_divideComplex, imaginary part

Percentage Accurate: 61.6% → 79.5%

Time: 9.6s

Alternatives: 9

Speedup: 1.0×

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: 61.6% 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: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \left(y.im \cdot \frac{1}{x.im}\right)} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

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 -6e+108)
     (- (* (/ 1.0 y.im) (/ y.re (/ y.im x.im))) (/ x.re y.im))
     (if (<= y.im -6.4e-58)
       t_0
       (if (<= y.im 1.2e-81)
         (+ (/ x.im y.re) (* y.im (* (/ x.re y.re) (/ -1.0 y.re))))
         (if (<= y.im 2.1e+118)
           t_0
           (- (/ y.re (* y.im (* y.im (/ 1.0 x.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 <= -6e+108) {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -6.4e-58) {
		tmp = t_0;
	} else if (y_46_im <= 1.2e-81) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else if (y_46_im <= 2.1e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im * (1.0 / x_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 <= (-6d+108)) then
        tmp = ((1.0d0 / y_46im) * (y_46re / (y_46im / x_46im))) - (x_46re / y_46im)
    else if (y_46im <= (-6.4d-58)) then
        tmp = t_0
    else if (y_46im <= 1.2d-81) then
        tmp = (x_46im / y_46re) + (y_46im * ((x_46re / y_46re) * ((-1.0d0) / y_46re)))
    else if (y_46im <= 2.1d+118) then
        tmp = t_0
    else
        tmp = (y_46re / (y_46im * (y_46im * (1.0d0 / x_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 <= -6e+108) {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -6.4e-58) {
		tmp = t_0;
	} else if (y_46_im <= 1.2e-81) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else if (y_46_im <= 2.1e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (y_46_im * (y_46_im * (1.0 / x_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 <= -6e+108:
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	elif y_46_im <= -6.4e-58:
		tmp = t_0
	elif y_46_im <= 1.2e-81:
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)))
	elif y_46_im <= 2.1e+118:
		tmp = t_0
	else:
		tmp = (y_46_re / (y_46_im * (y_46_im * (1.0 / x_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 <= -6e+108)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(y_46_re / Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -6.4e-58)
		tmp = t_0;
	elseif (y_46_im <= 1.2e-81)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) * Float64(-1.0 / y_46_re))));
	elseif (y_46_im <= 2.1e+118)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im * Float64(1.0 / x_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 <= -6e+108)
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	elseif (y_46_im <= -6.4e-58)
		tmp = t_0;
	elseif (y_46_im <= 1.2e-81)
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	elseif (y_46_im <= 2.1e+118)
		tmp = t_0;
	else
		tmp = (y_46_re / (y_46_im * (y_46_im * (1.0 / x_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, -6e+108], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6.4e-58], t$95$0, If[LessEqual[y$46$im, 1.2e-81], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.1e+118], t$95$0, N[(N[(y$46$re / N[(y$46$im * N[(y$46$im * N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.99999999999999968e108

   1. Initial program 21.3%

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

   2. Taylor expanded in y.re around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/l* 77.4%

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

   4. Simplified 77.4%

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

      1. pow2 77.4%

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

      2. div-inv 77.4%

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

      3. associate-*l* 83.0%

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

   6. Applied egg-rr 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. times-frac 84.8%

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

      3. un-div-inv 84.9%

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

   8. Applied egg-rr 84.9%

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

   if -5.99999999999999968e108 < y.im < -6.4000000000000002e-58 or 1.2e-81 < y.im < 2.1e118

   1. Initial program 75.6%

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

   if -6.4000000000000002e-58 < y.im < 1.2e-81

   1. Initial program 72.7%

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

   2. Taylor expanded in y.re around inf 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. unsub-neg 84.8%

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

      4. associate-/l* 81.9%

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

      5. associate-/r/ 80.0%

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

   4. Simplified 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. pow2 80.0%

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

      3. times-frac 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 2.1e118 < y.im

