_divideComplex, imaginary part

Percentage Accurate: 62.0% → 80.0%

Time: 11.2s

Alternatives: 7

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (/ (/ y.im y.re) (/ y.re x.re)))))
   (if (<= y.re -6.2e+87)
     t_0
     (if (<= y.re 1.16e-122)
       (* (/ 1.0 y.im) (- (* x.im (/ y.re y.im)) x.re))
       (if (<= y.re 1.15e+99)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	double tmp;
	if (y_46_re <= -6.2e+87) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-122) {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	} else if (y_46_re <= 1.15e+99) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	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 = (x_46im / y_46re) - ((y_46im / y_46re) / (y_46re / x_46re))
    if (y_46re <= (-6.2d+87)) then
        tmp = t_0
    else if (y_46re <= 1.16d-122) then
        tmp = (1.0d0 / y_46im) * ((x_46im * (y_46re / y_46im)) - x_46re)
    else if (y_46re <= 1.15d+99) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = t_0
    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 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	double tmp;
	if (y_46_re <= -6.2e+87) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-122) {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	} else if (y_46_re <= 1.15e+99) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re))
	tmp = 0
	if y_46_re <= -6.2e+87:
		tmp = t_0
	elif y_46_re <= 1.16e-122:
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re)
	elif y_46_re <= 1.15e+99:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_re)))
	tmp = 0.0
	if (y_46_re <= -6.2e+87)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-122)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re));
	elseif (y_46_re <= 1.15e+99)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	tmp = 0.0;
	if (y_46_re <= -6.2e+87)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-122)
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	elseif (y_46_re <= 1.15e+99)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = t_0;
	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[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e+87], t$95$0, If[LessEqual[y$46$re, 1.16e-122], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+99], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.1999999999999999e87 or 1.1500000000000001e99 < y.re

   1. Initial program 45.2%

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

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

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. unpow2 77.4%

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

      5. associate-/l* 78.6%

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

      6. associate-/r/ 78.6%

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

   4. Simplified 78.6%

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

      1. *-commutative 78.6%

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

      2. clear-num 78.6%

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

      3. un-div-inv 78.6%

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

      4. associate-/l* 87.0%

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

      5. clear-num 87.0%

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

      6. associate-/r/ 87.0%

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

      7. /-rgt-identity 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. associate-/r* 90.5%

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

   8. Simplified 90.5%

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

   if -6.1999999999999999e87 < y.re < 1.16000000000000001e-122

   1. Initial program 72.9%

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

      1. clear-num 72.8%

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

      2. associate-/r/ 72.0%

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

      3. fma-def 72.0%

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

   3. Applied egg-rr 72.0%

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

   4. Taylor expanded in y.re around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. associate-*r/ 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-/r/ 74.5%

