_divideComplex, real part

Average Error: 26.1 → 10.4

Time: 14.9s

Precision: binary64

Cost: 22088

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

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))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 -1e+305)
     (/ (/ y.re (hypot y.im y.re)) (/ (hypot y.im y.re) x.re))
     (if (<= t_0 2e+282)
       (*
        (/ 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.re 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));
}

↓

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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_0 <= -1e+305) {
		tmp = (y_46_re / hypot(y_46_im, y_46_re)) / (hypot(y_46_im, y_46_re) / x_46_re);
	} else if (t_0 <= 2e+282) {
		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_re / y_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(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)))
	tmp = 0.0
	if (t_0 <= -1e+305)
		tmp = Float64(Float64(y_46_re / hypot(y_46_im, y_46_re)) / Float64(hypot(y_46_im, y_46_re) / x_46_re));
	elseif (t_0 <= 2e+282)
		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(x_46_im / Float64(y_46_re * Float64(y_46_re / y_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[(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[t$95$0, -1e+305], N[(N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], 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[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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))) < -9.9999999999999994e304

   1. Initial program 62.7

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

   2. Taylor expanded in x.re around inf 62.8

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

   3. Simplified 42.9

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

      [Start] 62.8	\[ \frac{x.re \cdot y.re}{{y.re}^{2} + {y.im}^{2}} \]
      associate-/l* [=>] 42.2	\[ \color{blue}{\frac{x.re}{\frac{{y.re}^{2} + {y.im}^{2}}{y.re}}} \]
      associate-/r/ [=>] 42.9	\[ \color{blue}{\frac{x.re}{{y.re}^{2} + {y.im}^{2}} \cdot y.re} \]
      +-commutative [=>] 42.9	\[ \frac{x.re}{\color{blue}{{y.im}^{2} + {y.re}^{2}}} \cdot y.re \]
      unpow2 [=>] 42.9	\[ \frac{x.re}{\color{blue}{y.im \cdot y.im} + {y.re}^{2}} \cdot y.re \]
      fma-def [=>] 42.9	\[ \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot y.re \]
      unpow2 [=>] 42.9	\[ \frac{x.re}{\mathsf{fma}\left(y.im, y.im, \color{blue}{y.re \cdot y.re}\right)} \cdot y.re \]

   4. Applied egg-rr 33.5

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

   if -9.9999999999999994e304 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.00000000000000007e282

   1. Initial program 11.5

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

   2. Applied egg-rr 0.9

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

   if 2.00000000000000007e282 < (/.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 62.3

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

   2. Taylor expanded in y.re around inf 40.2

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

   3. Simplified 33.0

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

      [Start] 40.2	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      *-commutative [<=] 40.2	\[ \frac{x.re}{y.re} + \frac{\color{blue}{x.im \cdot y.im}}{{y.re}^{2}} \]
      unpow2 [=>] 40.2	\[ \frac{x.re}{y.re} + \frac{x.im \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 33.0	\[ \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{y.re} \cdot \frac{y.im}{y.re}} \]

   4. Applied egg-rr 34.3

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -1 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	13772

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	11.9
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-148}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ t_2 := \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}}\\ \mathbf{if}\;y.re \leq -9.8 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1364

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.6
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.05 \cdot 10^{+54} \lor \neg \left(y.re \leq -4 \cdot 10^{+15}\right) \land \left(y.re \leq -3.7 \cdot 10^{-32} \lor \neg \left(y.re \leq 3.8 \cdot 10^{+47}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -5.1 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	24.4
Cost	976

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	24.3
Cost	976

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Error	16.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-21} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 10
Error	16.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-21} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-82}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 11
Error	15.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{-21} \lor \neg \left(y.im \leq 6 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 12
Error	22.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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))))