_divideComplex, real part

Average Error: 26.7 → 11.1

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ t_1 := x.im + \frac{y.re}{y.im} \cdot x.re\\ t_2 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+113}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.3 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_1\\ \end{array} \]

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 (/ y.im y.re) (/ x.im y.re) (/ x.re y.re)))
        (t_1 (+ x.im (* (/ y.re y.im) x.re)))
        (t_2
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.im -5.8e+113)
     (* t_1 (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -1.32e+95)
       t_0
       (if (<= y.im -2.6e+30)
         (* (/ 1.0 (pow (hypot y.re y.im) 2.0)) (fma y.re x.re (* y.im x.im)))
         (if (<= y.im -0.3)
           t_0
           (if (<= y.im -3.5e-130)
             t_2
             (if (<= y.im 8e-131)
               (+
                (/ x.re y.re)
                (* (/ (/ y.im y.re) y.re) (- x.im (/ (* y.im x.re) y.re))))
               (if (<= y.im 9.3e+176)
                 t_2
                 (* (/ 1.0 (hypot y.re y.im)) 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((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	double t_1 = x_46_im + ((y_46_re / y_46_im) * x_46_re);
	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_im <= -5.8e+113) {
		tmp = t_1 * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.32e+95) {
		tmp = t_0;
	} else if (y_46_im <= -2.6e+30) {
		tmp = (1.0 / pow(hypot(y_46_re, y_46_im), 2.0)) * fma(y_46_re, x_46_re, (y_46_im * x_46_im));
	} else if (y_46_im <= -0.3) {
		tmp = t_0;
	} else if (y_46_im <= -3.5e-130) {
		tmp = t_2;
	} else if (y_46_im <= 8e-131) {
		tmp = (x_46_re / y_46_re) + (((y_46_im / y_46_re) / y_46_re) * (x_46_im - ((y_46_im * x_46_re) / y_46_re)));
	} else if (y_46_im <= 9.3e+176) {
		tmp = t_2;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * 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(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re))
	t_1 = Float64(x_46_im + Float64(Float64(y_46_re / y_46_im) * x_46_re))
	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_im <= -5.8e+113)
		tmp = Float64(t_1 * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1.32e+95)
		tmp = t_0;
	elseif (y_46_im <= -2.6e+30)
		tmp = Float64(Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)) * fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)));
	elseif (y_46_im <= -0.3)
		tmp = t_0;
	elseif (y_46_im <= -3.5e-130)
		tmp = t_2;
	elseif (y_46_im <= 8e-131)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(y_46_im / y_46_re) / y_46_re) * Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re))));
	elseif (y_46_im <= 9.3e+176)
		tmp = t_2;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * 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[(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]}, Block[{t$95$1 = N[(x$46$im + N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $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$46$im, -5.8e+113], N[(t$95$1 * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.32e+95], t$95$0, If[LessEqual[y$46$im, -2.6e+30], N[(N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -0.3], t$95$0, If[LessEqual[y$46$im, -3.5e-130], t$95$2, If[LessEqual[y$46$im, 8e-131], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.3e+176], t$95$2, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $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 6 regimes

2. if y.im < -5.79999999999999968e113

   1. Initial program 41.7

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

   2. Simplified 41.7

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

   3. Applied egg-rr 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.im around -inf 13.8

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

   5. Simplified 9.6

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

   if -5.79999999999999968e113 < y.im < -1.32e95 or -2.59999999999999988e30 < y.im < -0.299999999999999989

   1. Initial program 19.7

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

   2. Simplified 19.7

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

   3. Taylor expanded in y.re around inf 40.1

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

   4. Simplified 36.4

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

   if -1.32e95 < y.im < -2.59999999999999988e30

   1. Initial program 17.3

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

   2. Simplified 17.3

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

   3. Applied egg-rr 13.0

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

   4. Applied egg-rr 13.3

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

   5. Applied egg-rr 17.3

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

   if -0.299999999999999989 < y.im < -3.4999999999999999e-130 or 7.9999999999999999e-131 < y.im < 9.2999999999999997e176

   1. Initial program 18.3

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

   2. Simplified 18.3

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

   3. Applied egg-rr 12.1

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

   4. Applied egg-rr 12.2

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

   5. Applied egg-rr 11.9

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

   if -3.4999999999999999e-130 < y.im < 7.9999999999999999e-131

   1. Initial program 24.1

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

   2. Simplified 24.1

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

   3. Taylor expanded in y.re around inf 14.5

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

   4. Simplified 7.8

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

   if 9.2999999999999997e176 < y.im

   1. Initial program 42.7

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

   2. Simplified 42.7

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

   3. Applied egg-rr 29.3

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

   4. Taylor expanded in y.re around 0 11.7

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

   5. Simplified 6.4

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+113}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.3 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{y.re}{y.im} \cdot x.re\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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))))