_divideComplex, imaginary part

Percentage Accurate: 61.0% → 81.0%

Time: 12.1s

Alternatives: 8

Speedup: 1.1×

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: 61.0% 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: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -4.4e+112)
     (- (/ y.re (/ y.im (/ x.im y.im))) (/ x.re y.im))
     (if (<= y.im -2.15e-156)
       t_0
       (if (<= y.im 1.95e-115)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.im 9.6e+17)
           t_0
           (- (* (/ x.im y.im) (/ y.re 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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.4e+112) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -2.15e-156) {
		tmp = t_0;
	} else if (y_46_im <= 1.95e-115) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 9.6e+17) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-4.4d+112)) then
        tmp = (y_46re / (y_46im / (x_46im / y_46im))) - (x_46re / y_46im)
    else if (y_46im <= (-2.15d-156)) then
        tmp = t_0
    else if (y_46im <= 1.95d-115) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else if (y_46im <= 9.6d+17) then
        tmp = t_0
    else
        tmp = ((x_46im / y_46im) * (y_46re / 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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.4e+112) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -2.15e-156) {
		tmp = t_0;
	} else if (y_46_im <= 1.95e-115) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 9.6e+17) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -4.4e+112:
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im)
	elif y_46_im <= -2.15e-156:
		tmp = t_0
	elif y_46_im <= 1.95e-115:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	elif y_46_im <= 9.6e+17:
		tmp = t_0
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -4.4e+112)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / Float64(x_46_im / y_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -2.15e-156)
		tmp = t_0;
	elseif (y_46_im <= 1.95e-115)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_im <= 9.6e+17)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / 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)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -4.4e+112)
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	elseif (y_46_im <= -2.15e-156)
		tmp = t_0;
	elseif (y_46_im <= 1.95e-115)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	elseif (y_46_im <= 9.6e+17)
		tmp = t_0;
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.4e+112], N[(N[(y$46$re / N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.15e-156], t$95$0, If[LessEqual[y$46$im, 1.95e-115], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.6e+17], t$95$0, N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.3999999999999999e112

   1. Initial program 39.8%

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

   3. Taylor expanded in y.re around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-commutative 74.6%

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

      5. associate-/l* 80.0%

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

   5. Simplified 80.0%

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

      1. pow2 80.0%

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

      2. *-un-lft-identity 80.0%

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

      3. times-frac 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. /-rgt-identity 88.9%

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

      2. clear-num 88.8%

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

      3. un-div-inv 89.0%

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

   9. Applied egg-rr 89.0%

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

   if -4.3999999999999999e112 < y.im < -2.14999999999999989e-156 or 1.9499999999999999e-115 < y.im < 9.6e17

   1. Initial program 84.4%

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

   if -2.14999999999999989e-156 < y.im < 1.9499999999999999e-115

   1. Initial program 59.0%

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

   3. Taylor expanded in y.re around inf 77.5%

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

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-/l* 78.6%

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

      5. associate-/r/ 71.9%

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

   5. Simplified 71.9%

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

      1. pow2 71.9%

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

      2. associate-*l/ 77.5%

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

      3. associate-/r* 93.3%

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

   7. Applied egg-rr 93.3%

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

   if 9.6e17 < y.im

   1. Initial program 42.6%

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

   3. Taylor expanded in y.re around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. mul-1-neg 75.8%

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

      3. unsub-neg 75.8%

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

      4. *-commutative 75.8%

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

      5. associate-/l* 79.8%

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

   5. Simplified 79.8%

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

      1. pow2 79.8%

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

      2. *-un-lft-identity 79.8%

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

      3. times-frac 83.8%

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

   7. Applied egg-rr 83.8%

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

      1. /-rgt-identity 83.8%

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

      2. associate-/r* 86.2%

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

      3. div-inv 86.2%

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

      4. clear-num 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.15 \cdot 10^{-156}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.8e-29)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 1.3e-14)
     (/ x.im y.re)
     (- (* (/ x.im y.im) (/ y.re 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 <= -6.8e-29) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.3e-14) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= (-6.8d-29)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 1.3d-14) then
        tmp = x_46im / y_46re
    else
        tmp = ((x_46im / y_46im) * (y_46re / 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 <= -6.8e-29) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1.3e-14) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -6.8e-29:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 1.3e-14:
		tmp = x_46_im / y_46_re
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -6.8e-29)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 1.3e-14)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / 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 <= -6.8e-29)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 1.3e-14)
		tmp = x_46_im / y_46_re;
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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, -6.8e-29], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e-14], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / 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 < -6.79999999999999945e-29

   1. Initial program 62.5%

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

   3. Taylor expanded in y.re around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. *-un-lft-identity 77.1%

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

      3. times-frac 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. /-rgt-identity 82.4%

