_divideComplex, real part

Average Accuracy: 58.0% → 83.4%

Time: 14.6s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -5.7 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* x.im y.im)) (hypot y.re y.im))))
        (t_1 (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))))
   (if (<= y.re -5.7e+181)
     t_1
     (if (<= y.re -2.1e-137)
       t_0
       (if (<= y.re 1.1e-130)
         (+ (/ x.im y.im) (/ 1.0 (* (/ y.im y.re) (/ y.im x.re))))
         (if (<= y.re 5e+44) 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, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -5.7e+181) {
		tmp = t_1;
	} else if (y_46_re <= -2.1e-137) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-130) {
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	} else if (y_46_re <= 5e+44) {
		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(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -5.7e+181)
		tmp = t_1;
	elseif (y_46_re <= -2.1e-137)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-130)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 5e+44)
		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[(x$46$im * y$46$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[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.7e+181], t$95$1, If[LessEqual[y$46$re, -2.1e-137], t$95$0, If[LessEqual[y$46$re, 1.1e-130], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.7000000000000002e181 or 4.9999999999999996e44 < y.re

   1. Initial program 40.0%

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

   2. Taylor expanded in y.re around inf 74.9%

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

   3. Simplified 83.9%

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

      [Start] 74.9	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      *-commutative [<=] 74.9	\[ \frac{x.re}{y.re} + \frac{\color{blue}{x.im \cdot y.im}}{{y.re}^{2}} \]
      unpow2 [=>] 74.9	\[ \frac{x.re}{y.re} + \frac{x.im \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 83.9	\[ \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{y.re} \cdot \frac{y.im}{y.re}} \]

   if -5.7000000000000002e181 < y.re < -2.09999999999999992e-137 or 1.0999999999999999e-130 < y.re < 4.9999999999999996e44

   1. Initial program 70.3%

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

   2. Applied egg-rr 80.5%

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

      [Start] 70.3	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 70.3	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 70.3	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 70.3	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 70.3	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 70.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 80.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -2.09999999999999992e-137 < y.re < 1.0999999999999999e-130

   1. Initial program 62.5%

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

   2. Taylor expanded in y.re around 0 84.7%

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

   3. Simplified 86.8%

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

      [Start] 84.7	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 84.7	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 84.7	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 84.7	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 86.8	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   4. Applied egg-rr 87.2%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{x.re} \cdot \frac{y.im}{y.re}}} \]
      Proof

      [Start] 86.8	\[ \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im} \]
      associate-*r/ [=>] 90.3	\[ \frac{x.im}{y.im} + \color{blue}{\frac{\frac{y.re}{y.im} \cdot x.re}{y.im}} \]
      associate-/l* [=>] 87.3	\[ \frac{x.im}{y.im} + \color{blue}{\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}} \]
      clear-num [=>] 87.2	\[ \frac{x.im}{y.im} + \frac{\color{blue}{\frac{1}{\frac{y.im}{y.re}}}}{\frac{y.im}{x.re}} \]
      associate-/l/ [=>] 87.2	\[ \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{x.re} \cdot \frac{y.im}{y.re}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.5%
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.72 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	74.5%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -28000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{-234}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 6.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	74.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -58000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	74.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+18} \lor \neg \left(y.re \leq 5.9 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 5
Accuracy	74.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.9 \cdot 10^{+18} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 6
Accuracy	75.0%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 7
Accuracy	67.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y.im \leq -85000000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Accuracy	65.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -55000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	41.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	41.5%
Cost	192

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

Reproduce

herbie shell --seed 2023126 
(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))))