_divideComplex, real part

Average Error: 26.1 → 5.9

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -9.581051578847133 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, 0.5, 0.5 \cdot \frac{x.im}{{\left(\frac{y.re}{y.im}\right)}^{3}} - \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.523261181904555 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re \cdot x.re, \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, {\left(\sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im}}}\right)}^{3}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{y.re} + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right), x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

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
 (if (<= y.re -9.581051578847133e+165)
   (/
    (fma
     (* (/ x.re y.re) (/ y.im (/ y.re y.im)))
     0.5
     (-
      (* 0.5 (/ x.im (pow (/ y.re y.im) 3.0)))
      (fma (/ y.im y.re) x.im x.re)))
    (hypot y.re y.im))
   (if (<= y.re 4.523261181904555e+181)
     (/
      (fma
       (* y.re x.re)
       (/ 1.0 (hypot y.re y.im))
       (pow (cbrt (/ y.im (/ (hypot y.im y.re) x.im))) 3.0))
      (hypot y.re y.im))
     (/
      (+
       (* x.im (/ y.im y.re))
       (fma -0.5 (* (/ x.re y.re) (* y.im (/ y.im y.re))) x.re))
      (hypot 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 tmp;
	if (y_46_re <= -9.581051578847133e+165) {
		tmp = fma(((x_46_re / y_46_re) * (y_46_im / (y_46_re / y_46_im))), 0.5, ((0.5 * (x_46_im / pow((y_46_re / y_46_im), 3.0))) - fma((y_46_im / y_46_re), x_46_im, x_46_re))) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 4.523261181904555e+181) {
		tmp = fma((y_46_re * x_46_re), (1.0 / hypot(y_46_re, y_46_im)), pow(cbrt((y_46_im / (hypot(y_46_im, y_46_re) / x_46_im))), 3.0)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = ((x_46_im * (y_46_im / y_46_re)) + fma(-0.5, ((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re))), x_46_re)) / hypot(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)
	tmp = 0.0
	if (y_46_re <= -9.581051578847133e+165)
		tmp = Float64(fma(Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / Float64(y_46_re / y_46_im))), 0.5, Float64(Float64(0.5 * Float64(x_46_im / (Float64(y_46_re / y_46_im) ^ 3.0))) - fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re))) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 4.523261181904555e+181)
		tmp = Float64(fma(Float64(y_46_re * x_46_re), Float64(1.0 / hypot(y_46_re, y_46_im)), (cbrt(Float64(y_46_im / Float64(hypot(y_46_im, y_46_re) / x_46_im))) ^ 3.0)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) + fma(-0.5, Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im * Float64(y_46_im / y_46_re))), x_46_re)) / hypot(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_] := If[LessEqual[y$46$re, -9.581051578847133e+165], N[(N[(N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.5 * N[(x$46$im / N[Power[N[(y$46$re / y$46$im), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.523261181904555e+181], N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] * N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(y$46$im / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.58105157884713253e165

   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. Simplified 45.1

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

   3. Applied egg-rr 31.5

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

   4. Applied egg-rr 31.5

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

   5. Taylor expanded in y.re around -inf 25.0

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + \left(0.5 \cdot \frac{{y.im}^{3} \cdot x.im}{{y.re}^{3}} + -1 \cdot x.re\right)\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   6. Simplified 6.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, 0.5, \frac{x.im}{{\left(\frac{y.re}{y.im}\right)}^{3}} \cdot 0.5 - \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -9.58105157884713253e165 < y.re < 4.52326118190455499e181

   1. Initial program 20.7

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

   2. Simplified 20.7

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

   3. Applied egg-rr 13.3

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

   4. Applied egg-rr 13.2

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

   5. Applied egg-rr 13.3

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

   6. Applied egg-rr 5.8

      \[\leadsto \frac{\mathsf{fma}\left(x.re \cdot y.re, \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{{\left(\sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im}}}\right)}^{3}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if 4.52326118190455499e181 < 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. Simplified 42.8

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

   3. Applied egg-rr 29.0

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

   4. Applied egg-rr 29.0

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

   5. Applied egg-rr 29.0

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

   6. Taylor expanded in y.re around inf 19.7

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   7. Simplified 5.9

      \[\leadsto \frac{\color{blue}{\frac{y.im}{y.re} \cdot x.im + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \left(\frac{y.im}{y.re} \cdot y.im\right), x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.581051578847133 \cdot 10^{+165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}, 0.5, 0.5 \cdot \frac{x.im}{{\left(\frac{y.re}{y.im}\right)}^{3}} - \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.523261181904555 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re \cdot x.re, \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, {\left(\sqrt[3]{\frac{y.im}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im}}}\right)}^{3}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{y.re} + \mathsf{fma}\left(-0.5, \frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right), x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))