_divideComplex, real part

Percentage Accurate: 60.7% → 79.8%

Time: 6.9s

Alternatives: 10

Speedup: 2.1×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 60.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 79.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+137}:\\ \;\;\;\;t_0 \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)\\ \mathbf{elif}\;y.re \leq -1.18 \cdot 10^{-65}:\\ \;\;\;\;t_0 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))))
   (if (<= y.re -4e+137)
     (*
      t_0
      (-
       (-
        (* (* (/ x.re y.re) (* y.im (/ y.im y.re))) 0.5)
        (* y.im (/ x.im y.re)))
       x.re))
     (if (<= y.re -1.18e-65)
       (* t_0 (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))
       (if (<= y.re 2.2e+54)
         (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
         (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -4e+137) {
		tmp = t_0 * (((((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))) * 0.5) - (y_46_im * (x_46_im / y_46_re))) - x_46_re);
	} else if (y_46_re <= -1.18e-65) {
		tmp = t_0 * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 2.2e+54) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -4e+137)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im * Float64(y_46_im / y_46_re))) * 0.5) - Float64(y_46_im * Float64(x_46_im / y_46_re))) - x_46_re));
	elseif (y_46_re <= -1.18e-65)
		tmp = Float64(t_0 * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 2.2e+54)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4e+137], N[(t$95$0 * N[(N[(N[(N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.18e-65], N[(t$95$0 * N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+54], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.0000000000000001e137

   1. Initial program 25.8%

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

      1. *-un-lft-identity 25.8%

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

      2. add-sqr-sqrt 25.8%

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

      3. times-frac 25.7%

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

      4. hypot-def 25.7%

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

      5. fma-def 25.7%

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

      6. hypot-def 62.4%

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

   3. Applied egg-rr 62.4%

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

   4. Taylor expanded in y.re around -inf 76.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + \color{blue}{\left(-x.re\right)}\right)\right) \]

      2. associate-+r+ 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + -1 \cdot \frac{y.im \cdot x.im}{y.re}\right) + \left(-x.re\right)\right)} \]

      3. unsub-neg 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + -1 \cdot \frac{y.im \cdot x.im}{y.re}\right) - x.re\right)} \]

      4. mul-1-neg 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \color{blue}{\left(-\frac{y.im \cdot x.im}{y.re}\right)}\right) - x.re\right) \]

      5. unsub-neg 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} - \frac{y.im \cdot x.im}{y.re}\right)} - x.re\right) \]

      6. *-commutative 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\color{blue}{\frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} \cdot 0.5} - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      7. unpow2 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\frac{x.re \cdot \color{blue}{\left(y.im \cdot y.im\right)}}{{y.re}^{2}} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      8. unpow2 76.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\frac{x.re \cdot \left(y.im \cdot y.im\right)}{\color{blue}{y.re \cdot y.re}} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      9. times-frac 82.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\color{blue}{\left(\frac{x.re}{y.re} \cdot \frac{y.im \cdot y.im}{y.re}\right)} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      10. associate-*r/ 90.9%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \color{blue}{\left(y.im \cdot \frac{y.im}{y.re}\right)}\right) \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      11. *-lft-identity 90.9%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \frac{y.im \cdot x.im}{\color{blue}{1 \cdot y.re}}\right) - x.re\right) \]

      12. times-frac 97.0%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \color{blue}{\frac{y.im}{1} \cdot \frac{x.im}{y.re}}\right) - x.re\right) \]

      13. /-rgt-identity 97.0%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \color{blue}{y.im} \cdot \frac{x.im}{y.re}\right) - x.re\right) \]

