_divideComplex, real part

Percentage Accurate: 60.7% → 77.7%

Time: 8.3s

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: 77.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 660000000:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+30}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.7e+35)
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (if (<= y.re 8.2e-113)
     (+ (/ (/ y.re y.im) (/ y.im x.re)) (/ x.im y.im))
     (if (<= y.re 660000000.0)
       (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 1.12e+30)
         (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
         (+ (/ x.re y.re) (* y.im (/ (/ 1.0 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 <= -4.7e+35) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 8.2e-113) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	} else if (y_46_re <= 660000000.0) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 1.12e+30) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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 <= -4.7e+35)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 8.2e-113)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)) + Float64(x_46_im / y_46_im));
	elseif (y_46_re <= 660000000.0)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.12e+30)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(1.0 / 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_] := If[LessEqual[y$46$re, -4.7e+35], 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, 8.2e-113], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 660000000.0], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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, 1.12e+30], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.70000000000000033e35

   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 73.4%

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

      1. unpow2 73.4%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -4.70000000000000033e35 < y.re < 8.1999999999999999e-113

   1. Initial program 67.2%

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

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

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. associate-/l* 78.2%

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

      4. associate-/r/ 78.0%

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

   4. Simplified 78.0%

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

      1. +-commutative 78.0%

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

      2. *-un-lft-identity 78.0%

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

      3. fma-def 78.0%

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

      4. *-commutative 78.0%

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

      5. clear-num 77.6%

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

      6. un-div-inv 78.4%

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

      7. *-un-lft-identity 78.4%

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

      8. times-frac 83.6%

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

      9. /-rgt-identity 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. fma-udef 83.6%

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

      2. *-lft-identity 83.6%

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

      3. associate-/r* 87.9%

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

   8. Simplified 87.9%

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

   if 8.1999999999999999e-113 < y.re < 6.6e8

   1. Initial program 83.1%

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

   2. Taylor expanded in y.re around 0 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

      3. fma-udef 83.2%

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

   4. Simplified 83.2%

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

   if 6.6e8 < y.re < 1.12e30

   1. Initial program 55.9%

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

   2. Taylor expanded in y.re around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. associate-/l* 76.2%

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

      4. associate-/r/ 76.1%

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

   4. Simplified 76.1%

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

      1. +-commutative 76.1%

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

      2. *-un-lft-identity 76.1%

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

      3. fma-def 76.1%

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

      4. *-commutative 76.1%

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

      5. clear-num 76.1%

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

      6. un-div-inv 76.2%

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

      7. *-un-lft-identity 76.2%

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

      8. times-frac 76.2%

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

      9. /-rgt-identity 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. fma-udef 76.2%

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

      2. *-lft-identity 76.2%

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

      3. associate-/r* 76.2%

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

   8. Simplified 76.2%

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

      1. associate-/r/ 76.2%

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

   10. Applied egg-rr 76.2%

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

   if 1.12e30 < y.re

   1. Initial program 34.5%

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

   2. Taylor expanded in y.re around inf 80.3%

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

      1. unpow2 80.3%

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

      2. times-frac 85.2%

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

   4. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. div-inv 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. times-frac 85.3%

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

   8. Applied egg-rr 85.3%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{1} \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 660000000:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+30}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+307)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re 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 ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+307) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, 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 (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))) <= 2e+307)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, 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_] := If[LessEqual[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], 2e+307], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $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 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.99999999999999997e307

   1. Initial program 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. add-sqr-sqrt 78.1%

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

      3. times-frac 78.2%

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

      4. hypot-def 78.2%

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

      5. fma-def 78.2%

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

      6. hypot-def 95.1%

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

   3. Applied egg-rr 95.1%

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

   if 1.99999999999999997e307 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 6.4%

