_divideComplex, real part

Percentage Accurate: 61.9% → 80.7%

Time: 11.9s

Alternatives: 9

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: 61.9% 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: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ t_1 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{y.im}{\frac{t_1}{x.im}} + \frac{y.re}{\frac{t_1}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ 1.0 (/ y.re (* y.im (/ x.im y.re))))))
        (t_1 (fma y.im y.im (* y.re y.re))))
   (if (<= y.re -4e+103)
     t_0
     (if (<= y.re -2.2e-80)
       (+ (/ y.im (/ t_1 x.im)) (/ y.re (/ t_1 x.re)))
       (if (<= y.re 1.35e+52)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         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) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	double t_1 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -4e+103) {
		tmp = t_0;
	} else if (y_46_re <= -2.2e-80) {
		tmp = (y_46_im / (t_1 / x_46_im)) + (y_46_re / (t_1 / x_46_re));
	} else if (y_46_re <= 1.35e+52) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} 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(1.0 / Float64(y_46_re / Float64(y_46_im * Float64(x_46_im / y_46_re)))))
	t_1 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -4e+103)
		tmp = t_0;
	elseif (y_46_re <= -2.2e-80)
		tmp = Float64(Float64(y_46_im / Float64(t_1 / x_46_im)) + Float64(y_46_re / Float64(t_1 / x_46_re)));
	elseif (y_46_re <= 1.35e+52)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	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[(1.0 / N[(y$46$re / N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4e+103], t$95$0, If[LessEqual[y$46$re, -2.2e-80], N[(N[(y$46$im / N[(t$95$1 / x$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$re / N[(t$95$1 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+52], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4e103 or 1.35e52 < y.re

   1. Initial program 42.8%

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

   2. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. associate-/l* 79.5%

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

   4. Simplified 79.5%

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

      1. clear-num 79.5%

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

      2. inv-pow 79.5%

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

      3. associate-/l* 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. unpow-1 83.2%

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

      2. associate-/l/ 86.8%

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

   8. Simplified 86.8%

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

   if -4e103 < y.re < -2.2000000000000001e-80

   1. Initial program 79.9%

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

   2. Taylor expanded in x.re around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-/l* 85.9%

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

      3. unpow2 85.9%

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

      4. unpow2 85.9%

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

      5. fma-udef 85.9%

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

      6. *-commutative 85.9%

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

      7. associate-/l* 88.2%

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

      8. unpow2 88.2%

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

      9. unpow2 88.2%

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

      10. fma-udef 88.2%

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

   4. Simplified 88.2%

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

   if -2.2000000000000001e-80 < y.re < 1.35e52

   1. Initial program 67.0%

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

      1. *-un-lft-identity 67.0%

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

      2. add-sqr-sqrt 67.0%

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

      3. times-frac 67.0%

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

      4. hypot-def 67.0%

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

      5. fma-def 67.0%

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

      6. hypot-def 81.7%

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

   3. Applied egg-rr 81.7%

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

   4. Taylor expanded in y.re around 0 47.8%

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

      1. associate-/l* 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in y.re around 0 85.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{y.im}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.im}} + \frac{y.re}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \end{array} \]

Alternative 2: 84.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+270}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

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)))
      1e+270)
   (*
    (/ 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) (* (/ x.im y.re) (/ y.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+270) {
		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) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 1e+270)
		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(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$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], 1e+270], 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[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $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))) < 1e270

