_divideComplex, real part

Average Error: 26.6 → 9.7

Time: 12.3s

Precision: binary64

Cost: 22088

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}\\ \end{array} \]

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.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (+ (/ x.re y.re) (/ x.im (/ y.re (/ y.im y.re))))
     (if (<= t_0 2e+291)
       (*
        (/ 1.0 (hypot y.re y.im))
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
       (/ (/ 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_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_46_im / y_46_re)));
	} else if (t_0 <= 2e+291) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} 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(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_0 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re / Float64(y_46_im / y_46_re))));
	elseif (t_0 <= 2e+291)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	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[(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$0, (-Infinity)], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+291], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], 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

   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 y.re around inf 43.4

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

   3. Simplified 32.9

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 50 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 36.7

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

   5. Taylor expanded in y.im around 0 43.4

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

   6. Simplified 31.6

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}} \]
      Proof
      (/.f64 x.im (/.f64 y.re (/.f64 y.im y.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im (/.f64 y.im y.re)) y.re)): 23 points increase in error, 37 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x.im y.re) (/.f64 y.im y.re))): 30 points increase in error, 27 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.im y.im) (*.f64 y.re y.re))): 67 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) (*.f64 y.re y.re)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))): 0 points increase in error, 0 points decrease in error

   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))) < 1.9999999999999999e291

   1. Initial program 12.1

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

   2. Applied egg-rr 0.9

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

   if 1.9999999999999999e291 < (/.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 63.0

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

   2. Taylor expanded in x.re around 0 62.5

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

   3. Applied egg-rr 31.8

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

   4. Applied egg-rr 31.8

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{y.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.2
Cost	13508

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 2
Error	13.1
Cost	7828

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 3
Error	13.2
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.4:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 4
Error	19.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 100:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Error	19.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 100:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 6
Error	18.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 100:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	14.7
Cost	968

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 20:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 10.8:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Error	23.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Error	23.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(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))))