_divideComplex, real part

Average Error: 26.0 → 10.6

Time: 14.1s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))))
        (t_1 (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))))
   (if (<= y.re -3.5e+72)
     t_1
     (if (<= y.re -1.2e-193)
       t_0
       (if (<= y.re 3.5e-92)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 2.4e+50) 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 = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -3.5e+72) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-193) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e-92) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+50) {
		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 = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.5e+72)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-193)
		tmp = t_0;
	elseif (y_46_re <= 3.5e-92)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+50)
		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[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+72], t$95$1, If[LessEqual[y$46$re, -1.2e-193], t$95$0, If[LessEqual[y$46$re, 3.5e-92], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.5000000000000001e72 or 2.4000000000000002e50 < y.re

   1. Initial program 35.9

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

   2. Taylor expanded in y.re around inf 17.0

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

   3. Simplified 16.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (*.f64 y.re y.re)) x.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= associate-/r/_binary64 (/.f64 y.im (/.f64 (*.f64 y.re y.re) x.im)))): 0 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) x.im))): 0 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)))): 4 points increase in error, 0 points decrease in error

   4. Applied egg-rr 11.6

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

   if -3.5000000000000001e72 < y.re < -1.2e-193 or 3.5e-92 < y.re < 2.4000000000000002e50

   1. Initial program 15.9

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

   2. Applied egg-rr 10.1

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

   if -1.2e-193 < y.re < 3.5e-92

   1. Initial program 22.3

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

   2. Taylor expanded in y.re around 0 11.1

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

   3. Simplified 9.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (*.f64 y.re y.re)) x.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= associate-/r/_binary64 (/.f64 y.im (/.f64 (*.f64 y.re y.re) x.im)))): 0 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 y.im (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) x.im))): 0 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)))): 4 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	14280

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -1.08 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t_0} + \frac{y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	16.9
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-17} \lor \neg \left(y.re \leq 7 \cdot 10^{-56} \lor \neg \left(y.re \leq 3.6 \cdot 10^{+36}\right) \land y.re \leq 1.9 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	15.9
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-9} \lor \neg \left(y.re \leq 4.2 \cdot 10^{-56} \lor \neg \left(y.re \leq 1.35 \cdot 10^{+36}\right) \land y.re \leq 3.25 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	17.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6.3 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.8
Cost	1100

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{y.re} \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	20.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-94} \lor \neg \left(y.re \leq 5.6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	24.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y.re \leq -850000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	24.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1350000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	22.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(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))))