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

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

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

      1. unpow2 41.1%

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

      2. times-frac 57.6%

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

   4. Simplified 57.6%

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

      1. clear-num 57.6%

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

      2. un-div-inv 57.6%

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

   6. Applied egg-rr 57.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 3: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 580000000:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4e+35)
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (if (<= y.re 3.2e-113)
     (+ (/ (/ y.re y.im) (/ y.im x.re)) (/ x.im y.im))
     (if (<= y.re 580000000.0)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 9.8e+27)
         (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
         (+ (/ x.re y.re) (* y.im (/ (/ 1.0 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 <= -4e+35) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 3.2e-113) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	} else if (y_46_re <= 580000000.0) {
		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));
	} else if (y_46_re <= 9.8e+27) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / y_46_re) / (y_46_re / x_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 <= (-4d+35)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else if (y_46re <= 3.2d-113) then
        tmp = ((y_46re / y_46im) / (y_46im / x_46re)) + (x_46im / y_46im)
    else if (y_46re <= 580000000.0d0) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 9.8d+27) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else
        tmp = (x_46re / y_46re) + (y_46im * ((1.0d0 / y_46re) / (y_46re / x_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 <= -4e+35) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 3.2e-113) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	} else if (y_46_re <= 580000000.0) {
		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));
	} else if (y_46_re <= 9.8e+27) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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 <= -4e+35:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	elif y_46_re <= 3.2e-113:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im)
	elif y_46_re <= 580000000.0:
		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))
	elif y_46_re <= 9.8e+27:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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 <= -4e+35)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 3.2e-113)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)) + Float64(x_46_im / y_46_im));
	elseif (y_46_re <= 580000000.0)
		tmp = 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)));
	elseif (y_46_re <= 9.8e+27)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(1.0 / 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 <= -4e+35)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	elseif (y_46_re <= 3.2e-113)
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	elseif (y_46_re <= 580000000.0)
		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));
	elseif (y_46_re <= 9.8e+27)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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, -4e+35], 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, 3.2e-113], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 580000000.0], 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], If[LessEqual[y$46$re, 9.8e+27], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.9999999999999999e35

   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 73.4%

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

      1. unpow2 73.4%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -3.9999999999999999e35 < y.re < 3.2000000000000002e-113

   1. Initial program 67.2%

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

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

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. associate-/l* 78.2%

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

      4. associate-/r/ 78.0%

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

   4. Simplified 78.0%

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

      1. +-commutative 78.0%

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

      2. *-un-lft-identity 78.0%

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

      3. fma-def 78.0%

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

      4. *-commutative 78.0%

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

      5. clear-num 77.6%

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

      6. un-div-inv 78.4%

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

      7. *-un-lft-identity 78.4%

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

      8. times-frac 83.6%

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

      9. /-rgt-identity 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. fma-udef 83.6%

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

      2. *-lft-identity 83.6%

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

      3. associate-/r* 87.9%

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

   8. Simplified 87.9%

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

   if 3.2000000000000002e-113 < y.re < 5.8e8

   1. Initial program 83.1%

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

   if 5.8e8 < y.re < 9.8000000000000003e27

   1. Initial program 55.9%

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

   2. Taylor expanded in y.re around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. associate-/l* 76.2%

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

      4. associate-/r/ 76.1%

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

   4. Simplified 76.1%

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

      1. +-commutative 76.1%

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

      2. *-un-lft-identity 76.1%

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

      3. fma-def 76.1%

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

      4. *-commutative 76.1%

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

      5. clear-num 76.1%

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

      6. un-div-inv 76.2%

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

      7. *-un-lft-identity 76.2%

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

      8. times-frac 76.2%

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

      9. /-rgt-identity 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. fma-udef 76.2%

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

      2. *-lft-identity 76.2%

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

      3. associate-/r* 76.2%

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

   8. Simplified 76.2%

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

      1. associate-/r/ 76.2%

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

   10. Applied egg-rr 76.2%

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

   if 9.8000000000000003e27 < y.re

   1. Initial program 34.5%

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

   2. Taylor expanded in y.re around inf 80.3%

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

      1. unpow2 80.3%

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

      2. times-frac 85.2%

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

   4. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. div-inv 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. times-frac 85.3%

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

   8. Applied egg-rr 85.3%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{1} \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 580000000:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 4: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.7e+35)
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (if (<= y.re 3.2e+27)
     (+ (/ (/ y.re y.im) (/ y.im x.re)) (/ x.im y.im))
     (+ (/ x.re y.re) (* y.im (/ (/ 1.0 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 <= -2.7e+35) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 3.2e+27) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / y_46_re) / (y_46_re / x_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 <= (-2.7d+35)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else if (y_46re <= 3.2d+27) then
        tmp = ((y_46re / y_46im) / (y_46im / x_46re)) + (x_46im / y_46im)
    else
        tmp = (x_46re / y_46re) + (y_46im * ((1.0d0 / y_46re) / (y_46re / x_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 <= -2.7e+35) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 3.2e+27) {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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 <= -2.7e+35:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	elif y_46_re <= 3.2e+27:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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 <= -2.7e+35)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 3.2e+27)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)) + Float64(x_46_im / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(Float64(1.0 / 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 <= -2.7e+35)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	elseif (y_46_re <= 3.2e+27)
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_46_im / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + (y_46_im * ((1.0 / 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, -2.7e+35], 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, 3.2e+27], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.70000000000000003e35

