_divideComplex, real part

Average Accuracy: 59.2% → 81.9%

Time: 15.4s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))))
        (t_1 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -2.05e+123)
     t_1
     (if (<= y.im -1.4e-48)
       t_0
       (if (<= y.im 1.05e-58)
         (+ (/ x.re y.re) (* (/ 1.0 y.re) (* y.im (/ x.im y.re))))
         (if (<= y.im 1.35e+143)
           t_0
           (if (<= y.im 7.5e+205)
             (* (/ y.im (hypot y.im y.re)) (/ x.im (hypot y.im y.re)))
             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 = (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 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 <= -2.05e+123) {
		tmp = t_1;
	} else if (y_46_im <= -1.4e-48) {
		tmp = t_0;
	} else if (y_46_im <= 1.05e-58) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_im <= 1.35e+143) {
		tmp = t_0;
	} else if (y_46_im <= 7.5e+205) {
		tmp = (y_46_im / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	} 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(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)))
	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 <= -2.05e+123)
		tmp = t_1;
	elseif (y_46_im <= -1.4e-48)
		tmp = t_0;
	elseif (y_46_im <= 1.05e-58)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_im <= 1.35e+143)
		tmp = t_0;
	elseif (y_46_im <= 7.5e+205)
		tmp = Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(x_46_im / hypot(y_46_im, y_46_re)));
	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[(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]}, 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, -2.05e+123], t$95$1, If[LessEqual[y$46$im, -1.4e-48], t$95$0, If[LessEqual[y$46$im, 1.05e-58], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+143], t$95$0, If[LessEqual[y$46$im, 7.5e+205], N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.04999999999999995e123 or 7.5000000000000003e205 < y.im

   1. Initial program 36.0%

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

   2. Taylor expanded in y.re around 0 76.0%

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

   3. Simplified 88.6%

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

      [Start] 76.0	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 76.0	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 76.0	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 76.0	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 88.6	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   if -2.04999999999999995e123 < y.im < -1.40000000000000002e-48 or 1.04999999999999994e-58 < y.im < 1.3500000000000001e143

   1. Initial program 71.8%

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

   2. Applied egg-rr 79.4%

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

      [Start] 71.8	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 71.8	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 71.8	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 71.8	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 71.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 71.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 79.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -1.40000000000000002e-48 < y.im < 1.04999999999999994e-58

   1. Initial program 68.4%

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

   2. Taylor expanded in y.re around inf 77.7%

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

   3. Simplified 74.3%

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

      [Start] 77.7	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      associate-/l* [=>] 74.3	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{\frac{{y.re}^{2}}{x.im}}} \]
      unpow2 [=>] 74.3	\[ \frac{x.re}{y.re} + \frac{y.im}{\frac{\color{blue}{y.re \cdot y.re}}{x.im}} \]

   4. Applied egg-rr 80.7%

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

      [Start] 74.3	\[ \frac{x.re}{y.re} + \frac{y.im}{\frac{y.re \cdot y.re}{x.im}} \]
      clear-num [=>] 74.2	\[ \frac{x.re}{y.re} + \color{blue}{\frac{1}{\frac{\frac{y.re \cdot y.re}{x.im}}{y.im}}} \]
      associate-/l* [=>] 76.8	\[ \frac{x.re}{y.re} + \frac{1}{\frac{\color{blue}{\frac{y.re}{\frac{x.im}{y.re}}}}{y.im}} \]
      associate-/l/ [=>] 80.6	\[ \frac{x.re}{y.re} + \frac{1}{\color{blue}{\frac{y.re}{y.im \cdot \frac{x.im}{y.re}}}} \]
      associate-/r/ [=>] 80.7	\[ \frac{x.re}{y.re} + \color{blue}{\frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)} \]

   if 1.3500000000000001e143 < y.im < 7.5000000000000003e205

   1. Initial program 27.3%

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

   2. Taylor expanded in x.re around 0 25.4%

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

   3. Applied egg-rr 74.0%

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

      [Start] 25.4	\[ \frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 25.4	\[ \frac{y.im \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 29.5	\[ \color{blue}{\frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      +-commutative [=>] 29.5	\[ \frac{y.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 29.5	\[ \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      +-commutative [=>] 29.5	\[ \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \]
      hypot-def [=>] 74.0	\[ \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.7%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.25 \cdot 10^{-48}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 47000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 2
Accuracy	65.1%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;y.re \cdot \frac{x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 3
Accuracy	75.6%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 16200000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 4
Accuracy	70.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-48} \lor \neg \left(y.im \leq 580\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	70.0%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1400:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 6
Accuracy	70.0%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5300:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 7
Accuracy	75.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1550:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 8
Accuracy	63.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+34}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	41.0%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(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))))