   6. Simplified 97.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)} \]

   if -4.0000000000000001e137 < y.re < -1.18e-65

   1. Initial program 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. add-sqr-sqrt 79.2%

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

      3. times-frac 79.2%

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

      4. hypot-def 79.2%

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

      5. fma-def 79.2%

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

      6. hypot-def 87.3%

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

   3. Applied egg-rr 87.3%

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

   if -1.18e-65 < y.re < 2.1999999999999999e54

   1. Initial program 64.9%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 2.1999999999999999e54 < y.re

   1. Initial program 43.9%

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

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

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

      1. associate-/l* 82.8%

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

      2. associate-/r/ 82.8%

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

      3. unpow2 82.8%

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

   4. Simplified 82.8%

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

      1. associate-*l/ 79.6%

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

      2. frac-times 89.5%

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

      3. clear-num 89.5%

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

      4. un-div-inv 89.5%

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

   6. Applied egg-rr 89.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)\\ \mathbf{elif}\;y.re \leq -1.18 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 2: 78.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.5e+153)
   (*
    (/ 1.0 (hypot y.re y.im))
    (-
     (-
      (* (* (/ x.re y.re) (* y.im (/ y.im y.re))) 0.5)
      (* y.im (/ x.im y.re)))
     x.re))
   (if (<= y.re -3.8e-64)
     (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.85e+55)
       (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
       (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.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+153) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))) * 0.5) - (y_46_im * (x_46_im / y_46_re))) - x_46_re);
	} else if (y_46_re <= -3.8e-64) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.85e+55) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e+153) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (((((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))) * 0.5) - (y_46_im * (x_46_im / y_46_re))) - x_46_re);
	} else if (y_46_re <= -3.8e-64) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.85e+55) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_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+153:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (((((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))) * 0.5) - (y_46_im * (x_46_im / y_46_re))) - x_46_re)
	elif y_46_re <= -3.8e-64:
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.85e+55:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_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+153)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im * Float64(y_46_im / y_46_re))) * 0.5) - Float64(y_46_im * Float64(x_46_im / y_46_re))) - x_46_re));
	elseif (y_46_re <= -3.8e-64)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.85e+55)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_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+153)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))) * 0.5) - (y_46_im * (x_46_im / y_46_re))) - x_46_re);
	elseif (y_46_re <= -3.8e-64)
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.85e+55)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.5e+153], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.8e-64], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+55], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.50000000000000009e153

   1. Initial program 15.9%

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

      1. *-un-lft-identity 15.9%

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

      2. add-sqr-sqrt 15.9%

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

      3. times-frac 15.9%

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

      4. hypot-def 15.9%

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

      5. fma-def 15.9%

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

      6. hypot-def 57.5%

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

   3. Applied egg-rr 57.5%

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

   4. Taylor expanded in y.re around -inf 72.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + \color{blue}{\left(-x.re\right)}\right)\right) \]

      2. associate-+r+ 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + -1 \cdot \frac{y.im \cdot x.im}{y.re}\right) + \left(-x.re\right)\right)} \]

      3. unsub-neg 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + -1 \cdot \frac{y.im \cdot x.im}{y.re}\right) - x.re\right)} \]

      4. mul-1-neg 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \color{blue}{\left(-\frac{y.im \cdot x.im}{y.re}\right)}\right) - x.re\right) \]

      5. unsub-neg 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} - \frac{y.im \cdot x.im}{y.re}\right)} - x.re\right) \]

      6. *-commutative 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\color{blue}{\frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} \cdot 0.5} - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      7. unpow2 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\frac{x.re \cdot \color{blue}{\left(y.im \cdot y.im\right)}}{{y.re}^{2}} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      8. unpow2 72.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\frac{x.re \cdot \left(y.im \cdot y.im\right)}{\color{blue}{y.re \cdot y.re}} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      9. times-frac 79.8%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\color{blue}{\left(\frac{x.re}{y.re} \cdot \frac{y.im \cdot y.im}{y.re}\right)} \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      10. associate-*r/ 89.7%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \color{blue}{\left(y.im \cdot \frac{y.im}{y.re}\right)}\right) \cdot 0.5 - \frac{y.im \cdot x.im}{y.re}\right) - x.re\right) \]

      11. *-lft-identity 89.7%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \frac{y.im \cdot x.im}{\color{blue}{1 \cdot y.re}}\right) - x.re\right) \]

