_divideComplex, real part

Average Error: 26.0 → 9.8

Time: 12.3s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

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
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))))
   (if (<= y.im -1.35e+130)
     (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re)))
     (if (<= y.im -1.4e-132)
       t_0
       (if (<= y.im 2.2e-93)
         (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
         (if (<= y.im 9.2e+79)
           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 = (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 <= -1.35e+130) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= -1.4e-132) {
		tmp = t_0;
	} else if (y_46_im <= 2.2e-93) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_im <= 9.2e+79) {
		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(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 <= -1.35e+130)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	elseif (y_46_im <= -1.4e-132)
		tmp = t_0;
	elseif (y_46_im <= 2.2e-93)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	elseif (y_46_im <= 9.2e+79)
		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[(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, -1.35e+130], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-132], t$95$0, If[LessEqual[y$46$im, 2.2e-93], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.2e+79], 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 < -1.3499999999999999e130

   1. Initial program 43.7

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

   2. Taylor expanded in y.re around 0 16.3

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

   3. Simplified 9.2

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 38 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y.re around 0 16.3

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

   5. Simplified 9.3

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}} \]
      Proof
      (/.f64 (/.f64 x.re y.im) (/.f64 y.im y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 x.re (*.f64 y.im (/.f64 y.im y.re)))): 34 points increase in error, 25 points decrease in error
      (/.f64 x.re (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y.im y.im) y.re))): 23 points increase in error, 13 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re y.re) (*.f64 y.im y.im))): 31 points increase in error, 23 points decrease in error
      (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))): 0 points increase in error, 0 points decrease in error

   if -1.3499999999999999e130 < y.im < -1.40000000000000001e-132 or 2.19999999999999996e-93 < y.im < 9.2000000000000002e79

   1. Initial program 16.3

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

   2. Applied egg-rr 11.3

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

   if -1.40000000000000001e-132 < y.im < 2.19999999999999996e-93

   1. Initial program 20.9

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

   2. Taylor expanded in y.re around inf 10.2

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

   3. Simplified 11.4

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im} \]
      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)))): 15 points increase in error, 10 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 21 points increase in error, 15 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 6.9

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

   if 9.2000000000000002e79 < y.im

   1. Initial program 37.0

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

   2. Taylor expanded in y.re around 0 17.0

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

   3. Simplified 11.5

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 38 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	13768

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.9
Cost	1224

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 4
Error	19.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.22 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 5
Error	16.2
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.8
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.9
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -1.08 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.8
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.14 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Error	36.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2022332 
(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))))