_divideComplex, real part

Average Error: 26.1 → 9.5

Time: 17.1s

Precision: binary64

Cost: 21960

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ t_1 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}\\ \end{array} \]

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.re y.re) (/ x.im (* y.re (/ y.re y.im)))))
        (t_1
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 5e+284)
       (/
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
        (hypot y.re y.im))
       (if (<= t_1 INFINITY)
         t_0
         (/ (/ x.im (hypot y.im y.re)) (/ (hypot y.im 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 t_0 = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	double t_1 = ((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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= 5e+284) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / hypot(y_46_im, y_46_re)) / (hypot(y_46_im, 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)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))))
	t_1 = 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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= 5e+284)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / hypot(y_46_im, y_46_re)) / Float64(hypot(y_46_im, 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_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, 5e+284], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$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[t$95$1, Infinity], t$95$0, N[(N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -inf.0 or 4.9999999999999999e284 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 60.5

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

   2. Taylor expanded in y.re around inf 42.8

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

   3. Simplified 33.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)} \]
      Proof
      (fma.f64 (/.f64 y.im y.re) (/.f64 x.im y.re) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re)) (/.f64 x.re y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re))) (/.f64 x.re y.re)): 45 points increase in error, 13 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 33.4

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

   5. Applied egg-rr 31.9

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999999e284

   1. Initial program 11.7

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 0.8

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 64.0

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

   2. Taylor expanded in x.re around 0 62.9

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

   3. Applied egg-rr 31.9

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

   4. Applied egg-rr 31.9

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.2
Cost	13508

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -6.025858838213873 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot \left(x.im - \frac{y.im}{\frac{y.re}{x.re}}\right)}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.23647302120828 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0588169654702965 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)}{y.im}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	1884

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{if}\;y.im \leq -2.853498899881927 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 4.23647302120828 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0588169654702965 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 370751669.14618987:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.9736244675589573 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.3
Cost	1884

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{if}\;y.im \leq -2.853498899881927 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot \left(x.im - \frac{y.im}{\frac{y.re}{x.re}}\right)}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.23647302120828 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0588169654702965 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 370751669.14618987:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.9736244675589573 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.8
Cost	1748

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.853498899881927 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot \left(x.im - \frac{y.im}{\frac{y.re}{x.re}}\right)}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.23647302120828 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0588169654702965 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.7873953519218127 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.23647302120828 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.087015133523303 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.0188771669479473 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.3127502167617188 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.675275716212167 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.473740259983911 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.0188771669479473 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.3127502167617188 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.675275716212167 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.473740259983911 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.0188771669479473 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.3127502167617188 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 4.675275716212167 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.473740259983911 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.0188771669479473 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.570865415223703 \cdot 10^{-46}:\\ \;\;\;\;x.re \cdot \frac{y.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.51457990635922 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	24.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.7882807740781 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.0588169654702965 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	58.8
Cost	192

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

Alternative 12
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(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))))