_divideComplex, real part

Average Error: 26.1 → 10.9

Time: 12.1s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

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)))))
   (if (<= y.im -2.25e+123)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.im -7.5e-167)
       t_0
       (if (<= y.im 3.8e-55)
         (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))
         (if (<= y.im 7e+110)
           t_0
           (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))))))))

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 tmp;
	if (y_46_im <= -2.25e+123) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_im <= -7.5e-167) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-55) {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	} else if (y_46_im <= 7e+110) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	}
	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)))
	tmp = 0.0
	if (y_46_im <= -2.25e+123)
		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_im <= -7.5e-167)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-55)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	elseif (y_46_im <= 7e+110)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	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]}, If[LessEqual[y$46$im, -2.25e+123], 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$46$im, -7.5e-167], t$95$0, If[LessEqual[y$46$im, 3.8e-55], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e+110], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.24999999999999991e123

   1. Initial program 41.9

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

   2. Taylor expanded in y.re around 0 16.0

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

   3. Simplified 16.0

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

      [Start] 16.0	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 16.0	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 16.0	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 16.0	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]

   4. Applied egg-rr 9.1

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

   if -2.24999999999999991e123 < y.im < -7.5000000000000007e-167 or 3.7999999999999997e-55 < y.im < 6.9999999999999998e110

   1. Initial program 16.6

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

   2. Applied egg-rr 11.8

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

   if -7.5000000000000007e-167 < y.im < 3.7999999999999997e-55

   1. Initial program 21.8

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

   2. Taylor expanded in y.re around inf 12.3

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

   3. Simplified 15.0

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

      [Start] 12.3	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      associate-/l* [=>] 15.0	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{\frac{{y.re}^{2}}{x.im}}} \]
      unpow2 [=>] 15.0	\[ \frac{x.re}{y.re} + \frac{y.im}{\frac{\color{blue}{y.re \cdot y.re}}{x.im}} \]

   4. Applied egg-rr 11.1

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

   5. Applied egg-rr 11.1

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

   if 6.9999999999999998e110 < y.im

   1. Initial program 39.4

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

   2. Taylor expanded in y.re around 0 16.4

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

   3. Simplified 16.4

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

      [Start] 16.4	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 16.4	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 16.4	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 16.4	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]

   4. Applied egg-rr 9.7

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

   5. Applied egg-rr 10.5

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{\frac{y.im}{x.re} \cdot y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \left(y.re \cdot \frac{1}{x.im}\right)}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 2
Error	19.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-50} \lor \neg \left(y.im \leq 70000000000\right):\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 3
Error	19.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 21000000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 4
Error	19.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 18000000000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 5
Error	15.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 6
Error	15.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 7
Error	24.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	23.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	37.8
Cost	192

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

Reproduce

herbie shell --seed 2023057 
(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))))