_divideComplex, real part

Average Error: 26.6 → 6.1

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\right)\\ t_1 := \mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{if}\;y.im \leq -5.165129542917535 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -6.458426761032663 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.642430271924372 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.8787453566117075 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
          x.im
          (/ y.im (fma y.im y.im (* y.re y.re)))
          (* x.re (* (/ 1.0 (hypot y.im y.re)) (/ y.re (hypot y.im y.re))))))
        (t_1 (fma (/ y.re y.im) (/ x.re y.im) (/ x.im y.im))))
   (if (<= y.im -5.165129542917535e+152)
     t_1
     (if (<= y.im -6.458426761032663e-136)
       t_0
       (if (<= y.im 1.642430271924372e-175)
         (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
         (if (<= y.im 2.8787453566117075e+148) t_0 t_1))))))

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(x_46_im, (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))), (x_46_re * ((1.0 / hypot(y_46_im, y_46_re)) * (y_46_re / hypot(y_46_im, y_46_re)))));
	double t_1 = fma((y_46_re / y_46_im), (x_46_re / y_46_im), (x_46_im / y_46_im));
	double tmp;
	if (y_46_im <= -5.165129542917535e+152) {
		tmp = t_1;
	} else if (y_46_im <= -6.458426761032663e-136) {
		tmp = t_0;
	} else if (y_46_im <= 1.642430271924372e-175) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_im <= 2.8787453566117075e+148) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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(x_46_im, Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))), Float64(x_46_re * Float64(Float64(1.0 / hypot(y_46_im, y_46_re)) * Float64(y_46_re / hypot(y_46_im, y_46_re)))))
	t_1 = fma(Float64(y_46_re / y_46_im), Float64(x_46_re / y_46_im), Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -5.165129542917535e+152)
		tmp = t_1;
	elseif (y_46_im <= -6.458426761032663e-136)
		tmp = t_0;
	elseif (y_46_im <= 1.642430271924372e-175)
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_im <= 2.8787453566117075e+148)
		tmp = t_0;
	else
		tmp = t_1;
	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[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(1.0 / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$46$im, -5.165129542917535e+152], t$95$1, If[LessEqual[y$46$im, -6.458426761032663e-136], t$95$0, If[LessEqual[y$46$im, 1.642430271924372e-175], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8787453566117075e+148], t$95$0, t$95$1]]]]]]

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 3 regimes

2. if y.im < -5.1651295429175348e152 or 2.87874535661170751e148 < y.im

   1. Initial program 43.6

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

   2. Simplified 43.6

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

   3. Applied egg-rr 28.1

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

   4. Taylor expanded in y.re around 0 14.6

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

   5. Simplified 7.4

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

   if -5.1651295429175348e152 < y.im < -6.4584267610326632e-136 or 1.64243027192437199e-175 < y.im < 2.87874535661170751e148

   1. Initial program 18.1

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

   2. Simplified 18.1

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

   3. Taylor expanded in x.re around 0 18.1

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

   4. Simplified 13.5

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

   5. Applied egg-rr 4.2

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

   if -6.4584267610326632e-136 < y.im < 1.64243027192437199e-175

   1. Initial program 25.0

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

   2. Simplified 25.0

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

   3. Applied egg-rr 13.9

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

   4. Taylor expanded in y.re around inf 10.2

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

   5. Simplified 8.6

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

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.165129542917535 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -6.458426761032663 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\right)\\ \mathbf{elif}\;y.im \leq 1.642430271924372 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.8787453566117075 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.re \cdot \left(\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))))