_divideComplex, real part

Percentage Accurate: 62.6% → 84.4%

Time: 16.5s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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.im y.im) (/ (/ y.re (/ y.im x.re)) y.im))))
   (if (<= y.im -1.3e+64)
     t_1
     (if (<= y.im -3.1e-102)
       t_0
       (if (<= y.im 2.2e-205)
         (+ (/ x.re y.re) (/ (* y.im (* x.im (/ 1.0 y.re))) y.re))
         (if (<= y.im 7.2e+200) 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_im / y_46_im) + ((y_46_re / (y_46_im / x_46_re)) / y_46_im);
	double tmp;
	if (y_46_im <= -1.3e+64) {
		tmp = t_1;
	} else if (y_46_im <= -3.1e-102) {
		tmp = t_0;
	} else if (y_46_im <= 2.2e-205) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im * (1.0 / y_46_re))) / y_46_re);
	} else if (y_46_im <= 7.2e+200) {
		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_im / y_46_im) + Float64(Float64(y_46_re / Float64(y_46_im / x_46_re)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.3e+64)
		tmp = t_1;
	elseif (y_46_im <= -3.1e-102)
		tmp = t_0;
	elseif (y_46_im <= 2.2e-205)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im * Float64(1.0 / y_46_re))) / y_46_re));
	elseif (y_46_im <= 7.2e+200)
		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$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.3e+64], t$95$1, If[LessEqual[y$46$im, -3.1e-102], t$95$0, If[LessEqual[y$46$im, 2.2e-205], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+200], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.29999999999999998e64 or 7.1999999999999995e200 < y.im

   1. Initial program 41.3%

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

   2. Taylor expanded in y.re around 0 79.5%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Step-by-step derivation

      [Start] 79.5	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 79.5	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 79.5	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 79.5	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 89.8	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   4. Applied egg-rr 91.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}} \]
      Step-by-step derivation

      [Start] 89.8	\[ \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im} \]
      associate-*l/ [=>] 91.4	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re \cdot \frac{x.re}{y.im}}{y.im}} \]
      clear-num [=>] 91.4	\[ \frac{x.im}{y.im} + \frac{y.re \cdot \color{blue}{\frac{1}{\frac{y.im}{x.re}}}}{y.im} \]
      un-div-inv [=>] 91.4	\[ \frac{x.im}{y.im} + \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.re}}}}{y.im} \]

   if -1.29999999999999998e64 < y.im < -3.10000000000000013e-102 or 2.20000000000000009e-205 < y.im < 7.1999999999999995e200

   1. Initial program 80.3%

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

   2. Applied egg-rr 91.5%

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

      [Start] 80.3	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 80.3	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 80.3	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 80.4	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 80.4	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 80.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 91.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -3.10000000000000013e-102 < y.im < 2.20000000000000009e-205

   1. Initial program 66.4%

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

   2. Taylor expanded in y.re around inf 75.2%

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

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im} \]
      Step-by-step derivation

      [Start] 75.2	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      associate-/l* [=>] 75.4	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{\frac{{y.re}^{2}}{x.im}}} \]
      associate-/r/ [=>] 75.7	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{{y.re}^{2}} \cdot x.im} \]
      unpow2 [=>] 75.7	\[ \frac{x.re}{y.re} + \frac{y.im}{\color{blue}{y.re \cdot y.re}} \cdot x.im \]

   4. Applied egg-rr 88.5%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{y.im \cdot x.im}{y.re}}{y.re}} \]
      Step-by-step derivation

      [Start] 75.7	\[ \frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im \]
      associate-*l/ [=>] 75.2	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im \cdot x.im}{y.re \cdot y.re}} \]
      associate-/r* [=>] 88.5	\[ \frac{x.re}{y.re} + \color{blue}{\frac{\frac{y.im \cdot x.im}{y.re}}{y.re}} \]

   5. Applied egg-rr 90.9%

      \[\leadsto \frac{x.re}{y.re} + \frac{\color{blue}{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}}{y.re} \]
      Step-by-step derivation

      [Start] 88.5	\[ \frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re} \]
      div-inv [=>] 88.5	\[ \frac{x.re}{y.re} + \frac{\color{blue}{\left(y.im \cdot x.im\right) \cdot \frac{1}{y.re}}}{y.re} \]
      associate-*l* [=>] 90.9	\[ \frac{x.re}{y.re} + \frac{\color{blue}{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}}{y.re} \]

3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-205}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.6%
Cost	7496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{t_0}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+109}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	81.8%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	75.3%
Cost	1360

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -1.52 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \left(x.im \cdot \frac{1}{y.re}\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	76.0%
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.5 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-86} \lor \neg \left(y.im \leq 2.5 \cdot 10^{+34}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 5
Accuracy	76.2%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.52 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	76.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	75.8%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -1.66 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+30}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	71.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Accuracy	62.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 650000000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Accuracy	43.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+219}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Accuracy	42.9%
Cost	192

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

Reproduce

herbie shell --seed 2023160 
(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))))