_divideComplex, real part

Percentage Accurate: 61.6% → 79.7%

Time: 8.1s

Alternatives: 6

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: 61.6% 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: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))
   (if (<= y.re -2.35e+98)
     t_0
     (if (<= y.re -1.6e-9)
       (* (fma x.re y.re (* y.im x.im)) (/ 1.0 (fma y.re y.re (* y.im y.im))))
       (if (<= y.re 5.2e-123)
         (* (/ 1.0 y.im) (+ x.im (* x.re (/ y.re y.im))))
         (if (<= y.re 9.4e-54)
           (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.im y.im) (* y.re y.re)))
           (if (<= y.re 1.85e+56)
             (+ (/ 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) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -2.35e+98) {
		tmp = t_0;
	} else if (y_46_re <= -1.6e-9) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) * (1.0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_re <= 5.2e-123) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 9.4e-54) {
		tmp = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_re <= 1.85e+56) {
		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(x_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.35e+98)
		tmp = t_0;
	elseif (y_46_re <= -1.6e-9)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) * Float64(1.0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_re <= 5.2e-123)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 9.4e-54)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 1.85e+56)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * x_46_re) / Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.35e+98], t$95$0, If[LessEqual[y$46$re, -1.6e-9], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e-123], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.4e-54], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.85e+56], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.34999999999999985e98 or 1.84999999999999998e56 < y.re

   1. Initial program 35.1%

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

   2. Taylor expanded in y.re around inf 78.5%

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

      1. *-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. times-frac 87.4%

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

   4. Simplified 87.4%

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

   if -2.34999999999999985e98 < y.re < -1.60000000000000006e-9

   1. Initial program 72.3%

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

      1. div-inv 72.4%

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

      2. fma-def 72.4%

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

      3. fma-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   if -1.60000000000000006e-9 < y.re < 5.1999999999999999e-123

   1. Initial program 69.3%

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

   2. Taylor expanded in y.re around 0 88.2%

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

      1. *-commutative 88.2%

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

      2. unpow2 88.2%

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

   4. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. associate-/r* 95.5%

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

      3. div-inv 95.6%

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

      4. div-inv 95.4%

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

      5. distribute-rgt-out 96.4%

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

      6. div-inv 96.4%

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

      7. *-commutative 96.4%

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

      8. associate-*l* 96.4%

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

      9. div-inv 96.4%

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

   6. Applied egg-rr 96.4%

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

   if 5.1999999999999999e-123 < y.re < 9.4e-54

   1. Initial program 80.6%

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

   if 9.4e-54 < y.re < 1.84999999999999998e56

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. unpow2 69.9%

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

   4. Simplified 69.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 2: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.im y.im) (* y.re y.re))))
        (t_1 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))
   (if (<= y.re -3.2e+97)
     t_1
     (if (<= y.re -1.6e-9)
       t_0
       (if (<= y.re 1.5e-122)
         (* (/ 1.0 y.im) (+ x.im (* x.re (/ y.re y.im))))
         (if (<= y.re 2.25e-43)
           t_0
           (if (<= y.re 9e+56)
             (+ (/ x.im y.im) (/ (* y.re x.re) (* y.im y.im)))
             t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -3.2e+97) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-9) {
		tmp = t_0;
	} else if (y_46_re <= 1.5e-122) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 2.25e-43) {
		tmp = t_0;
	} else if (y_46_re <= 9e+56) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46im * x_46im) + (y_46re * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    t_1 = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    if (y_46re <= (-3.2d+97)) then
        tmp = t_1
    else if (y_46re <= (-1.6d-9)) then
        tmp = t_0
    else if (y_46re <= 1.5d-122) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re * (y_46re / y_46im)))
    else if (y_46re <= 2.25d-43) then
        tmp = t_0
    else if (y_46re <= 9d+56) then
        tmp = (x_46im / y_46im) + ((y_46re * x_46re) / (y_46im * y_46im))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -3.2e+97) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-9) {
		tmp = t_0;
	} else if (y_46_re <= 1.5e-122) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 2.25e-43) {
		tmp = t_0;
	} else if (y_46_re <= 9e+56) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -3.2e+97:
		tmp = t_1
	elif y_46_re <= -1.6e-9:
		tmp = t_0
	elif y_46_re <= 1.5e-122:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)))
	elif y_46_re <= 2.25e-43:
		tmp = t_0
	elif y_46_re <= 9e+56:
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -3.2e+97)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-9)
		tmp = t_0;
	elseif (y_46_re <= 1.5e-122)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 2.25e-43)
		tmp = t_0;
	elseif (y_46_re <= 9e+56)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * x_46_re) / Float64(y_46_im * y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -3.2e+97)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-9)
		tmp = t_0;
	elseif (y_46_re <= 1.5e-122)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	elseif (y_46_re <= 2.25e-43)
		tmp = t_0;
	elseif (y_46_re <= 9e+56)
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$re, -3.2e+97], t$95$1, If[LessEqual[y$46$re, -1.6e-9], t$95$0, If[LessEqual[y$46$re, 1.5e-122], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.25e-43], t$95$0, If[LessEqual[y$46$re, 9e+56], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.20000000000000016e97 or 9.0000000000000006e56 < y.re

