_divideComplex, imaginary part

Percentage Accurate: 61.5% → 85.6%

Time: 13.1s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% 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: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(x.re, y.im, x.im \cdot \left(-y.re\right)\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+300)
   (/
    (/ (- (fma x.re y.im (* x.im (- y.re)))) (hypot y.im y.re))
    (hypot y.im y.re))
   (- (/ 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 ((((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+300) {
		tmp = (-fma(x_46_re, y_46_im, (x_46_im * -y_46_re)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
	} 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 (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))) <= 2e+300)
		tmp = Float64(Float64(Float64(-fma(x_46_re, y_46_im, Float64(x_46_im * Float64(-y_46_re)))) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 2e+300], N[(N[((-N[(x$46$re * y$46$im + N[(x$46$im * (-y$46$re)), $MachinePrecision]), $MachinePrecision]) / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e300

   1. Initial program 75.5%

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

      1. frac-2neg 75.5%

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

      2. div-inv 75.5%

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

   3. Applied egg-rr 75.5%

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

      1. *-commutative 75.5%

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

      2. neg-mul-1 75.5%

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

      3. associate-/r* 75.5%

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

      4. metadata-eval 75.5%

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

   5. Simplified 75.5%

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

      1. associate-*r/ 75.5%

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

      2. unpow2 75.5%

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

      3. associate-/r* 96.3%

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

      4. fma-neg 96.3%

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

      5. distribute-rgt-neg-in 96.3%

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

      6. hypot-udef 75.7%

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

      7. +-commutative 75.7%

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

      8. hypot-def 96.3%

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

      9. hypot-udef 75.7%

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

      10. +-commutative 75.7%

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

      11. hypot-def 96.3%

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

   7. Applied egg-rr 96.3%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 12.0%

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

   2. Taylor expanded in y.re around inf 49.9%

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

      1. *-un-lft-identity 49.9%

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

      2. metadata-eval 49.9%

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

      3. unpow2 49.9%

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

      4. times-frac 56.9%

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

      5. metadata-eval 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. *-commutative 56.9%

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

      2. div-inv 56.8%

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

      3. associate-/l* 66.5%

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

      4. associate-/r/ 66.6%

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

   6. Applied egg-rr 66.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(x.re, y.im, x.im \cdot \left(-y.re\right)\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.35 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (/ (* y.im (/ x.re y.re)) y.re))))
   (if (<= y.re -3.35e+84)
     t_0
     (if (<= y.re -3e-170)
       (/ (fma x.im y.re (* x.re (- y.im))) (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 7.8e-185)
         (/ (- x.re) y.im)
         (if (<= y.re 5.2e+43)
           (/ (- (* x.im y.re) (* x.re y.im)) (+ (* 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 * (x_46_re / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -3.35e+84) {
		tmp = t_0;
	} else if (y_46_re <= -3e-170) {
		tmp = fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 7.8e-185) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 5.2e+43) {
		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));
	} 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 * Float64(x_46_re / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.35e+84)
		tmp = t_0;
	elseif (y_46_re <= -3e-170)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 7.8e-185)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 5.2e+43)
		tmp = 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)));
	else
		tmp = t_0;
	end
	return 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 * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.35e+84], t$95$0, If[LessEqual[y$46$re, -3e-170], N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e-185], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+43], 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], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.3500000000000002e84 or 5.20000000000000042e43 < y.re

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. metadata-eval 74.2%

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

      3. unpow2 74.2%

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

      4. times-frac 81.9%

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

      5. metadata-eval 81.9%

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

   4. Applied egg-rr 81.9%

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

      1. *-commutative 81.9%

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

      2. div-inv 81.9%

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

      3. associate-/l* 87.0%

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

      4. associate-/r/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if -3.3500000000000002e84 < y.re < -3.00000000000000013e-170

