_divideComplex, real part

Average Error: 59.4% → 84.3%

Time: 16.6s

Precision: binary64

Cost: 20432.00

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

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
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.im -1.1e+116)
     (+ (/ x.im y.im) (/ (* (/ x.re y.im) y.re) y.im))
     (if (<= y.im -1.35e-207)
       t_0
       (if (<= y.im 9.8e-92)
         (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re)))
         (if (<= y.im 3.8e+96)
           t_0
           (+ (/ x.im y.im) (* (/ x.re 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 = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.1e+116) {
		tmp = (x_46_im / y_46_im) + (((x_46_re / y_46_im) * y_46_re) / y_46_im);
	} else if (y_46_im <= -1.35e-207) {
		tmp = t_0;
	} else if (y_46_im <= 9.8e-92) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	} else if (y_46_im <= 3.8e+96) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / 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(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.1e+116)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re / y_46_im) * y_46_re) / y_46_im));
	elseif (y_46_im <= -1.35e-207)
		tmp = t_0;
	elseif (y_46_im <= 9.8e-92)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	elseif (y_46_im <= 3.8e+96)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(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 + N[(y$46$im * x$46$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[y$46$im, -1.1e+116], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.35e-207], t$95$0, If[LessEqual[y$46$im, 9.8e-92], 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$46$im, 3.8e+96], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.1e116

   1. Initial program 36.5

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

   2. Simplified 36.5

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

      [Start] 36.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 36.5	\[ \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 36.5	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Taylor expanded in y.re around 0 74.1

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

   4. Simplified 77.4

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

      [Start] 74.1	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 74.1	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      unpow2 [=>] 74.1	\[ \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]
      associate-*l/ [<=] 77.4	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{y.im \cdot y.im} \cdot y.re} \]
      *-commutative [=>] 77.4	\[ \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{x.re}{y.im \cdot y.im}} \]

   5. Applied egg-rr 86.4

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

   if -1.1e116 < y.im < -1.35e-207 or 9.8e-92 < y.im < 3.8000000000000002e96

   1. Initial program 74.0

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

   2. Applied egg-rr 82.4

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

   3. Applied egg-rr 82.6

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

   if -1.35e-207 < y.im < 9.8e-92

   1. Initial program 65.7

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

   2. Taylor expanded in y.re around inf 84.5

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

   3. Simplified 86.3

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

      [Start] 84.5	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      *-commutative [<=] 84.5	\[ \frac{x.re}{y.re} + \frac{\color{blue}{x.im \cdot y.im}}{{y.re}^{2}} \]
      unpow2 [=>] 84.5	\[ \frac{x.re}{y.re} + \frac{x.im \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 86.3	\[ \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{y.re} \cdot \frac{y.im}{y.re}} \]

   if 3.8000000000000002e96 < y.im

   1. Initial program 38.8

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

   2. Taylor expanded in y.re around 0 73.1

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

   3. Simplified 84.0

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

      [Start] 73.1	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 73.1	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 73.1	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 73.1	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 84.0	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	80.5%
Cost	14036.00

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1850:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 2
Error	80.5%
Cost	1488.00

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 3
Error	73.5%
Cost	1232.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 4
Error	74.9%
Cost	1232.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \end{array} \]

Alternative 5
Error	75.2%
Cost	1232.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.65 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \end{array} \]

Alternative 6
Error	72.5%
Cost	969.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+78} \lor \neg \left(y.im \leq 4.48 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \end{array} \]

Alternative 7
Error	72.7%
Cost	968.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 8
Error	72.8%
Cost	968.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{y.re}{\frac{y.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 9
Error	73.0%
Cost	968.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.56 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im} \cdot y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 10
Error	66.7%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+152} \lor \neg \left(y.im \leq 4.48 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \end{array} \]

Alternative 11
Error	62.0%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -13500000:\\ \;\;\;\;y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12
Error	64.2%
Cost	456.00

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2500:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 4.48 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13
Error	41.8%
Cost	192.00

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

Reproduce

herbie shell --seed 2023097 
(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))))