   1. Initial program 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. add-sqr-sqrt 78.8%

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

      3. times-frac 78.7%

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

      4. hypot-def 78.7%

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

      5. fma-def 78.7%

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

      6. hypot-def 96.5%

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

   3. Applied egg-rr 96.5%

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

   if 1e270 < (/.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 9.0%

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

   2. Taylor expanded in y.re around inf 48.4%

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

      1. *-commutative 48.4%

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

      2. unpow2 48.4%

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

      3. associate-/l* 51.7%

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

   4. Simplified 51.7%

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

   5. Taylor expanded in y.re around 0 51.7%

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

      1. unpow2 51.7%

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

      2. associate-*l/ 57.1%

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

   7. Simplified 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. times-frac 59.8%

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

      3. clear-num 59.8%

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

   9. Applied egg-rr 59.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+270}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 3: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ 1.0 (/ y.re (* y.im (/ x.im y.re)))))))
   (if (<= y.re -1.7e+89)
     t_0
     (if (<= y.re -1.45e-93)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.25e+49)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         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) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	double tmp;
	if (y_46_re <= -1.7e+89) {
		tmp = t_0;
	} else if (y_46_re <= -1.45e-93) {
		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 <= 1.25e+49) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + (1.0d0 / (y_46re / (y_46im * (x_46im / y_46re))))
    if (y_46re <= (-1.7d+89)) then
        tmp = t_0
    else if (y_46re <= (-1.45d-93)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1.25d+49) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	double tmp;
	if (y_46_re <= -1.7e+89) {
		tmp = t_0;
	} else if (y_46_re <= -1.45e-93) {
		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 <= 1.25e+49) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))))
	tmp = 0
	if y_46_re <= -1.7e+89:
		tmp = t_0
	elif y_46_re <= -1.45e-93:
		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 <= 1.25e+49:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	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(1.0 / Float64(y_46_re / Float64(y_46_im * Float64(x_46_im / y_46_re)))))
	tmp = 0.0
	if (y_46_re <= -1.7e+89)
		tmp = t_0;
	elseif (y_46_re <= -1.45e-93)
		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 <= 1.25e+49)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.7e+89)
		tmp = t_0;
	elseif (y_46_re <= -1.45e-93)
		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 <= 1.25e+49)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.7e+89], t$95$0, If[LessEqual[y$46$re, -1.45e-93], 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, 1.25e+49], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.7000000000000001e89 or 1.2500000000000001e49 < y.re

   1. Initial program 43.1%

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

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

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. associate-/l* 79.3%

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

   4. Simplified 79.3%

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

      1. clear-num 79.3%

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

      2. inv-pow 79.3%

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

      3. associate-/l* 82.9%

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

   6. Applied egg-rr 82.9%

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

      1. unpow-1 82.9%

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

      2. associate-/l/ 86.4%

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

   8. Simplified 86.4%

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

   if -1.7000000000000001e89 < y.re < -1.4499999999999999e-93

   1. Initial program 85.3%

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

   if -1.4499999999999999e-93 < y.re < 1.2500000000000001e49

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 66.2%

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

      4. hypot-def 66.2%

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

      5. fma-def 66.2%

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

      6. hypot-def 81.3%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in y.re around 0 48.0%

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

      1. associate-/l* 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in y.re around 0 85.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \end{array} \]

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-53} \lor \neg \left(y.re \leq 9.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e-53) (not (<= y.re 9.6e+48)))
   (+ (/ x.re y.re) (/ 1.0 (/ y.re (* y.im (/ x.im y.re)))))
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.5e-53) || !(y_46_re <= 9.6e+48)) {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6.5d-53)) .or. (.not. (y_46re <= 9.6d+48))) then
        tmp = (x_46re / y_46re) + (1.0d0 / (y_46re / (y_46im * (x_46im / y_46re))))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.5e-53) || !(y_46_re <= 9.6e+48)) {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6.5e-53) or not (y_46_re <= 9.6e+48):
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.5e-53) || !(y_46_re <= 9.6e+48))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(y_46_im * 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_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.5e-53) || ~((y_46_re <= 9.6e+48)))
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (y_46_im * (x_46_im / y_46_re))));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6.5e-53], N[Not[LessEqual[y$46$re, 9.6e+48]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.4999999999999997e-53 or 9.6000000000000004e48 < y.re

   1. Initial program 51.0%

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

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

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

      1. *-commutative 75.7%

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

      2. unpow2 75.7%

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

      3. associate-/l* 77.4%

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

   4. Simplified 77.4%

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

      1. clear-num 77.0%

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

      2. inv-pow 77.0%

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

      3. associate-/l* 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. unpow-1 79.9%

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

      2. associate-/l/ 82.7%

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

   8. Simplified 82.7%

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

   if -6.4999999999999997e-53 < y.re < 9.6000000000000004e48

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.6%

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

      4. hypot-def 67.6%

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

      5. fma-def 67.6%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around 0 46.8%

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

      1. associate-/l* 46.8%

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

   6. Simplified 46.8%

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

   7. Taylor expanded in y.re around 0 84.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-53} \lor \neg \left(y.re \leq 9.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.6 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.16e-47) (not (<= y.re 1.6e+50)))
   (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.16e-47) || !(y_46_re <= 1.6e+50)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.16d-47)) .or. (.not. (y_46re <= 1.6d+50))) then
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im / y_46re) / y_46re))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.16e-47) || !(y_46_re <= 1.6e+50)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.16e-47) or not (y_46_re <= 1.6e+50):
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.16e-47) || !(y_46_re <= 1.6e+50))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_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_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.16e-47) || ~((y_46_re <= 1.6e+50)))
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.16e-47], N[Not[LessEqual[y$46$re, 1.6e+50]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.1600000000000001e-47 or 1.59999999999999991e50 < y.re