   1. Initial program 87.5%

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

      1. fma-neg 87.5%

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

      2. distribute-rgt-neg-out 87.5%

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

      3. fma-def 87.5%

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

   3. Simplified 87.5%

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

   if -3.00000000000000013e-170 < y.re < 7.7999999999999999e-185

   1. Initial program 56.5%

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

   2. Taylor expanded in y.re around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   4. Simplified 83.8%

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

   if 7.7999999999999999e-185 < y.re < 5.20000000000000042e43

   1. Initial program 82.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.35 \cdot 10^{+84}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-184}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.25 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ (* y.im (/ x.re y.re)) y.re))))
   (if (<= y.re -5.2e+83)
     t_1
     (if (<= y.re -8e-165)
       t_0
       (if (<= y.re 1.85e-184)
         (/ (- x.re) y.im)
         (if (<= y.re 4.25e+43) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -5.2e+83) {
		tmp = t_1;
	} else if (y_46_re <= -8e-165) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-184) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.25e+43) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((y_46im * (x_46re / y_46re)) / y_46re)
    if (y_46re <= (-5.2d+83)) then
        tmp = t_1
    else if (y_46re <= (-8d-165)) then
        tmp = t_0
    else if (y_46re <= 1.85d-184) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 4.25d+43) then
        tmp = t_0
    else
        tmp = t_1
    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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -5.2e+83) {
		tmp = t_1;
	} else if (y_46_re <= -8e-165) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-184) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.25e+43) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -5.2e+83:
		tmp = t_1
	elif y_46_re <= -8e-165:
		tmp = t_0
	elif y_46_re <= 1.85e-184:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 4.25e+43:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -5.2e+83)
		tmp = t_1;
	elseif (y_46_re <= -8e-165)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-184)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 4.25e+43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.2e+83)
		tmp = t_1;
	elseif (y_46_re <= -8e-165)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-184)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 4.25e+43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.2e+83], t$95$1, If[LessEqual[y$46$re, -8e-165], t$95$0, If[LessEqual[y$46$re, 1.85e-184], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.25e+43], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.2000000000000002e83 or 4.25e43 < y.re

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. metadata-eval 74.2%

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

      3. unpow2 74.2%

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

      4. times-frac 81.9%

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

      5. metadata-eval 81.9%

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

   4. Applied egg-rr 81.9%

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

      1. *-commutative 81.9%

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

      2. div-inv 81.9%

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

      3. associate-/l* 87.0%

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

      4. associate-/r/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if -5.2000000000000002e83 < y.re < -8.0000000000000001e-165 or 1.8499999999999999e-184 < y.re < 4.25e43

   1. Initial program 85.0%

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

   if -8.0000000000000001e-165 < y.re < 1.8499999999999999e-184

   1. Initial program 56.5%

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

   2. Taylor expanded in y.re around 0 83.8%

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

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-165}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-184}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.im y.re) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -4.6e+94)
     (/ x.im y.re)
     (if (<= y.re -3.2e-86)
       t_0
       (if (<= y.re 6.6e-67)
         (/ (- x.re) y.im)
         (if (<= y.re 2.4e+42) t_0 (/ x.im y.re)))))))

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_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -4.6e+94) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-86) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-67) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.4e+42) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-4.6d+94)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-3.2d-86)) then
        tmp = t_0
    else if (y_46re <= 6.6d-67) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 2.4d+42) then
        tmp = t_0
    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 t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -4.6e+94) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-86) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-67) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.4e+42) {
		tmp = t_0;
	} 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):
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -4.6e+94:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -3.2e-86:
		tmp = t_0
	elif y_46_re <= 6.6e-67:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 2.4e+42:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x_46_im * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -4.6e+94)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.2e-86)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-67)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 2.4e+42)
		tmp = t_0;
	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)
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -4.6e+94)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -3.2e-86)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-67)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 2.4e+42)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+94], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.2e-86], t$95$0, If[LessEqual[y$46$re, 6.6e-67], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+42], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.5999999999999999e94 or 2.3999999999999999e42 < y.re