      12. times-frac 96.7%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \color{blue}{\frac{y.im}{1} \cdot \frac{x.im}{y.re}}\right) - x.re\right) \]

      13. /-rgt-identity 96.7%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - \color{blue}{y.im} \cdot \frac{x.im}{y.re}\right) - x.re\right) \]

   6. Simplified 96.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)} \]

   if -2.50000000000000009e153 < y.re < -3.8000000000000002e-64

   1. Initial program 81.5%

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

   if -3.8000000000000002e-64 < y.re < 1.8500000000000001e55

   1. Initial program 64.9%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 1.8500000000000001e55 < y.re

   1. Initial program 43.9%

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

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

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

      1. associate-/l* 82.8%

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

      2. associate-/r/ 82.8%

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

      3. unpow2 82.8%

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

   4. Simplified 82.8%

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

      1. associate-*l/ 79.6%

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

      2. frac-times 89.5%

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

      3. clear-num 89.5%

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

      4. un-div-inv 89.5%

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

   6. Applied egg-rr 89.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(\left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) \cdot 0.5 - y.im \cdot \frac{x.im}{y.re}\right) - x.re\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 3: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -0.0031:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))))
   (if (<= y.re -1.1e+42)
     (/ x.re y.re)
     (if (<= y.re -0.0031)
       t_0
       (if (<= y.re -5.5e-62)
         (/ x.re y.re)
         (if (<= y.re -1.3e-147)
           (/ x.im y.im)
           (if (<= y.re 5.8e+53) t_0 (/ x.re 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_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_re <= -1.1e+42) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -0.0031) {
		tmp = t_0;
	} else if (y_46_re <= -5.5e-62) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.3e-147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 5.8e+53) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    if (y_46re <= (-1.1d+42)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-0.0031d0)) then
        tmp = t_0
    else if (y_46re <= (-5.5d-62)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-1.3d-147)) then
        tmp = x_46im / y_46im
    else if (y_46re <= 5.8d+53) then
        tmp = t_0
    else
        tmp = 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 t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_re <= -1.1e+42) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -0.0031) {
		tmp = t_0;
	} else if (y_46_re <= -5.5e-62) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.3e-147) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 5.8e+53) {
		tmp = t_0;
	} else {
		tmp = x_46_re / 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_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	tmp = 0
	if y_46_re <= -1.1e+42:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -0.0031:
		tmp = t_0
	elif y_46_re <= -5.5e-62:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -1.3e-147:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 5.8e+53:
		tmp = t_0
	else:
		tmp = x_46_re / 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_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.1e+42)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -0.0031)
		tmp = t_0;
	elseif (y_46_re <= -5.5e-62)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.3e-147)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 5.8e+53)
		tmp = t_0;
	else
		tmp = Float64(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)
	t_0 = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.1e+42)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -0.0031)
		tmp = t_0;
	elseif (y_46_re <= -5.5e-62)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -1.3e-147)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 5.8e+53)
		tmp = t_0;
	else
		tmp = 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_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.1e+42], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -0.0031], t$95$0, If[LessEqual[y$46$re, -5.5e-62], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.3e-147], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+53], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1000000000000001e42 or -0.00309999999999999989 < y.re < -5.50000000000000022e-62 or 5.8000000000000004e53 < y.re

   1. Initial program 46.8%

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

   2. Taylor expanded in y.re around inf 75.1%

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

   if -1.1000000000000001e42 < y.re < -0.00309999999999999989 or -1.2999999999999999e-147 < y.re < 5.8000000000000004e53