   1. Initial program 51.0%

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

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

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

      1. *-commutative 75.7%

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

      2. unpow2 75.7%

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

      3. associate-/l* 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in y.im around 0 75.7%

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

      1. associate-*r/ 77.4%

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

      2. unpow2 77.4%

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

      3. associate-/r* 78.6%

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

   7. Simplified 78.6%

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

   if -1.1600000000000001e-47 < y.re < 1.59999999999999991e50

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.6%

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

      4. hypot-def 67.6%

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

      5. fma-def 67.6%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around 0 46.8%

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

      1. associate-/l* 46.8%

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

   6. Simplified 46.8%

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

   7. Taylor expanded in y.re around 0 84.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47} \lor \neg \left(y.re \leq 1.6 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 6: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-53} \lor \neg \left(y.re \leq 7.2 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.2e-53) (not (<= y.re 7.2e+48)))
   (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.2e-53) || !(y_46_re <= 7.2e+48)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6.2d-53)) .or. (.not. (y_46re <= 7.2d+48))) then
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.2e-53) || !(y_46_re <= 7.2e+48)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6.2e-53) or not (y_46_re <= 7.2e+48):
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.2e-53) || !(y_46_re <= 7.2e+48))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_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_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.2e-53) || ~((y_46_re <= 7.2e+48)))
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6.2e-53], N[Not[LessEqual[y$46$re, 7.2e+48]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$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$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.20000000000000031e-53 or 7.19999999999999967e48 < y.re

   1. Initial program 51.0%

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

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

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

      1. *-commutative 75.7%

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

      2. unpow2 75.7%

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

      3. associate-/l* 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in y.re around 0 77.4%

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

      1. unpow2 77.4%

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

      2. associate-*l/ 80.3%

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

   7. Simplified 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. times-frac 81.7%

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

      3. clear-num 81.8%

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

   9. Applied egg-rr 81.8%

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

   if -6.20000000000000031e-53 < y.re < 7.19999999999999967e48

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.6%

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

      4. hypot-def 67.6%

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

      5. fma-def 67.6%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around 0 46.8%

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

      1. associate-/l* 46.8%

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

   6. Simplified 46.8%

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

   7. Taylor expanded in y.re around 0 84.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-53} \lor \neg \left(y.re \leq 7.2 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 7: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.2e-75)
   (/ x.re y.re)
   (if (<= y.re 6.2e+49)
     (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.2e-75) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.2e+49) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} 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 <= (-5.2d-75)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 6.2d+49) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    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 <= -5.2e-75) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.2e+49) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} 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 <= -5.2e-75:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 6.2e+49:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	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 <= -5.2e-75)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 6.2e+49)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	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 <= -5.2e-75)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 6.2e+49)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	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, -5.2e-75], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+49], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.2e-75 or 6.19999999999999985e49 < y.re

   1. Initial program 51.8%

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

   2. Taylor expanded in y.re around inf 70.6%

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

   if -5.2e-75 < y.re < 6.19999999999999985e49

   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. Step-by-step derivation

      1. *-un-lft-identity 67.2%

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

      2. add-sqr-sqrt 67.2%

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

      3. times-frac 67.3%

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

      4. hypot-def 67.3%

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

      5. fma-def 67.3%

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

      6. hypot-def 81.9%

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

   3. Applied egg-rr 81.9%

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

   4. Taylor expanded in y.re around 0 47.4%

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

      1. associate-/l* 47.4%

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

   6. Simplified 47.4%

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

   7. Taylor expanded in y.re around 0 85.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8: 63.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.75 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.16e-47)
   (/ x.re y.re)
   (if (<= y.re 2.75e+52) (/ 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 <= -1.16e-47) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.75e+52) {
		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 <= (-1.16d-47)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.75d+52) 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 <= -1.16e-47) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.75e+52) {
		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 <= -1.16e-47:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.75e+52:
		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 <= -1.16e-47)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.75e+52)
		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 <= -1.16e-47)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.75e+52)
		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, -1.16e-47], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.75e+52], 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 < -1.1600000000000001e-47 or 2.74999999999999998e52 < y.re

   1. Initial program 51.0%

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

   2. Taylor expanded in y.re around inf 71.2%

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

   if -1.1600000000000001e-47 < y.re < 2.74999999999999998e52

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 72.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.75 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9: 42.5% 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 59.2%

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

2. Taylor expanded in y.re around 0 44.9%

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

3. Final simplification 44.9%

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

Reproduce

herbie shell --seed 2023283 
(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))))