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

      2. *-un-lft-identity 82.4%

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

      3. times-frac 82.4%

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

   9. Applied egg-rr 82.4%

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

       1. associate-*l/ 82.4%

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

       2. *-un-lft-identity 82.4%

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

       3. sub-div 82.4%

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

       4. div-inv 82.4%

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

       5. clear-num 82.4%

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

   11. Applied egg-rr 82.4%

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

   if -6.79999999999999945e-29 < y.im < 1.29999999999999998e-14

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around inf 69.3%

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

   if 1.29999999999999998e-14 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. mul-1-neg 75.2%

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

      3. unsub-neg 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-/l* 78.9%

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

   5. Simplified 78.9%

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

      1. pow2 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. /-rgt-identity 82.6%

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

      2. associate-/r* 84.8%

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

      3. div-inv 84.8%

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

      4. clear-num 84.8%

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

   9. Applied egg-rr 84.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.4e-25)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 8e-13)
     (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
     (- (* (/ x.im y.im) (/ y.re 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 <= -2.4e-25) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 8e-13) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= (-2.4d-25)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 8d-13) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else
        tmp = ((x_46im / y_46im) * (y_46re / 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 <= -2.4e-25) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 8e-13) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -2.4e-25:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 8e-13:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -2.4e-25)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 8e-13)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / 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 <= -2.4e-25)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 8e-13)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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, -2.4e-25], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 8e-13], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / 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 < -2.40000000000000009e-25

   1. Initial program 62.5%

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

   3. Taylor expanded in y.re around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. *-un-lft-identity 77.1%

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

      3. times-frac 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. /-rgt-identity 82.4%

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

      2. *-un-lft-identity 82.4%

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

      3. times-frac 82.4%

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

   9. Applied egg-rr 82.4%

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

       1. associate-*l/ 82.4%

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

       2. *-un-lft-identity 82.4%

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

       3. sub-div 82.4%

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

       4. div-inv 82.4%

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

       5. clear-num 82.4%

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

   11. Applied egg-rr 82.4%

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

   if -2.40000000000000009e-25 < y.im < 8.0000000000000002e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around inf 73.3%

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

      1. +-commutative 73.3%

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

      2. mul-1-neg 73.3%

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

      3. unsub-neg 73.3%

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

      4. associate-/l* 73.9%

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

      5. associate-/r/ 70.2%

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

   5. Simplified 70.2%

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

      1. pow2 70.2%

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

      2. associate-*l/ 73.3%

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

      3. associate-/r* 82.1%

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

   7. Applied egg-rr 82.1%

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

   if 8.0000000000000002e-13 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. mul-1-neg 75.2%

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

      3. unsub-neg 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-/l* 78.9%

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

   5. Simplified 78.9%

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

      1. pow2 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. /-rgt-identity 82.6%

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

      2. associate-/r* 84.8%

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

      3. div-inv 84.8%

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

      4. clear-num 84.8%

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

   9. Applied egg-rr 84.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.2e-23)
   (- (/ y.re (/ y.im (/ x.im y.im))) (/ x.re y.im))
   (if (<= y.im 9.6e-15)
     (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
     (- (* (/ x.im y.im) (/ y.re 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 <= -7.2e-23) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 9.6e-15) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= (-7.2d-23)) then
        tmp = (y_46re / (y_46im / (x_46im / y_46im))) - (x_46re / y_46im)
    else if (y_46im <= 9.6d-15) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else
        tmp = ((x_46im / y_46im) * (y_46re / 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 <= -7.2e-23) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= 9.6e-15) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -7.2e-23:
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im)
	elif y_46_im <= 9.6e-15:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	else:
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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 <= -7.2e-23)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / Float64(x_46_im / y_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 9.6e-15)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	else
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / 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 <= -7.2e-23)
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	elseif (y_46_im <= 9.6e-15)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	else
		tmp = ((x_46_im / y_46_im) * (y_46_re / 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, -7.2e-23], N[(N[(y$46$re / N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.6e-15], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / 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 < -7.1999999999999996e-23

   1. Initial program 62.5%

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

   3. Taylor expanded in y.re around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. *-un-lft-identity 77.1%

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

      3. times-frac 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. /-rgt-identity 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.4%

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

   9. Applied egg-rr 82.4%

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

   if -7.1999999999999996e-23 < y.im < 9.5999999999999998e-15

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around inf 73.3%

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

      1. +-commutative 73.3%

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

      2. mul-1-neg 73.3%

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

      3. unsub-neg 73.3%

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

      4. associate-/l* 73.9%

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

      5. associate-/r/ 70.2%

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

   5. Simplified 70.2%

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

      1. pow2 70.2%

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

      2. associate-*l/ 73.3%

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

      3. associate-/r* 82.1%

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

   7. Applied egg-rr 82.1%

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

   if 9.5999999999999998e-15 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. mul-1-neg 75.2%