   1. Initial program 66.1%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 85.8%

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

   4. Simplified 85.8%

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

      1. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

   if -5.50000000000000022e-62 < y.re < -1.2999999999999999e-147

   1. Initial program 56.9%

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

   2. Taylor expanded in y.re around 0 85.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -0.0031:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))))
   (if (<= y.re -4.5e+151)
     t_0
     (if (<= y.re -2.75e-61)
       (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 2.3e+53)
         (+ (/ x.im y.im) (/ (* y.re (/ x.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_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -4.5e+151) {
		tmp = t_0;
	} else if (y_46_re <= -2.75e-61) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.3e+53) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    if (y_46re <= (-4.5d+151)) then
        tmp = t_0
    else if (y_46re <= (-2.75d-61)) then
        tmp = ((y_46im * x_46im) + (y_46re * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 2.3d+53) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_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_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -4.5e+151) {
		tmp = t_0;
	} else if (y_46_re <= -2.75e-61) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.3e+53) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	tmp = 0
	if y_46_re <= -4.5e+151:
		tmp = t_0
	elif y_46_re <= -2.75e-61:
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 2.3e+53:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -4.5e+151)
		tmp = t_0;
	elseif (y_46_re <= -2.75e-61)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.3e+53)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / 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_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	tmp = 0.0;
	if (y_46_re <= -4.5e+151)
		tmp = t_0;
	elseif (y_46_re <= -2.75e-61)
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 2.3e+53)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.5e+151], t$95$0, If[LessEqual[y$46$re, -2.75e-61], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+53], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.4999999999999999e151 or 2.3000000000000002e53 < y.re

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 80.5%

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

      1. associate-/l* 82.8%

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

      2. associate-/r/ 82.9%

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

      3. unpow2 82.9%

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

   4. Simplified 82.9%

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

      1. associate-*l/ 80.5%

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

      2. frac-times 91.4%

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

      3. clear-num 91.4%

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

      4. un-div-inv 91.4%

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

   6. Applied egg-rr 91.4%

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

   if -4.4999999999999999e151 < y.re < -2.7499999999999998e-61

   1. Initial program 81.5%

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

   if -2.7499999999999998e-61 < y.re < 2.3000000000000002e53

   1. Initial program 64.9%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 5: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+41} \lor \neg \left(y.re \leq 3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e+41) (not (<= y.re 3e+53)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6e+41) || !(y_46_re <= 3e+53)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_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 <= (-6d+41)) .or. (.not. (y_46re <= 3d+53))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_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 <= -6e+41) || !(y_46_re <= 3e+53)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6e+41) or not (y_46_re <= 3e+53):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6e+41) || !(y_46_re <= 3e+53))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6e+41) || ~((y_46_re <= 3e+53)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6e+41], N[Not[LessEqual[y$46$re, 3e+53]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.9999999999999997e41 or 2.99999999999999998e53 < y.re

   1. Initial program 43.1%

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

   2. Taylor expanded in y.re around inf 75.8%

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

   if -5.9999999999999997e41 < y.re < 2.99999999999999998e53

   1. Initial program 66.4%

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

   2. Taylor expanded in y.re around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. *-commutative 79.0%

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

      3. unpow2 79.0%

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

      4. times-frac 80.6%

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

   4. Simplified 80.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+41} \lor \neg \left(y.re \leq 3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 6: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-60} \lor \neg \left(y.re \leq 2.35 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.1e-60) (not (<= y.re 2.35e+55)))
   (+ (/ x.re y.re) (* x.im (/ y.im (* y.re y.re))))
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) 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 <= -1.1e-60) || !(y_46_re <= 2.35e+55)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= (-1.1d-60)) .or. (.not. (y_46re <= 2.35d+55))) then
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re * y_46re)))
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / 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 <= -1.1e-60) || !(y_46_re <= 2.35e+55)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= -1.1e-60) or not (y_46_re <= 2.35e+55):
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= -1.1e-60) || !(y_46_re <= 2.35e+55))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / 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 <= -1.1e-60) || ~((y_46_re <= 2.35e+55)))
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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, -1.1e-60], N[Not[LessEqual[y$46$re, 2.35e+55]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.0999999999999999e-60 or 2.35e55 < y.re