   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 73.4%

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

      1. unpow2 73.4%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -2.70000000000000003e35 < y.re < 3.20000000000000015e27

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around 0 76.3%

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

      1. +-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. associate-/l* 75.8%

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

      4. associate-/r/ 75.7%

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

   4. Simplified 75.7%

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

      1. +-commutative 75.7%

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

      2. *-un-lft-identity 75.7%

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

      3. fma-def 75.7%

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

      4. *-commutative 75.7%

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

      5. clear-num 75.4%

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

      6. un-div-inv 76.0%

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

      7. *-un-lft-identity 76.0%

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

      8. times-frac 80.1%

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

      9. /-rgt-identity 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. fma-udef 80.1%

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

      2. *-lft-identity 80.1%

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

      3. associate-/r* 83.5%

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

   8. Simplified 83.5%

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

   if 3.20000000000000015e27 < y.re

   1. Initial program 34.5%

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

   2. Taylor expanded in y.re around inf 80.3%

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

      1. unpow2 80.3%

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

      2. times-frac 85.2%

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

   4. Simplified 85.2%

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

      1. clear-num 85.2%

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

      2. un-div-inv 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. div-inv 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. times-frac 85.3%

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

   8. Applied egg-rr 85.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{\frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 5: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+35} \lor \neg \left(y.re \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.7e+35) (not (<= y.re 2.8e+30)))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (* (+ x.im (/ x.re (/ y.im y.re))) (/ 1.0 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 <= -2.7e+35) || !(y_46_re <= 2.8e+30)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= (-2.7d+35)) .or. (.not. (y_46re <= 2.8d+30))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) * (1.0d0 / 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 <= -2.7e+35) || !(y_46_re <= 2.8e+30)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -2.7e+35) or not (y_46_re <= 2.8e+30):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -2.7e+35) || !(y_46_re <= 2.8e+30))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) * Float64(1.0 / 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 <= -2.7e+35) || ~((y_46_re <= 2.8e+30)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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, -2.7e+35], N[Not[LessEqual[y$46$re, 2.8e+30]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.70000000000000003e35 or 2.79999999999999983e30 < y.re

   1. Initial program 39.3%

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

   2. Taylor expanded in y.re around inf 76.5%

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

      1. unpow2 76.5%

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

      2. times-frac 84.1%

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

   4. Simplified 84.1%

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

   if -2.70000000000000003e35 < y.re < 2.79999999999999983e30

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around 0 55.8%

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

      1. unpow2 55.8%

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

   4. Simplified 55.8%

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

      1. associate-/r* 69.1%

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

      2. div-inv 69.1%

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

      3. fma-def 69.1%

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in x.re around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-/l* 83.4%

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

   9. Simplified 83.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+35} \lor \neg \left(y.re \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 6: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+35} \lor \neg \left(y.re \leq 2.9 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.15e+35) (not (<= y.re 2.9e+30)))
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (* (+ x.im (/ x.re (/ y.im y.re))) (/ 1.0 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 <= -2.15e+35) || !(y_46_re <= 2.9e+30)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= (-2.15d+35)) .or. (.not. (y_46re <= 2.9d+30))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) * (1.0d0 / 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 <= -2.15e+35) || !(y_46_re <= 2.9e+30)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -2.15e+35) or not (y_46_re <= 2.9e+30):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -2.15e+35) || !(y_46_re <= 2.9e+30))
		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 + Float64(x_46_re / Float64(y_46_im / y_46_re))) * Float64(1.0 / 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 <= -2.15e+35) || ~((y_46_re <= 2.9e+30)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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, -2.15e+35], N[Not[LessEqual[y$46$re, 2.9e+30]], $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 + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.1499999999999999e35 or 2.8999999999999998e30 < y.re

   1. Initial program 39.3%

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

   2. Taylor expanded in y.re around inf 76.5%

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

      1. unpow2 76.5%

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

      2. times-frac 84.1%

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

   4. Simplified 84.1%

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

      1. clear-num 84.1%

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

      2. 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 -2.1499999999999999e35 < y.re < 2.8999999999999998e30

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around 0 55.8%

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

      1. unpow2 55.8%

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

   4. Simplified 55.8%

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

      1. associate-/r* 69.1%

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

      2. div-inv 69.1%

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

      3. fma-def 69.1%

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in x.re around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-/l* 83.4%

