_divideComplex, real part

Average Error: 26.0 → 10.6

Time: 11.1s

Precision: binary64

Cost: 20432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-255}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-193}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{y.re}{y.im \cdot \left(-\frac{y.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im)))
        (t_1 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))
   (if (<= y.re -2.5e+99)
     t_1
     (if (<= y.re -2.6e-255)
       t_0
       (if (<= y.re 8e-193)
         (- (/ x.im y.im) (/ y.re (* y.im (- (/ y.im x.re)))))
         (if (<= y.re 3.45e+52) 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 = (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 t_1 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -2.5e+99) {
		tmp = t_1;
	} else if (y_46_re <= -2.6e-255) {
		tmp = t_0;
	} else if (y_46_re <= 8e-193) {
		tmp = (x_46_im / y_46_im) - (y_46_re / (y_46_im * -(y_46_im / x_46_re)));
	} else if (y_46_re <= 3.45e+52) {
		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(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))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.5e+99)
		tmp = t_1;
	elseif (y_46_re <= -2.6e-255)
		tmp = t_0;
	elseif (y_46_re <= 8e-193)
		tmp = Float64(Float64(x_46_im / y_46_im) - Float64(y_46_re / Float64(y_46_im * Float64(-Float64(y_46_im / x_46_re)))));
	elseif (y_46_re <= 3.45e+52)
		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[(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]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e+99], t$95$1, If[LessEqual[y$46$re, -2.6e-255], t$95$0, If[LessEqual[y$46$re, 8e-193], 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], If[LessEqual[y$46$re, 3.45e+52], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.50000000000000004e99 or 3.44999999999999998e52 < y.re

   1. Initial program 37.4

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

   2. Taylor expanded in y.re around inf 17.0

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

   3. Simplified 11.1

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 34 points increase in error, 6 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

   if -2.50000000000000004e99 < y.re < -2.60000000000000021e-255 or 8.0000000000000004e-193 < y.re < 3.44999999999999998e52

   1. Initial program 16.8

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

   2. Applied egg-rr 10.3

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

   3. Applied egg-rr 10.2

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

   if -2.60000000000000021e-255 < y.re < 8.0000000000000004e-193

   1. Initial program 23.5

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

   2. Taylor expanded in y.re around 0 8.2

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

   3. Simplified 8.2

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im))): 0 points increase in error, 0 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. Applied egg-rr 8.7

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

   5. Simplified 10.7

      \[\leadsto \color{blue}{\frac{x.im}{y.im} - \frac{y.re}{\left(-y.im\right) \cdot \frac{y.im}{x.re}}} \]
      Proof
      (-.f64 (/.f64 x.im y.im) (/.f64 y.re (*.f64 (neg.f64 y.im) (/.f64 y.im x.re)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.im) (/.f64 y.re (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (neg.f64 y.im) y.im) x.re)))): 32 points increase in error, 5 points decrease in error
      (-.f64 (/.f64 x.im y.im) (/.f64 y.re (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im (neg.f64 y.im))) x.re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.im) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im (neg.f64 y.im))))): 22 points increase in error, 13 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 x.im y.im) (neg.f64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im (neg.f64 y.im)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 (*.f64 y.re x.re)) (*.f64 y.im (neg.f64 y.im))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 y.re (neg.f64 x.re))) (*.f64 y.im (neg.f64 y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 y.re (neg.f64 x.re)) 1)) (*.f64 y.im (neg.f64 y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 y.re (neg.f64 x.re)) (/.f64 1 (*.f64 y.im (neg.f64 y.im)))))): 10 points increase in error, 6 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 y.re (neg.f64 x.re)) (/.f64 1 (*.f64 y.im (neg.f64 y.im)))) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-udef_binary64 (fma.f64 (*.f64 y.re (neg.f64 x.re)) (/.f64 1 (*.f64 y.im (neg.f64 y.im))) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-255}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-193}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{y.re}{y.im \cdot \left(-\frac{y.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{y.re}{y.im \cdot \left(-\frac{y.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.5
Cost	1296

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 6800000000:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{y.re}{y.im \cdot \left(-\frac{y.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.5
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 4800000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.9
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 6
Error	15.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 7
Error	22.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	37.5
Cost	192

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

Reproduce

herbie shell --seed 2022328 
(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))))