   1. Initial program 35.0%

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

   2. Taylor expanded in y.re around inf 73.5%

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

   if -4.5999999999999999e94 < y.re < -3.20000000000000006e-86 or 6.6000000000000003e-67 < y.re < 2.3999999999999999e42

   1. Initial program 90.3%

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

   2. Taylor expanded in x.im around inf 66.7%

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

      1. *-commutative 66.7%

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

   4. Simplified 66.7%

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

   if -3.20000000000000006e-86 < y.re < 6.6000000000000003e-67

   1. Initial program 66.4%

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

   2. Taylor expanded in y.re around 0 70.7%

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

      1. associate-*r/ 70.7%

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

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 5: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -24000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.im y.re) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))
   (if (<= y.re -24000.0)
     t_1
     (if (<= y.re -2.5e-87)
       t_0
       (if (<= y.re 1.85e-67)
         (/ (- x.re) y.im)
         (if (<= y.re 4.3e+36) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -24000.0) {
		tmp = t_1;
	} else if (y_46_re <= -2.5e-87) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-67) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.3e+36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((x_46re / y_46re) * (y_46im / y_46re))
    if (y_46re <= (-24000.0d0)) then
        tmp = t_1
    else if (y_46re <= (-2.5d-87)) then
        tmp = t_0
    else if (y_46re <= 1.85d-67) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 4.3d+36) then
        tmp = t_0
    else
        tmp = t_1
    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_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -24000.0) {
		tmp = t_1;
	} else if (y_46_re <= -2.5e-87) {
		tmp = t_0;
	} else if (y_46_re <= 1.85e-67) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.3e+36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -24000.0:
		tmp = t_1
	elif y_46_re <= -2.5e-87:
		tmp = t_0
	elif y_46_re <= 1.85e-67:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 4.3e+36:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -24000.0)
		tmp = t_1;
	elseif (y_46_re <= -2.5e-87)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-67)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 4.3e+36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -24000.0)
		tmp = t_1;
	elseif (y_46_re <= -2.5e-87)
		tmp = t_0;
	elseif (y_46_re <= 1.85e-67)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 4.3e+36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -24000.0], t$95$1, If[LessEqual[y$46$re, -2.5e-87], t$95$0, If[LessEqual[y$46$re, 1.85e-67], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.3e+36], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -24000 or 4.30000000000000005e36 < 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 75.1%

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

      1. *-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. times-frac 86.4%

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

   4. Applied egg-rr 86.4%

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

   if -24000 < y.re < -2.50000000000000021e-87 or 1.85e-67 < y.re < 4.30000000000000005e36

   1. Initial program 88.3%

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

   2. Taylor expanded in x.im around inf 68.5%

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

      1. *-commutative 68.5%

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

   4. Simplified 68.5%

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

   if -2.50000000000000021e-87 < y.re < 1.85e-67

   1. Initial program 66.4%

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

   2. Taylor expanded in y.re around 0 70.7%

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

      1. associate-*r/ 70.7%

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

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -24000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 6: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -9000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -6.5 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-66}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.im y.re) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ (* y.im (/ x.re y.re)) y.re))))
   (if (<= y.re -9000.0)
     t_1
     (if (<= y.re -6.5e-84)
       t_0
       (if (<= y.re 1.16e-66)
         (/ (- x.re) y.im)
         (if (<= y.re 2.6e+38) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -9000.0) {
		tmp = t_1;
	} else if (y_46_re <= -6.5e-84) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-66) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.6e+38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - ((y_46im * (x_46re / y_46re)) / y_46re)
    if (y_46re <= (-9000.0d0)) then
        tmp = t_1
    else if (y_46re <= (-6.5d-84)) then
        tmp = t_0
    else if (y_46re <= 1.16d-66) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 2.6d+38) then
        tmp = t_0
    else
        tmp = t_1
    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_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	double tmp;
	if (y_46_re <= -9000.0) {
		tmp = t_1;
	} else if (y_46_re <= -6.5e-84) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-66) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.6e+38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re)
	tmp = 0
	if y_46_re <= -9000.0:
		tmp = t_1
	elif y_46_re <= -6.5e-84:
		tmp = t_0
	elif y_46_re <= 1.16e-66:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 2.6e+38:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) / y_46_re))
	tmp = 0.0
	if (y_46_re <= -9000.0)
		tmp = t_1;
	elseif (y_46_re <= -6.5e-84)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-66)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 2.6e+38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - ((y_46_im * (x_46_re / y_46_re)) / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -9000.0)
		tmp = t_1;
	elseif (y_46_re <= -6.5e-84)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-66)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 2.6e+38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9000.0], t$95$1, If[LessEqual[y$46$re, -6.5e-84], t$95$0, If[LessEqual[y$46$re, 1.16e-66], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.6e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9e3 or 2.5999999999999999e38 < 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 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. metadata-eval 75.1%

