   1. Initial program 71.4%

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

   2. Taylor expanded in y.re around inf 72.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+48} \lor \neg \left(y.im \leq 1.65 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 8: 46.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.5e+117)
   (/ x.re y.im)
   (if (<= y.im 6.6e+144) (/ x.im y.re) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.5e+117) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 6.6e+144) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-4.5d+117)) then
        tmp = x_46re / y_46im
    else if (y_46im <= 6.6d+144) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.5e+117) {
		tmp = x_46_re / y_46_im;
	} else if (y_46_im <= 6.6e+144) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.5e+117:
		tmp = x_46_re / y_46_im
	elif y_46_im <= 6.6e+144:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.5e+117)
		tmp = Float64(x_46_re / y_46_im);
	elseif (y_46_im <= 6.6e+144)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.5e+117)
		tmp = x_46_re / y_46_im;
	elseif (y_46_im <= 6.6e+144)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.5e+117], N[(x$46$re / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 6.6e+144], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.5e117 or 6.6e144 < y.im

   1. Initial program 34.8%

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

      1. *-un-lft-identity 34.8%

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

      2. add-sqr-sqrt 34.8%

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

      3. times-frac 34.8%

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

      4. hypot-def 34.8%

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

      5. hypot-def 62.0%

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

   3. Applied egg-rr 62.0%

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

   4. Taylor expanded in y.re around 0 59.6%

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

      1. neg-mul-1 59.6%

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

      2. unsub-neg 59.6%

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

      3. *-lft-identity 59.6%

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

      4. times-frac 62.0%

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

      5. /-rgt-identity 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in y.im around -inf 26.9%

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

   if -4.5e117 < y.im < 6.6e144

   1. Initial program 71.0%

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

   2. Taylor expanded in y.re around inf 59.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 9: 9.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.7%

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

   1. *-un-lft-identity 59.7%

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

   2. add-sqr-sqrt 59.7%

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

   3. times-frac 59.6%

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

   4. hypot-def 59.7%

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

   5. hypot-def 73.6%

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

3. Applied egg-rr 73.6%

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

4. Taylor expanded in y.re around 0 33.8%

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

   1. neg-mul-1 33.8%

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

   2. unsub-neg 33.8%

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

   3. *-lft-identity 33.8%

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

   4. times-frac 34.2%

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

   5. /-rgt-identity 34.2%

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

6. Simplified 34.2%

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

7. Taylor expanded in y.re around inf 10.2%

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

8. Final simplification 10.2%

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

Alternative 10: 42.5% 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 59.7%

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

2. Taylor expanded in y.re around inf 44.9%

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

3. Final simplification 44.9%

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

Reproduce

herbie shell --seed 2023172 
(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))))