   1. Initial program 19.2%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

      1. pow2 90.9%

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

      2. div-inv 90.9%

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

      3. associate-*l* 96.9%

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

   6. Applied egg-rr 96.9%

      \[\leadsto \frac{y.re}{\color{blue}{y.im \cdot \left(y.im \cdot \frac{1}{x.im}\right)}} - \frac{x.re}{y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \left(y.im \cdot \frac{1}{x.im}\right)} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+17} \lor \neg \left(y.re \leq 950\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.3e+17) (not (<= y.re 950.0)))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
   (- (* (/ 1.0 y.im) (/ y.re (/ y.im x.im))) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.3e+17) || !(y_46_re <= 950.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.3d+17)) .or. (.not. (y_46re <= 950.0d0))) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else
        tmp = ((1.0d0 / y_46im) * (y_46re / (y_46im / x_46im))) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.3e+17) || !(y_46_re <= 950.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.3e+17) or not (y_46_re <= 950.0):
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	else:
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.3e+17) || !(y_46_re <= 950.0))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(y_46_re / Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.3e+17) || ~((y_46_re <= 950.0)))
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	else
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.3e+17], N[Not[LessEqual[y$46$re, 950.0]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.3e17 or 950 < y.re

   1. Initial program 43.0%

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

   2. Taylor expanded in y.re around inf 68.8%

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

      1. +-commutative 68.8%

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

      2. mul-1-neg 68.8%

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

      3. unsub-neg 68.8%

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

      4. associate-/l* 68.4%

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

      5. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. pow2 70.1%

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

      3. times-frac 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

   8. Applied egg-rr 73.3%

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

   if -3.3e17 < y.re < 950

   1. Initial program 69.8%

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

   2. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. *-commutative 71.6%

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

      5. associate-/l* 72.2%

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

   4. Simplified 72.2%

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

      1. pow2 72.2%

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

      2. div-inv 72.1%

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

      3. associate-*l* 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. *-un-lft-identity 74.9%

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

      2. times-frac 77.2%

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

      3. un-div-inv 77.2%

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

   8. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}}} - \frac{x.re}{y.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+17} \lor \neg \left(y.re \leq 950\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -940000000 \lor \neg \left(y.re \leq 4800\right):\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -940000000.0) (not (<= y.re 4800.0)))
   (+ (/ x.im y.re) (* y.im (* (/ x.re y.re) (/ -1.0 y.re))))
   (- (* (/ 1.0 y.im) (/ y.re (/ y.im x.im))) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -940000000.0) || !(y_46_re <= 4800.0)) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-940000000.0d0)) .or. (.not. (y_46re <= 4800.0d0))) then
        tmp = (x_46im / y_46re) + (y_46im * ((x_46re / y_46re) * ((-1.0d0) / y_46re)))
    else
        tmp = ((1.0d0 / y_46im) * (y_46re / (y_46im / x_46im))) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -940000000.0) || !(y_46_re <= 4800.0)) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else {
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -940000000.0) or not (y_46_re <= 4800.0):
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)))
	else:
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -940000000.0) || !(y_46_re <= 4800.0))
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) * Float64(-1.0 / y_46_re))));
	else
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(y_46_re / Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -940000000.0) || ~((y_46_re <= 4800.0)))
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	else
		tmp = ((1.0 / y_46_im) * (y_46_re / (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -940000000.0], N[Not[LessEqual[y$46$re, 4800.0]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.4e8 or 4800 < y.re

   1. Initial program 43.0%

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

   2. Taylor expanded in y.re around inf 68.8%

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

      1. +-commutative 68.8%

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

      2. mul-1-neg 68.8%

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

      3. unsub-neg 68.8%

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

      4. associate-/l* 68.4%

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

      5. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. pow2 70.1%

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

      3. times-frac 73.3%

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

   6. Applied egg-rr 73.3%

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

   if -9.4e8 < y.re < 4800

   1. Initial program 69.8%

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

   2. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. *-commutative 71.6%

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

      5. associate-/l* 72.2%

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

   4. Simplified 72.2%

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

      1. pow2 72.2%

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

      2. div-inv 72.1%

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

      3. associate-*l* 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. *-un-lft-identity 74.9%