   1. Initial program 47.4%

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

   2. Taylor expanded in y.re around inf 75.9%

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

      1. associate-/l* 77.8%

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

      2. associate-/r/ 76.8%

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

      3. unpow2 76.8%

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

   4. Simplified 76.8%

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

   if -1.0999999999999999e-60 < y.re < 2.35e55

   1. Initial program 64.9%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-60} \lor \neg \left(y.re \leq 2.35 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 7: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-61} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.6e-61) (not (<= y.re 1.7e+56)))
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) 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 <= -2.6e-61) || !(y_46_re <= 1.7e+56)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= (-2.6d-61)) .or. (.not. (y_46re <= 1.7d+56))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / 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 <= -2.6e-61) || !(y_46_re <= 1.7e+56)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= -2.6e-61) or not (y_46_re <= 1.7e+56):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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 <= -2.6e-61) || !(y_46_re <= 1.7e+56))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / 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 <= -2.6e-61) || ~((y_46_re <= 1.7e+56)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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, -2.6e-61], N[Not[LessEqual[y$46$re, 1.7e+56]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.6000000000000001e-61 or 1.7e56 < y.re

   1. Initial program 47.4%

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

   2. Taylor expanded in y.re around inf 75.9%

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

      1. associate-/l* 77.8%

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

      2. associate-/r/ 76.8%

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

      3. unpow2 76.8%

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

   4. Simplified 76.8%

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

      1. associate-*l/ 75.9%

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

      2. frac-times 84.0%

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

      3. clear-num 84.0%

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

      4. un-div-inv 84.1%

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

   6. Applied egg-rr 84.1%

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

   if -2.6000000000000001e-61 < y.re < 1.7e56

   1. Initial program 64.9%

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

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

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

      1. +-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. unpow2 83.8%

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

      4. times-frac 84.9%

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

   4. Simplified 84.9%

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

      1. associate-*l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-61} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8: 62.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.35e-62)
   (/ x.re y.re)
   (if (<= y.re 2.5e+53) (/ x.im y.im) (/ 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_re <= -1.35e-62) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.5e+53) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = 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_46re <= (-1.35d-62)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.5d+53) then
        tmp = x_46im / y_46im
    else
        tmp = 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_re <= -1.35e-62) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.5e+53) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = 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_re <= -1.35e-62:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.5e+53:
		tmp = x_46_im / y_46_im
	else:
		tmp = 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_re <= -1.35e-62)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.5e+53)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(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_re <= -1.35e-62)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.5e+53)
		tmp = x_46_im / y_46_im;
	else
		tmp = 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[LessEqual[y$46$re, -1.35e-62], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+53], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.3500000000000001e-62 or 2.5000000000000002e53 < y.re

   1. Initial program 47.4%

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

   2. Taylor expanded in y.re around inf 72.5%

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

   if -1.3500000000000001e-62 < y.re < 2.5000000000000002e53

   1. Initial program 64.9%

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

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

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 9.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 9.2e+158) (/ x.im 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 <= 9.2e+158) {
		tmp = x_46_im / 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 <= 9.2d+158) then
        tmp = x_46im / 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 <= 9.2e+158) {
		tmp = x_46_im / 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 <= 9.2e+158:
		tmp = x_46_im / 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 <= 9.2e+158)
		tmp = Float64(x_46_im / 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 <= 9.2e+158)
		tmp = x_46_im / 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, 9.2e+158], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 9.19999999999999942e158

   1. Initial program 61.1%

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

   2. Taylor expanded in y.re around 0 52.8%

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

   if 9.19999999999999942e158 < y.re

   1. Initial program 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. add-sqr-sqrt 30.5%

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

      3. times-frac 30.5%

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

      4. hypot-def 30.5%

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

      5. fma-def 30.5%

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

      6. hypot-def 61.8%

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

   3. Applied egg-rr 61.8%

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

   4. Taylor expanded in y.im around -inf 21.6%

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

      1. neg-mul-1 21.6%

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

   6. Simplified 21.6%

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

   7. Taylor expanded in y.re around -inf 21.5%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 9.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 42.7% 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 55.6%

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

2. Taylor expanded in y.re around 0 45.0%

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

3. Final simplification 45.0%

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

Reproduce

herbie shell --seed 2023240 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))