   1. Initial program 35.1%

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

   2. Taylor expanded in y.re around inf 78.5%

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

      1. *-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. times-frac 87.4%

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

   4. Simplified 87.4%

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

   if -3.20000000000000016e97 < y.re < -1.60000000000000006e-9 or 1.50000000000000002e-122 < y.re < 2.25000000000000012e-43

   1. Initial program 75.3%

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

   if -1.60000000000000006e-9 < y.re < 1.50000000000000002e-122

   1. Initial program 69.3%

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

   2. Taylor expanded in y.re around 0 88.2%

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

      1. *-commutative 88.2%

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

      2. unpow2 88.2%

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

   4. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. associate-/r* 95.5%

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

      3. div-inv 95.6%

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

      4. div-inv 95.4%

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

      5. distribute-rgt-out 96.4%

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

      6. div-inv 96.4%

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

      7. *-commutative 96.4%

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

      8. associate-*l* 96.4%

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

      9. div-inv 96.4%

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

   6. Applied egg-rr 96.4%

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

   if 2.25000000000000012e-43 < y.re < 9.0000000000000006e56

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. unpow2 69.9%

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

   4. Simplified 69.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-43}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 3: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+88} \lor \neg \left(y.re \leq 9.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e+88) (not (<= y.re 9.5e+57)))
   (/ x.re y.re)
   (* (/ 1.0 y.im) (+ x.im (* x.re (/ 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 <= -6.5e+88) || !(y_46_re <= 9.5e+57)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= (-6.5d+88)) .or. (.not. (y_46re <= 9.5d+57))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re * (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 <= -6.5e+88) || !(y_46_re <= 9.5e+57)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= -6.5e+88) or not (y_46_re <= 9.5e+57):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= -6.5e+88) || !(y_46_re <= 9.5e+57))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re * 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 <= -6.5e+88) || ~((y_46_re <= 9.5e+57)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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, -6.5e+88], N[Not[LessEqual[y$46$re, 9.5e+57]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5000000000000002e88 or 9.4999999999999997e57 < y.re

   1. Initial program 38.1%

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

   2. Taylor expanded in y.re around inf 77.1%

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

   if -6.5000000000000002e88 < y.re < 9.4999999999999997e57

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. unpow2 75.2%

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

   4. Simplified 75.2%

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

      1. +-commutative 75.2%

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

      2. associate-/r* 81.0%

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

      3. div-inv 81.0%

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

      4. div-inv 80.9%

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

      5. distribute-rgt-out 81.5%

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

      6. div-inv 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*l* 82.1%

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

      9. div-inv 82.1%

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

   6. Applied egg-rr 82.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+88} \lor \neg \left(y.re \leq 9.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 4: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.05 \cdot 10^{+88} \lor \neg \left(y.re \leq 10^{+57}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.05e+88) (not (<= y.re 1e+57)))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (* (/ 1.0 y.im) (+ x.im (* x.re (/ 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 <= -2.05e+88) || !(y_46_re <= 1e+57)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= (-2.05d+88)) .or. (.not. (y_46re <= 1d+57))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re * (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 <= -2.05e+88) || !(y_46_re <= 1e+57)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= -2.05e+88) or not (y_46_re <= 1e+57):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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 <= -2.05e+88) || !(y_46_re <= 1e+57))
		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(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re * 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 <= -2.05e+88) || ~((y_46_re <= 1e+57)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (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, -2.05e+88], N[Not[LessEqual[y$46$re, 1e+57]], $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[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.05000000000000014e88 or 1.00000000000000005e57 < y.re

   1. Initial program 38.1%

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

   2. Taylor expanded in y.re around inf 77.9%

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

      1. *-commutative 77.9%

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

      2. unpow2 77.9%

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

      3. times-frac 85.4%

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

   4. Simplified 85.4%

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

   if -2.05000000000000014e88 < y.re < 1.00000000000000005e57

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. unpow2 75.2%

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

   4. Simplified 75.2%

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

      1. +-commutative 75.2%

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

      2. associate-/r* 81.0%

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

      3. div-inv 81.0%

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

      4. div-inv 80.9%

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

      5. distribute-rgt-out 81.5%

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

      6. div-inv 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*l* 82.1%

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

      9. div-inv 82.1%

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

   6. Applied egg-rr 82.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.05 \cdot 10^{+88} \lor \neg \left(y.re \leq 10^{+57}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 5: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.6e-9)
   (/ x.re y.re)
   (if (<= y.re 3.1e+19) (/ 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 <= -1.6e-9) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+19) {
		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 <= (-1.6d-9)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.1d+19) 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 <= -1.6e-9) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+19) {
		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 <= -1.6e-9:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.1e+19:
		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 <= -1.6e-9)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.1e+19)
		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 <= -1.6e-9)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.1e+19)
		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, -1.6e-9], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.1e+19], 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 < -1.60000000000000006e-9 or 3.1e19 < y.re

   1. Initial program 44.7%

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

   2. Taylor expanded in y.re around inf 70.0%

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

   if -1.60000000000000006e-9 < y.re < 3.1e19

   1. Initial program 69.2%

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

   2. Taylor expanded in y.re around 0 75.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 6: 41.8% 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 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 46.4%

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

3. Final simplification 46.4%

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

Reproduce

herbie shell --seed 2023297 
(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))))