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

      2. times-frac 77.2%

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

      3. un-div-inv 77.2%

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

   8. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}}} - \frac{x.re}{y.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -940000000 \lor \neg \left(y.re \leq 4800\right):\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{y.re}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq 1.9 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.6e+128) (not (<= y.im 1.9e+101)))
   (/ (- x.re) y.im)
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 1.9e+101)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.6d+128)) .or. (.not. (y_46im <= 1.9d+101))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 1.9e+101)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.6e+128) or not (y_46_im <= 1.9e+101):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 1.9e+101))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.6e+128) || ~((y_46_im <= 1.9e+101)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.6e+128], N[Not[LessEqual[y$46$im, 1.9e+101]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5999999999999998e128 or 1.8999999999999999e101 < y.im

   1. Initial program 24.8%

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

   2. Taylor expanded in y.re around 0 84.3%

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

      1. associate-*r/ 84.3%

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

      2. neg-mul-1 84.3%

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

   4. Simplified 84.3%

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

   if -7.5999999999999998e128 < y.im < 1.8999999999999999e101

   1. Initial program 72.6%

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

   2. Taylor expanded in y.re around inf 66.3%

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

      1. +-commutative 66.3%

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

      2. mul-1-neg 66.3%

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

      3. unsub-neg 66.3%

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

      4. associate-/l* 65.2%

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

      5. associate-/r/ 63.7%

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

   4. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. pow2 63.7%

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

      3. times-frac 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. associate-*l/ 67.5%

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

      2. *-un-lft-identity 67.5%

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

   8. Applied egg-rr 67.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq 1.9 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+30} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.8e+30) (not (<= y.im 5.5e-35)))
   (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.8e+30) || !(y_46_im <= 5.5e-35)) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-3.8d+30)) .or. (.not. (y_46im <= 5.5d-35))) then
        tmp = (y_46re / (y_46im * (y_46im / x_46im))) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -3.8e+30) || !(y_46_im <= 5.5e-35)) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -3.8e+30) or not (y_46_im <= 5.5e-35):
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -3.8e+30) || !(y_46_im <= 5.5e-35))
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -3.8e+30) || ~((y_46_im <= 5.5e-35)))
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.8e+30], N[Not[LessEqual[y$46$im, 5.5e-35]], $MachinePrecision]], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.8000000000000001e30 or 5.4999999999999997e-35 < y.im

   1. Initial program 41.6%

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

   2. Taylor expanded in y.re around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-/l* 72.8%

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

   4. Simplified 72.8%

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

      1. pow2 72.8%

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

      2. div-inv 72.8%

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

      3. associate-*l* 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in y.im around 0 76.2%

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

   if -3.8000000000000001e30 < y.im < 5.4999999999999997e-35

   1. Initial program 74.4%

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

   2. Taylor expanded in y.re around inf 77.7%

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

      1. +-commutative 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

      4. associate-/l* 75.5%

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

      5. associate-/r/ 74.0%

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

   4. Simplified 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. pow2 74.0%

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

      3. times-frac 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. associate-*l/ 77.2%

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

      2. *-un-lft-identity 77.2%

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

   8. Applied egg-rr 77.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+30} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -17500000000 \lor \neg \left(y.re \leq 220000\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -17500000000.0) (not (<= y.re 220000.0)))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
   (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -17500000000.0) || !(y_46_re <= 220000.0)) {
		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_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-17500000000.0d0)) .or. (.not. (y_46re <= 220000.0d0))) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else
        tmp = ((x_46im * (y_46re / y_46im)) / y_46im) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -17500000000.0) || !(y_46_re <= 220000.0)) {
		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_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -17500000000.0) or not (y_46_re <= 220000.0):
		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_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -17500000000.0) || !(y_46_re <= 220000.0))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -17500000000.0) || ~((y_46_re <= 220000.0)))
		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_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -17500000000.0], N[Not[LessEqual[y$46$re, 220000.0]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.75e10 or 2.2e5 < y.re

   1. Initial program 43.0%

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

   2. Taylor expanded in y.re around inf 68.8%

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

      1. +-commutative 68.8%

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

      2. mul-1-neg 68.8%

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

      3. unsub-neg 68.8%

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

      4. associate-/l* 68.4%

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

      5. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. pow2 70.1%

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

      3. times-frac 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. associate-*l/ 73.3%

