_divideComplex, real part

Average Error: 25.6 → 9.2

Time: 12.5s

Precision: binary64

Cost: 20432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -7.2e+106)
     t_1
     (if (<= y.im -1.25e-154)
       t_0
       (if (<= y.im 4.3e-181)
         (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))
         (if (<= y.im 2.8e+112) 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_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -7.2e+106) {
		tmp = t_1;
	} else if (y_46_im <= -1.25e-154) {
		tmp = t_0;
	} else if (y_46_im <= 4.3e-181) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 2.8e+112) {
		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_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -7.2e+106)
		tmp = t_1;
	elseif (y_46_im <= -1.25e-154)
		tmp = t_0;
	elseif (y_46_im <= 4.3e-181)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 2.8e+112)
		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$46$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, -7.2e+106], t$95$1, If[LessEqual[y$46$im, -1.25e-154], t$95$0, If[LessEqual[y$46$im, 4.3e-181], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+112], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.2000000000000002e106 or 2.8000000000000001e112 < y.im

   1. Initial program 40.4

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

   2. Taylor expanded in y.re around 0 16.0

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

   3. Simplified 9.1

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      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	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 9.1	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   if -7.2000000000000002e106 < y.im < -1.25000000000000005e-154 or 4.3e-181 < y.im < 2.8000000000000001e112

   1. Initial program 15.7

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

   2. Applied egg-rr 10.9

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

   3. Applied egg-rr 10.8

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

   if -1.25000000000000005e-154 < y.im < 4.3e-181

   1. Initial program 23.3

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

   2. Simplified 23.3

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

      [Start] 23.3	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 23.3	\[ \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 23.3	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Taylor expanded in y.re around inf 9.0

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

   4. Simplified 8.6

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

      [Start] 9.0	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      *-commutative [<=] 9.0	\[ \frac{x.re}{y.re} + \frac{\color{blue}{x.im \cdot y.im}}{{y.re}^{2}} \]
      unpow2 [=>] 9.0	\[ \frac{x.re}{y.re} + \frac{x.im \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 8.6	\[ \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{y.re} \cdot \frac{y.im}{y.re}} \]

   5. Applied egg-rr 7.8

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

   6. Taylor expanded in x.im around 0 5.2

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

   7. Simplified 5.6

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

      [Start] 5.2	\[ \frac{1}{y.re} \cdot \left(\frac{y.im \cdot x.im}{y.re} + x.re\right) \]
      associate-*l/ [<=] 5.6	\[ \frac{1}{y.re} \cdot \left(\color{blue}{\frac{y.im}{y.re} \cdot x.im} + x.re\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.4
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{+44} \lor \neg \left(y.im \leq -1.15 \cdot 10^{+23}\right) \land y.im \leq 9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 3
Error	15.9
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.38 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-94} \lor \neg \left(y.im \leq 3.9 \cdot 10^{+34}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 4
Error	16.0
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-94} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+35}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 5
Error	18.8
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{+51}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Error	23.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 7
Error	37.0
Cost	192

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

Reproduce

herbie shell --seed 2023187 
(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))))