_divideComplex, real part

Average Error: 26.7 → 13.6

Time: 25.4s

Precision: binary64

Cost: 14032

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ t_1 := \frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -6.2 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.02 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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.im y.im) (/ (* y.re (/ x.re y.im)) y.im)))
        (t_1 (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)))
   (if (<= y.im -6.2e+65)
     t_0
     (if (<= y.im -1.02e-115)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 9.2e-151)
         t_1
         (if (<= y.im 6.5e-77)
           (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
           (if (<= y.im 5.3e+20) t_1 t_0)))))))

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_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	double t_1 = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_im <= -6.2e+65) {
		tmp = t_0;
	} else if (y_46_im <= -1.02e-115) {
		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_im <= 9.2e-151) {
		tmp = t_1;
	} else if (y_46_im <= 6.5e-77) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 5.3e+20) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im))
	t_1 = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_im <= -6.2e+65)
		tmp = t_0;
	elseif (y_46_im <= -1.02e-115)
		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_im <= 9.2e-151)
		tmp = t_1;
	elseif (y_46_im <= 6.5e-77)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 5.3e+20)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -6.2e+65], t$95$0, If[LessEqual[y$46$im, -1.02e-115], 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$46$im, 9.2e-151], t$95$1, If[LessEqual[y$46$im, 6.5e-77], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.3e+20], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.19999999999999981e65 or 5.3e20 < y.im

   1. Initial program 34.7

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

   2. Taylor expanded in y.im around inf 17.7

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + x.re \cdot \left({\left(\frac{1}{y.im}\right)}^{2} \cdot y.re\right)} \]

   3. Taylor expanded in y.im around inf 17.7

      \[\leadsto \frac{x.im}{y.im} + x.re \cdot \color{blue}{\left(y.re \cdot {\left(\frac{1}{y.im}\right)}^{2}\right)} \]

   4. Simplified 16.3

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

   5. Applied egg-rr 13.3

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

   if -6.19999999999999981e65 < y.im < -1.02e-115

   1. Initial program 17.2

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

   if -1.02e-115 < y.im < 9.19999999999999984e-151 or 6.4999999999999999e-77 < y.im < 5.3e20

   1. Initial program 22.4

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

   2. Taylor expanded in y.im around 0 30.2

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

   3. Simplified 30.2

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

   4. Taylor expanded in x.re around 0 30.2

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

   5. Taylor expanded in y.re around inf 16.2

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + {\left(\frac{1}{y.re}\right)}^{2} \cdot \left(y.im \cdot x.im\right)} \]

   6. Simplified 12.2

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

   if 9.19999999999999984e-151 < y.im < 6.4999999999999999e-77

   1. Initial program 15.1

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

   2. Simplified 15.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
3. Recombined 4 regimes into one program.

Alternatives

Alternative 1
Error	13.8
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ t_2 := \frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.5
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im + \frac{y.re}{y.im} \cdot x.re}{y.im}\\ t_1 := \frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.9 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im\\ \end{array} \]

Alternative 3
Error	18.4
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.im + \frac{y.re}{y.im} \cdot x.re}{y.im}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Error	17.2
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.im + \frac{y.re}{y.im} \cdot x.re}{y.im}\\ t_1 := \frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.1 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Error	37.7
Cost	192

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

Reproduce

herbie shell --seed 2023010 
(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))))