_divideComplex, real part

Average Error: 26.0 → 10.0

Time: 17.8s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.903477509570262 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_1}}\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.281947855931773 \cdot 10^{+159}:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \end{array} \]

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))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))))
   (if (<= y.re -4.903477509570262e+67)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (if (<= y.re -1e-115)
       (/ 1.0 (/ (hypot y.re y.im) t_1))
       (if (<= y.re 1e-130)
         (* (/ 1.0 y.im) (+ x.im (* x.re (/ y.re y.im))))
         (if (<= y.re 2.281947855931773e+159)
           (* t_1 t_0)
           (* t_0 (+ x.re (* (/ y.im y.re) x.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));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -4.903477509570262e+67) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= -1e-115) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / t_1);
	} else if (y_46_re <= 1e-130) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 2.281947855931773e+159) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * (x_46_re + ((y_46_im / y_46_re) * x_46_im));
	}
	return tmp;
}

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

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.903477509570262e+67)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= -1e-115)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / t_1));
	elseif (y_46_re <= 1e-130)
		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 <= 2.281947855931773e+159)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_0 * Float64(x_46_re + Float64(Float64(y_46_im / y_46_re) * x_46_im)));
	end
	return tmp
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]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.903477509570262e+67], 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, -1e-115], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e-130], 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, 2.281947855931773e+159], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(x$46$re + N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.9034775095702621e67

   1. Initial program 36.6

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

   2. Applied egg-rr 25.1

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

   3. Taylor expanded in y.re around inf 15.8

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

   4. Simplified 11.2

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 45 points increase in error, 14 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   if -4.9034775095702621e67 < y.re < -1.0000000000000001e-115

   1. Initial program 13.9

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

   2. Applied egg-rr 9.5

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

   3. Applied egg-rr 9.7

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

   if -1.0000000000000001e-115 < y.re < 1.0000000000000001e-130

   1. Initial program 22.3

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

   2. Applied egg-rr 12.5

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

   3. Taylor expanded in y.re around 0 31.6

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

   4. Simplified 31.5

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)} \]
      Proof
      (+.f64 x.im (*.f64 (/.f64 y.re y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 x.im (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 y.im x.re)))): 16 points increase in error, 24 points decrease in error
      (+.f64 x.im (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) y.im))): 23 points increase in error, 14 points decrease in error
      (+.f64 x.im (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) y.im) x.im)): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in y.re around 0 7.3

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

   if 1.0000000000000001e-130 < y.re < 2.281947855931773e159

   1. Initial program 18.3

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

   2. Applied egg-rr 13.6

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

   if 2.281947855931773e159 < y.re

   1. Initial program 45.1

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

   2. Applied egg-rr 29.3

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

   3. Taylor expanded in y.re around inf 10.8

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

   4. Simplified 6.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + \frac{y.im}{y.re} \cdot x.im\right)} \]
      Proof
      (+.f64 x.re (*.f64 (/.f64 y.im y.re) x.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) y.re))): 25 points increase in error, 28 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.903477509570262 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.281947855931773 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	20560

\[\begin{array}{l} t_0 := \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{if}\;y.re \leq -4.903477509570262 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.281947855931773 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \end{array} \]

Alternative 2
Error	11.4
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.57134307228816 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 7.2010048922993756 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \end{array} \]

Alternative 3
Error	11.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.57134307228816 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 5.346374350427911 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 4
Error	20.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -7.558892591188873 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4308316382960976 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -925777994571.0144:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 5
Error	16.7
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1.8151233042098852 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.0
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1.8151233042098852 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.8151233042098852 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 8
Error	25.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.145997256351953 \cdot 10^{+173}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.781394454418386 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq -1.8151233042098852 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.4263770608205756 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 26419729.609084405:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \end{array} \]

Alternative 10
Error	22.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.4263770608205756 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.0595451483748359 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	58.6
Cost	192

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

Alternative 12
Error	36.7
Cost	192

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

Reproduce

herbie shell --seed 2022317 
(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))))