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

      3. unpow2 75.1%

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

      4. times-frac 81.5%

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

      5. metadata-eval 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. div-inv 81.5%

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

      3. associate-/l* 85.8%

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

      4. associate-/r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

   if -9e3 < y.re < -6.50000000000000022e-84 or 1.16000000000000002e-66 < y.re < 2.5999999999999999e38

   1. Initial program 88.3%

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

   2. Taylor expanded in x.im around inf 68.5%

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

      1. *-commutative 68.5%

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

   4. Simplified 68.5%

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

   if -6.50000000000000022e-84 < y.re < 1.16000000000000002e-66

   1. Initial program 66.4%

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

   2. Taylor expanded in y.re around 0 70.7%

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

      1. associate-*r/ 70.7%

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

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-66}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.08 \cdot 10^{+86} \lor \neg \left(y.im \leq 5.3 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re \cdot y.im}{y.re} \cdot \frac{-1}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.08e+86) (not (<= y.im 5.3e+22)))
   (/ (- x.re) y.im)
   (+ (/ x.im y.re) (* (/ (* x.re y.im) y.re) (/ -1.0 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 <= -1.08e+86) || !(y_46_im <= 5.3e+22)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) + (((x_46_re * y_46_im) / y_46_re) * (-1.0 / 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 <= (-1.08d+86)) .or. (.not. (y_46im <= 5.3d+22))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im / y_46re) + (((x_46re * y_46im) / y_46re) * ((-1.0d0) / 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 <= -1.08e+86) || !(y_46_im <= 5.3e+22)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) + (((x_46_re * y_46_im) / y_46_re) * (-1.0 / 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 <= -1.08e+86) or not (y_46_im <= 5.3e+22):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im / y_46_re) + (((x_46_re * y_46_im) / y_46_re) * (-1.0 / 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 <= -1.08e+86) || !(y_46_im <= 5.3e+22))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(Float64(Float64(x_46_re * y_46_im) / y_46_re) * Float64(-1.0 / 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 <= -1.08e+86) || ~((y_46_im <= 5.3e+22)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im / y_46_re) + (((x_46_re * y_46_im) / y_46_re) * (-1.0 / 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, -1.08e+86], N[Not[LessEqual[y$46$im, 5.3e+22]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.07999999999999993e86 or 5.2999999999999998e22 < y.im

   1. Initial program 40.4%

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

   2. Taylor expanded in y.re around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   4. Simplified 68.0%

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

   if -1.07999999999999993e86 < y.im < 5.2999999999999998e22

   1. Initial program 72.0%

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

   2. Taylor expanded in y.re around inf 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. metadata-eval 72.3%

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

      3. unpow2 72.3%

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

      4. times-frac 78.9%

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

      5. metadata-eval 78.9%

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

   4. Applied egg-rr 78.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.08 \cdot 10^{+86} \lor \neg \left(y.im \leq 5.3 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re \cdot y.im}{y.re} \cdot \frac{-1}{y.re}\\ \end{array} \]