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

      6. mul-1-neg 74.5%

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

      7. sub-neg 74.5%

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

      8. associate-/r/ 78.9%

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

      9. *-commutative 78.9%

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

      10. associate-/r* 86.3%

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

   6. Simplified 86.3%

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

      1. associate-/l/ 78.9%

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

      2. associate-*r/ 78.8%

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

      3. *-commutative 78.8%

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

      4. sqr-neg 78.8%

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

      5. associate-/r* 87.1%

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

   8. Applied egg-rr 87.1%

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

      1. frac-2neg 87.1%

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

      2. remove-double-neg 87.1%

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

      3. div-inv 87.1%

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

      4. div-inv 86.9%

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

      5. distribute-rgt-out-- 87.8%

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

      6. associate-/l* 86.3%

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

      7. distribute-neg-frac 86.3%

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

      8. associate-/r/ 88.6%

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

      9. frac-2neg 88.6%

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

   10. Applied egg-rr 88.6%

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

   if 1.16000000000000001e-122 < y.re < 1.1500000000000001e99

   1. Initial program 72.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \end{array} \]

Alternative 2: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+89} \lor \neg \left(y.re \leq 3.8 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.25e+89) (not (<= y.re 3.8e+56)))
   (/ (- x.im (* (/ y.im y.re) x.re)) y.re)
   (* (/ 1.0 y.im) (- (* x.im (/ y.re y.im)) x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.25e+89) || !(y_46_re <= 3.8e+56)) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.25d+89)) .or. (.not. (y_46re <= 3.8d+56))) then
        tmp = (x_46im - ((y_46im / y_46re) * x_46re)) / y_46re
    else
        tmp = (1.0d0 / y_46im) * ((x_46im * (y_46re / y_46im)) - x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.25e+89) || !(y_46_re <= 3.8e+56)) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.25e+89) or not (y_46_re <= 3.8e+56):
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re
	else:
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.25e+89) || !(y_46_re <= 3.8e+56))
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.25e+89) || ~((y_46_re <= 3.8e+56)))
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	else
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_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$46$re, -2.25e+89], N[Not[LessEqual[y$46$re, 3.8e+56]], $MachinePrecision]], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.25e89 or 3.79999999999999996e56 < y.re

   1. Initial program 48.4%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. unpow2 76.0%

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

      5. associate-/l* 76.1%

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

      6. associate-/r/ 78.1%

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

   4. Simplified 78.1%

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

      1. sub-neg 78.1%

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

      2. div-inv 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 75.8%

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

      5. distribute-frac-neg 75.8%

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

      6. distribute-rgt-neg-out 75.8%

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

      7. associate-/r* 83.6%

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

      8. div-inv 83.6%

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

      9. distribute-rgt-out 83.6%

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

      10. *-commutative 83.6%

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

      11. associate-/l* 86.5%

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

      12. clear-num 86.4%

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

      13. frac-2neg 86.4%

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

      14. metadata-eval 86.4%

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

      15. associate-/r/ 86.4%

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

      16. neg-mul-1 86.4%

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

      17. *-commutative 86.4%

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

      18. associate-/l* 86.4%

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

      19. metadata-eval 86.4%

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

      20. /-rgt-identity 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. associate-*l/ 86.7%

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

      2. *-lft-identity 86.7%

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

      3. distribute-rgt-neg-out 86.7%

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

      4. unsub-neg 86.7%

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

   8. Simplified 86.7%

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

   if -2.25e89 < y.re < 3.79999999999999996e56

   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. Step-by-step derivation

      1. clear-num 72.6%

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

      2. associate-/r/ 71.9%

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

      3. fma-def 71.9%

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

   3. Applied egg-rr 71.9%

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

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

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

      1. +-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*r/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 70.9%

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

      6. mul-1-neg 70.9%

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

      7. sub-neg 70.9%

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

      8. associate-/r/ 74.3%

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

      9. *-commutative 74.3%

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

      10. associate-/r* 80.1%

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

   6. Simplified 80.1%

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

      1. associate-/l/ 74.3%

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

      2. associate-*r/ 74.2%

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

      3. *-commutative 74.2%

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

      4. sqr-neg 74.2%

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

      5. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. frac-2neg 80.8%

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

      2. remove-double-neg 80.8%

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

      3. div-inv 80.8%

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

      4. div-inv 80.6%

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

      5. distribute-rgt-out-- 81.4%

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

      6. associate-/l* 80.3%

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

      7. distribute-neg-frac 80.3%

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

      8. associate-/r/ 82.1%

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

      9. frac-2neg 82.1%

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+89} \lor \neg \left(y.re \leq 3.8 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \end{array} \]

Alternative 3: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+87} \lor \neg \left(y.re \leq 1.85 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.8e+87) (not (<= y.re 1.85e+56)))
   (- (/ x.im y.re) (/ (/ y.im y.re) (/ y.re x.re)))
   (* (/ 1.0 y.im) (- (* x.im (/ y.re y.im)) x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.8e+87) || !(y_46_re <= 1.85e+56)) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	} else {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.8d+87)) .or. (.not. (y_46re <= 1.85d+56))) then
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) / (y_46re / x_46re))
    else
        tmp = (1.0d0 / y_46im) * ((x_46im * (y_46re / y_46im)) - x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.8e+87) || !(y_46_re <= 1.85e+56)) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	} else {
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.8e+87) or not (y_46_re <= 1.85e+56):
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re))
	else:
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.8e+87) || !(y_46_re <= 1.85e+56))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_re)));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.8e+87) || ~((y_46_re <= 1.85e+56)))
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) / (y_46_re / x_46_re));
	else
		tmp = (1.0 / y_46_im) * ((x_46_im * (y_46_re / y_46_im)) - x_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$46$re, -1.8e+87], N[Not[LessEqual[y$46$re, 1.85e+56]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.79999999999999997e87 or 1.84999999999999998e56 < y.re