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

   9. Simplified 83.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+35} \lor \neg \left(y.re \leq 2.9 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+35} \lor \neg \left(y.re \leq 5.1 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.75e+35) (not (<= y.re 5.1e+29)))
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (+ (/ (/ y.re y.im) (/ y.im x.re)) (/ x.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 <= -1.75e+35) || !(y_46_re <= 5.1e+29)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_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 <= (-1.75d+35)) .or. (.not. (y_46re <= 5.1d+29))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else
        tmp = ((y_46re / y_46im) / (y_46im / x_46re)) + (x_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 <= -1.75e+35) || !(y_46_re <= 5.1e+29)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_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 <= -1.75e+35) or not (y_46_re <= 5.1e+29):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	else:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_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 <= -1.75e+35) || !(y_46_re <= 5.1e+29))
		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(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_re)) + Float64(x_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 <= -1.75e+35) || ~((y_46_re <= 5.1e+29)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	else
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_re)) + (x_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, -1.75e+35], N[Not[LessEqual[y$46$re, 5.1e+29]], $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[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.75e35 or 5.1000000000000001e29 < y.re

   1. Initial program 39.3%

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

   2. Taylor expanded in y.re around inf 76.5%

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

      1. unpow2 76.5%

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

      2. times-frac 84.1%

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

   4. Simplified 84.1%

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

      1. clear-num 84.1%

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

      2. 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 -1.75e35 < y.re < 5.1000000000000001e29

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around 0 76.3%

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

      1. +-commutative 76.3%

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

      2. unpow2 76.3%

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

      3. associate-/l* 75.8%

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

      4. associate-/r/ 75.7%

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

   4. Simplified 75.7%

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

      1. +-commutative 75.7%

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

      2. *-un-lft-identity 75.7%

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

      3. fma-def 75.7%

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

      4. *-commutative 75.7%

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

      5. clear-num 75.4%

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

      6. un-div-inv 76.0%

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

      7. *-un-lft-identity 76.0%

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

      8. times-frac 80.1%

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

      9. /-rgt-identity 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. fma-udef 80.1%

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

      2. *-lft-identity 80.1%

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

      3. associate-/r* 83.5%

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

   8. Simplified 83.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+35} \lor \neg \left(y.re \leq 5.1 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \end{array} \]

Alternative 8: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+29}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.9e+35)
   (/ x.re y.re)
   (if (<= y.re 3.2e+29)
     (* (+ x.im (/ x.re (/ y.im y.re))) (/ 1.0 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 <= -3.9e+35) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.2e+29) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= (-3.9d+35)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.2d+29) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) * (1.0d0 / 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 <= -3.9e+35) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.2e+29) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -3.9e+35:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.2e+29:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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 <= -3.9e+35)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.2e+29)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) * Float64(1.0 / 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 <= -3.9e+35)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.2e+29)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (1.0 / 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, -3.9e+35], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.2e+29], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.8999999999999999e35 or 3.19999999999999987e29 < y.re

   1. Initial program 39.3%

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

   2. Taylor expanded in y.re around inf 74.0%

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

   if -3.8999999999999999e35 < y.re < 3.19999999999999987e29

   1. Initial program 68.7%

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

   2. Taylor expanded in y.re around 0 55.8%

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

      1. unpow2 55.8%

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

   4. Simplified 55.8%

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

      1. associate-/r* 69.1%

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

      2. div-inv 69.1%

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

      3. fma-def 69.1%

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in x.re around 0 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-/l* 83.4%

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

   9. Simplified 83.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+29}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9: 63.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+33)
   (/ x.re y.re)
   (if (<= y.re 2.3e+28) (/ 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 <= -1.15e+33) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.3e+28) {
		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 <= (-1.15d+33)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.3d+28) 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 <= -1.15e+33) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.3e+28) {
		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 <= -1.15e+33:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.3e+28:
		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 <= -1.15e+33)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.3e+28)
		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 <= -1.15e+33)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.3e+28)
		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, -1.15e+33], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+28], 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 < -1.15000000000000005e33 or 2.29999999999999984e28 < y.re

   1. Initial program 38.9%

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

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

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

   if -1.15000000000000005e33 < y.re < 2.29999999999999984e28

   1. Initial program 69.1%

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

   2. Taylor expanded in y.re around 0 68.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 42.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in y.re around 0 49.4%

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

3. Final simplification 49.4%

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

Reproduce

herbie shell --seed 2023199 
(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))))