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

      2. *-un-lft-identity 73.3%

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

   8. Applied egg-rr 73.3%

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

   if -1.75e10 < y.re < 2.2e5

   1. Initial program 69.8%

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

   2. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. *-commutative 71.6%

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

      5. associate-/l* 72.2%

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

   4. Simplified 72.2%

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

      1. pow2 72.2%

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

      2. div-inv 72.1%

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

      3. associate-*l* 74.9%

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

   6. Applied egg-rr 74.9%

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

      1. *-un-lft-identity 74.9%

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

      2. times-frac 77.2%

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

      3. un-div-inv 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. associate-*l/ 77.2%

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

      2. *-lft-identity 77.2%

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

      3. associate-/r/ 77.2%

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

   10. Simplified 77.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -17500000000 \lor \neg \left(y.re \leq 220000\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 61.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq -440\right) \land \left(y.im \leq -1.4 \cdot 10^{-60} \lor \neg \left(y.im \leq 8.2 \cdot 10^{+99}\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.6e+128)
         (and (not (<= y.im -440.0))
              (or (<= y.im -1.4e-60) (not (<= y.im 8.2e+99)))))
   (/ (- 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 <= -7.6e+128) || (!(y_46_im <= -440.0) && ((y_46_im <= -1.4e-60) || !(y_46_im <= 8.2e+99)))) {
		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 <= (-7.6d+128)) .or. (.not. (y_46im <= (-440.0d0))) .and. (y_46im <= (-1.4d-60)) .or. (.not. (y_46im <= 8.2d+99))) 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 <= -7.6e+128) || (!(y_46_im <= -440.0) && ((y_46_im <= -1.4e-60) || !(y_46_im <= 8.2e+99)))) {
		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 <= -7.6e+128) or (not (y_46_im <= -440.0) and ((y_46_im <= -1.4e-60) or not (y_46_im <= 8.2e+99))):
		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 <= -7.6e+128) || (!(y_46_im <= -440.0) && ((y_46_im <= -1.4e-60) || !(y_46_im <= 8.2e+99))))
		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 <= -7.6e+128) || (~((y_46_im <= -440.0)) && ((y_46_im <= -1.4e-60) || ~((y_46_im <= 8.2e+99)))))
		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, -7.6e+128], And[N[Not[LessEqual[y$46$im, -440.0]], $MachinePrecision], Or[LessEqual[y$46$im, -1.4e-60], N[Not[LessEqual[y$46$im, 8.2e+99]], $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 < -7.5999999999999998e128 or -440 < y.im < -1.4000000000000001e-60 or 8.19999999999999959e99 < y.im

   1. Initial program 31.7%

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

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

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   4. Simplified 81.9%

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

   if -7.5999999999999998e128 < y.im < -440 or -1.4000000000000001e-60 < y.im < 8.19999999999999959e99

   1. Initial program 71.6%

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

   2. Taylor expanded in y.re around inf 58.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq -440\right) \land \left(y.im \leq -1.4 \cdot 10^{-60} \lor \neg \left(y.im \leq 8.2 \cdot 10^{+99}\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 8: 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 57.1%

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

2. Taylor expanded in x.im around inf 36.0%

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

   1. associate-/l* 40.2%

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

4. Simplified 40.2%

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

   1. div-inv 40.1%

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

   2. pow2 40.1%

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

   3. +-commutative 40.1%

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

   4. add-sqr-sqrt 40.1%

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

   5. associate-*l* 40.1%

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

   6. +-commutative 40.1%

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

   7. pow2 40.1%

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

   8. hypot-def 40.1%

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

   9. +-commutative 40.1%

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

   10. pow2 40.1%

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

   11. hypot-def 53.6%

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

6. Applied egg-rr 53.6%

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

7. Taylor expanded in y.im around inf 13.1%

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

8. Taylor expanded in y.im around 0 7.9%

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

9. Final simplification 7.9%

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

Alternative 9: 42.3% 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 57.1%

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

2. Taylor expanded in y.re around inf 41.1%

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

3. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023333 
(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))))