   1. Initial program 48.4%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. unpow2 76.0%

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

      5. associate-/l* 76.1%

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

      6. associate-/r/ 78.1%

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

   4. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. clear-num 78.1%

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

      3. un-div-inv 78.1%

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

      4. associate-/l* 85.5%

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

      5. clear-num 85.5%

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

      6. associate-/r/ 85.5%

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

      7. /-rgt-identity 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. associate-/r* 86.7%

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

   8. Simplified 86.7%

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

   if -1.79999999999999997e87 < y.re < 1.84999999999999998e56

   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. Step-by-step derivation

      1. clear-num 72.6%

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

      2. associate-/r/ 71.9%

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

      3. fma-def 71.9%

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

   3. Applied egg-rr 71.9%

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

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

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

      1. +-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*r/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 70.9%

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

      6. mul-1-neg 70.9%

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

      7. sub-neg 70.9%

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

      8. associate-/r/ 74.3%

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

      9. *-commutative 74.3%

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

      10. associate-/r* 80.1%

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

   6. Simplified 80.1%

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

      1. associate-/l/ 74.3%

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

      2. associate-*r/ 74.2%

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

      3. *-commutative 74.2%

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

      4. sqr-neg 74.2%

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

      5. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. frac-2neg 80.8%

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

      2. remove-double-neg 80.8%

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

      3. div-inv 80.8%

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

      4. div-inv 80.6%

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

      5. distribute-rgt-out-- 81.4%

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

      6. associate-/l* 80.3%

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

      7. distribute-neg-frac 80.3%

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

      8. associate-/r/ 82.1%

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

      9. frac-2neg 82.1%

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

   10. Applied egg-rr 82.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+87} \lor \neg \left(y.re \leq 1.85 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im \cdot \frac{y.re}{y.im} - x.re\right)\\ \end{array} \]

Alternative 4: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+27} \lor \neg \left(y.im \leq 3.9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.95e+27) (not (<= y.im 3.9e+43)))
   (/ (- x.re) y.im)
   (/ (- x.im (* (/ y.im y.re) x.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 <= -1.95e+27) || !(y_46_im <= 3.9e+43)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_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 <= (-1.95d+27)) .or. (.not. (y_46im <= 3.9d+43))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - ((y_46im / y_46re) * x_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 <= -1.95e+27) || !(y_46_im <= 3.9e+43)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_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 <= -1.95e+27) or not (y_46_im <= 3.9e+43):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_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 <= -1.95e+27) || !(y_46_im <= 3.9e+43))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_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 <= -1.95e+27) || ~((y_46_im <= 3.9e+43)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_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, -1.95e+27], N[Not[LessEqual[y$46$im, 3.9e+43]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.9499999999999999e27 or 3.9000000000000001e43 < y.im

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   4. Simplified 75.7%

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

   if -1.9499999999999999e27 < y.im < 3.9000000000000001e43

   1. Initial program 73.6%

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

   2. Taylor expanded in y.re around inf 69.1%

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

      1. +-commutative 69.1%

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

      2. mul-1-neg 69.1%

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

      3. unsub-neg 69.1%

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

      4. unpow2 69.1%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 67.9%

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

   4. Simplified 67.9%

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

      1. sub-neg 67.9%

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

      2. div-inv 67.7%

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

      3. *-commutative 67.7%

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

      4. associate-*r/ 68.9%

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

      5. distribute-frac-neg 68.9%

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

      6. distribute-rgt-neg-out 68.9%

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

      7. associate-/r* 73.7%

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

      8. div-inv 73.7%

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

      9. distribute-rgt-out 75.9%

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

      10. *-commutative 75.9%

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

      11. associate-/l* 75.3%

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

      12. clear-num 75.3%

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

      13. frac-2neg 75.3%

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

      14. metadata-eval 75.3%

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

      15. associate-/r/ 75.3%

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

      16. neg-mul-1 75.3%

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

      17. *-commutative 75.3%

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

      18. associate-/l* 75.3%

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

      19. metadata-eval 75.3%

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

      20. /-rgt-identity 75.3%

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

   6. Applied egg-rr 75.3%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

      3. distribute-rgt-neg-out 75.5%

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

      4. unsub-neg 75.5%

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

   8. Simplified 75.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+27} \lor \neg \left(y.im \leq 3.9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \end{array} \]

