_divideComplex, real part

Average Error: 26.3 → 11.2

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+157}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) - t_0\right)\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_0\\ \end{array} \]

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 (fma (/ y.im y.re) x.im x.re))
        (t_1 (/ 1.0 (hypot y.re y.im)))
        (t_2
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.re -1.1e+157)
     (* t_1 (- (* 0.5 (* (/ x.re y.re) (* y.im (/ y.im y.re)))) t_0))
     (if (<= y.re -4.3e-27)
       t_2
       (if (<= y.re 1.65e-284)
         (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))
         (if (<= y.re 4.6e+108) t_2 (* 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 = fma((y_46_im / y_46_re), x_46_im, x_46_re);
	double t_1 = 1.0 / hypot(y_46_re, y_46_im);
	double t_2 = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -1.1e+157) {
		tmp = t_1 * ((0.5 * ((x_46_re / y_46_re) * (y_46_im * (y_46_im / y_46_re)))) - t_0);
	} else if (y_46_re <= -4.3e-27) {
		tmp = t_2;
	} else if (y_46_re <= 1.65e-284) {
		tmp = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	} else if (y_46_re <= 4.6e+108) {
		tmp = t_2;
	} else {
		tmp = t_1 * 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 = fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re)
	t_1 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_2 = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.1e+157)
		tmp = Float64(t_1 * Float64(Float64(0.5 * Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im * Float64(y_46_im / y_46_re)))) - t_0));
	elseif (y_46_re <= -4.3e-27)
		tmp = t_2;
	elseif (y_46_re <= 1.65e-284)
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im));
	elseif (y_46_re <= 4.6e+108)
		tmp = t_2;
	else
		tmp = Float64(t_1 * 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[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.1e+157], N[(t$95$1 * N[(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]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.3e-27], t$95$2, If[LessEqual[y$46$re, 1.65e-284], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.6e+108], t$95$2, N[(t$95$1 * t$95$0), $MachinePrecision]]]]]]]]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 4 regimes

2. if y.re < -1.1000000000000001e157

   1. Initial program 45.3

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

   2. Simplified 45.3

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

   3. Applied egg-rr 29.6

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

   4. Taylor expanded in y.re around -inf 21.6

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

   5. Simplified 6.7

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

   if -1.1000000000000001e157 < y.re < -4.30000000000000002e-27 or 1.65000000000000004e-284 < y.re < 4.5999999999999998e108

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   3. Applied egg-rr 12.3

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

   4. Applied egg-rr 12.6

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

   5. Applied egg-rr 12.2

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

   if -4.30000000000000002e-27 < y.re < 1.65000000000000004e-284

   1. Initial program 20.2

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

   2. Simplified 20.2

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

   3. Applied egg-rr 11.7

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

   4. Taylor expanded in y.re around 0 14.5

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

   5. Simplified 12.9

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

   if 4.5999999999999998e108 < y.re

   1. Initial program 40.7

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

   2. Simplified 40.7

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

   3. Applied egg-rr 27.3

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

   4. Taylor expanded in y.re around inf 13.6

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

   5. Simplified 9.8

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(0.5 \cdot \left(\frac{x.re}{y.re} \cdot \left(y.im \cdot \frac{y.im}{y.re}\right)\right) - \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))