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

      3. unsub-neg 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-/l* 78.9%

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

   5. Simplified 78.9%

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

      1. pow2 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 82.6%

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

   7. Applied egg-rr 82.6%

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

      1. /-rgt-identity 82.6%

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

      2. associate-/r* 84.8%

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

      3. div-inv 84.8%

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

      4. clear-num 84.8%

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

   9. Applied egg-rr 84.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-25} \lor \neg \left(y.im \leq 1.15 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.05e-25) (not (<= y.im 1.15e-12)))
   (/ (- (* y.re (/ x.im y.im)) 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 <= -2.05e-25) || !(y_46_im <= 1.15e-12)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - 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 <= (-2.05d-25)) .or. (.not. (y_46im <= 1.15d-12))) then
        tmp = ((y_46re * (x_46im / y_46im)) - 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 <= -2.05e-25) || !(y_46_im <= 1.15e-12)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - 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 <= -2.05e-25) or not (y_46_im <= 1.15e-12):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - 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 <= -2.05e-25) || !(y_46_im <= 1.15e-12))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - 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 <= -2.05e-25) || ~((y_46_im <= 1.15e-12)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - 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, -2.05e-25], N[Not[LessEqual[y$46$im, 1.15e-12]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.04999999999999994e-25 or 1.14999999999999995e-12 < y.im

   1. Initial program 53.4%

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

   3. Taylor expanded in y.re around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-commutative 74.6%

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

      5. associate-/l* 78.1%

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

   5. Simplified 78.1%

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

      1. pow2 78.1%

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

      2. *-un-lft-identity 78.1%

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

      3. times-frac 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. /-rgt-identity 82.5%

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

      2. *-un-lft-identity 82.5%

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

      3. times-frac 83.7%

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

   9. Applied egg-rr 83.7%

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

       1. associate-*l/ 83.7%

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

       2. *-un-lft-identity 83.7%

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

       3. sub-div 83.7%

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

       4. div-inv 83.7%

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

       5. clear-num 83.7%

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

   11. Applied egg-rr 83.7%

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

   if -2.04999999999999994e-25 < y.im < 1.14999999999999995e-12

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around inf 69.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-25} \lor \neg \left(y.im \leq 1.15 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{-26} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9e-26) (not (<= y.im 5.5e-14)))
   (/ (- 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 <= -9e-26) || !(y_46_im <= 5.5e-14)) {
		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 <= (-9d-26)) .or. (.not. (y_46im <= 5.5d-14))) 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 <= -9e-26) || !(y_46_im <= 5.5e-14)) {
		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 <= -9e-26) or not (y_46_im <= 5.5e-14):
		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 <= -9e-26) || !(y_46_im <= 5.5e-14))
		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 <= -9e-26) || ~((y_46_im <= 5.5e-14)))
		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, -9e-26], N[Not[LessEqual[y$46$im, 5.5e-14]], $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 < -8.9999999999999998e-26 or 5.49999999999999991e-14 < y.im

   1. Initial program 53.4%

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

   3. Taylor expanded in y.re around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. neg-mul-1 71.1%

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

   5. Simplified 71.1%

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

   if -8.9999999999999998e-26 < y.im < 5.49999999999999991e-14

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around inf 69.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{-26} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 9.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. *-un-lft-identity 59.9%

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

   2. fma-def 59.9%

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

   3. add-sqr-sqrt 59.9%

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

   4. times-frac 59.9%

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

   5. fma-def 59.9%

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

   6. hypot-def 59.9%

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

   7. cancel-sign-sub-inv 59.9%

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

   8. add-sqr-sqrt 27.1%

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

   9. sqrt-unprod 41.2%

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

   10. sqr-neg 41.2%

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

   11. sqrt-unprod 18.4%

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

   12. add-sqr-sqrt 36.7%

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

   13. fma-def 36.7%

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

   14. fma-def 36.7%

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

   15. hypot-def 44.7%

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

4. Applied egg-rr 44.7%

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

5. Taylor expanded in y.im around -inf 32.7%

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

   1. neg-mul-1 32.7%

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

   2. +-commutative 32.7%

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

   3. unsub-neg 32.7%

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

   4. mul-1-neg 32.7%

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

   5. associate-/l* 33.5%

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

   6. distribute-neg-frac 33.5%

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

7. Simplified 33.5%

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

8. Taylor expanded in y.re around -inf 9.3%

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

9. Final simplification 9.3%

   \[\leadsto \frac{x.im}{y.im} \]
10. Add Preprocessing

Alternative 8: 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 59.9%

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

3. Taylor expanded in y.re around inf 40.7%

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

4. Final simplification 40.7%

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

Reproduce

herbie shell --seed 2024017 
(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))))