Alternative 5: 76.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+87} \lor \neg \left(y.re \leq 4.4 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.85e+87) (not (<= y.re 4.4e+57)))
   (/ (- x.im (* (/ y.im y.re) x.re)) y.re)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.85e+87) || !(y_46_re <= 4.4e+57)) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.85d+87)) .or. (.not. (y_46re <= 4.4d+57))) then
        tmp = (x_46im - ((y_46im / y_46re) * x_46re)) / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.85e+87) || !(y_46_re <= 4.4e+57)) {
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.85e+87) or not (y_46_re <= 4.4e+57):
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.85e+87) || !(y_46_re <= 4.4e+57))
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im / y_46_re) * x_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.85e+87) || ~((y_46_re <= 4.4e+57)))
		tmp = (x_46_im - ((y_46_im / y_46_re) * x_46_re)) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.85e+87], N[Not[LessEqual[y$46$re, 4.4e+57]], $MachinePrecision]], N[(N[(x$46$im - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.85000000000000001e87 or 4.4000000000000001e57 < y.re

   1. Initial program 48.4%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. unpow2 76.0%

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

      5. associate-/l* 76.1%

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

      6. associate-/r/ 78.1%

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

   4. Simplified 78.1%

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

      1. sub-neg 78.1%

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

      2. div-inv 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 75.8%

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

      5. distribute-frac-neg 75.8%

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

      6. distribute-rgt-neg-out 75.8%

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

      7. associate-/r* 83.6%

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

      8. div-inv 83.6%

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

      9. distribute-rgt-out 83.6%

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

      10. *-commutative 83.6%

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

      11. associate-/l* 86.5%

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

      12. clear-num 86.4%

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

      13. frac-2neg 86.4%

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

      14. metadata-eval 86.4%

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

      15. associate-/r/ 86.4%

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

      16. neg-mul-1 86.4%

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

      17. *-commutative 86.4%

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

      18. associate-/l* 86.4%

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

      19. metadata-eval 86.4%

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

      20. /-rgt-identity 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. associate-*l/ 86.7%

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

      2. *-lft-identity 86.7%

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

      3. distribute-rgt-neg-out 86.7%

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

      4. unsub-neg 86.7%

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

   8. Simplified 86.7%

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

   if -1.85000000000000001e87 < y.re < 4.4000000000000001e57

   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. Step-by-step derivation

      1. clear-num 72.6%

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

      2. associate-/r/ 71.9%

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

      3. fma-def 71.9%

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

   3. Applied egg-rr 71.9%

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

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

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

      1. +-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. associate-*r/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 70.9%

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

      6. mul-1-neg 70.9%

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

      7. sub-neg 70.9%

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

      8. associate-/r/ 74.3%

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

      9. *-commutative 74.3%

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

      10. associate-/r* 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in x.im around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. sub-neg 74.2%

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

      4. *-commutative 74.2%

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

      5. unpow2 74.2%

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

      6. associate-/r* 80.8%

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

      7. associate-*r/ 79.0%

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

      8. div-sub 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+87} \lor \neg \left(y.re \leq 4.4 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 6: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.2e+91)
   (/ x.im y.re)
   (if (<= y.re 3.2e+57) (/ (- x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.2e+91) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 3.2e+57) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3.2d+91)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 3.2d+57) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.2e+91) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 3.2e+57) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.2e+91:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 3.2e+57:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.2e+91)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 3.2e+57)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.2e+91)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 3.2e+57)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.2e+91], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.2e+57], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.19999999999999989e91 or 3.20000000000000029e57 < y.re

   1. Initial program 47.3%

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

   2. Taylor expanded in y.re around inf 75.7%

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

   if -3.19999999999999989e91 < y.re < 3.20000000000000029e57

   1. Initial program 73.0%

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

   2. Taylor expanded in y.re around 0 67.5%

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

      1. associate-*r/ 67.5%

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

      2. neg-mul-1 67.5%

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

   4. Simplified 67.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 7: 43.3% 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 63.4%

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

2. Taylor expanded in y.re around inf 39.6%

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

3. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2023297 
(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))))