Alternative 8: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-85} \lor \neg \left(y.re \leq 3 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.5e-85) (not (<= y.re 3e-41)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.5e-85) || !(y_46_re <= 3e-41)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.5d-85)) .or. (.not. (y_46re <= 3d-41))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.5e-85) || !(y_46_re <= 3e-41)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.5e-85) or not (y_46_re <= 3e-41):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.5e-85) || !(y_46_re <= 3e-41))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.5e-85) || ~((y_46_re <= 3e-41)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.5e-85], N[Not[LessEqual[y$46$re, 3e-41]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.5000000000000001e-85 or 2.99999999999999989e-41 < y.re

   1. Initial program 55.3%

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

   2. Taylor expanded in y.re around inf 64.1%

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

   if -2.5000000000000001e-85 < y.re < 2.99999999999999989e-41

   1. Initial program 67.7%

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

   2. Taylor expanded in y.re around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

   4. Simplified 69.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-85} \lor \neg \left(y.re \leq 3 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 9: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+196}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.8e+196) (/ x.re y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+196) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.8d+196)) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+196) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.8e+196:
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.8e+196)
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.8e+196)
		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$46$im, -2.8e+196], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.8000000000000002e196

   1. Initial program 30.0%

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

      1. frac-2neg 30.0%

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

      2. div-inv 30.0%

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

   3. Applied egg-rr 30.0%

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

      1. *-commutative 30.0%

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

      2. neg-mul-1 30.0%

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

      3. associate-/r* 30.0%

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

      4. metadata-eval 30.0%

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

   5. Simplified 30.0%

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

      1. associate-*r/ 30.0%

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

      2. unpow2 30.0%

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

      3. associate-/r* 63.5%

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

      4. fma-neg 63.5%

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

      5. distribute-rgt-neg-in 63.5%

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

      6. hypot-udef 30.0%

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

      7. +-commutative 30.0%

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

      8. hypot-def 63.5%

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

      9. hypot-udef 30.0%

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

      10. +-commutative 30.0%

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

      11. hypot-def 63.5%

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

   7. Applied egg-rr 63.5%

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

   8. Taylor expanded in y.im around -inf 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*r/ 94.4%

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

      3. mul-1-neg 94.4%

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

      4. *-commutative 94.4%

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

      5. distribute-rgt-neg-out 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in y.re around 0 29.8%

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

   if -2.8000000000000002e196 < y.im

   1. Initial program 62.4%

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

   2. Taylor expanded in y.re around inf 46.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+196}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 9.1% 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 60.4%

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

   1. frac-2neg 60.4%

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

   2. div-inv 60.3%

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

3. Applied egg-rr 60.3%

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

   1. *-commutative 60.3%

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

   2. neg-mul-1 60.3%

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

   3. associate-/r* 60.3%

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

   4. metadata-eval 60.3%

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

5. Simplified 60.3%

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

   1. associate-*r/ 60.4%

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

   2. unpow2 60.4%

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

   3. associate-/r* 77.5%

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

   4. fma-neg 77.5%

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

   5. distribute-rgt-neg-in 77.5%

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

   6. hypot-udef 60.5%

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

   7. +-commutative 60.5%

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

   8. hypot-def 77.5%

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

   9. hypot-udef 60.5%

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

   10. +-commutative 60.5%

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

   11. hypot-def 77.5%

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

7. Applied egg-rr 77.5%

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

8. Taylor expanded in y.im around -inf 24.0%

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

   1. *-commutative 24.0%

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

   2. associate-*r/ 24.0%

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

   3. mul-1-neg 24.0%

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

   4. *-commutative 24.0%

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

   5. distribute-rgt-neg-out 24.0%

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

10. Simplified 24.0%

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

11. Taylor expanded in y.re around -inf 7.7%

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

12. Final simplification 7.7%

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

Alternative 11: 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 60.4%

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

2. Taylor expanded in y.re around inf 43.6%

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

3. Final simplification 43.6%

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

Reproduce

herbie shell --seed 2023305 
(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))))