_divideComplex, real part

Average Error: 59.4% → 97.6%

Time: 27.0s

Precision: binary64

Cost: 39360.00

Math

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

↓

\[{\left(\sqrt[3]{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{3} \]

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
 (pow
  (cbrt
   (/
    (fma x.im (/ y.im (hypot y.re y.im)) (/ x.re (/ (hypot y.re y.im) y.re)))
    (hypot y.re y.im)))
  3.0))

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) {
	return pow(cbrt((fma(x_46_im, (y_46_im / hypot(y_46_re, y_46_im)), (x_46_re / (hypot(y_46_re, y_46_im) / y_46_re))) / hypot(y_46_re, y_46_im))), 3.0);
}

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)
	return cbrt(Float64(fma(x_46_im, Float64(y_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re / Float64(hypot(y_46_re, y_46_im) / y_46_re))) / hypot(y_46_re, y_46_im))) ^ 3.0
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_] := N[Power[N[Power[N[(N[(x$46$im * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 59.4

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

2. Applied egg-rr 73.9

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

3. Applied egg-rr 73.9

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

4. Simplified 85.3

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

   [Start] 73.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   +-commutative [<=] 73.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   associate-/l* [=>] 85.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   *-commutative [=>] 85.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

5. Applied egg-rr 97.6

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{3}} \]

6. Final simplification 97.6

   \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{3} \]

Alternatives

Alternative 1
Error	89.3%
Cost	21961.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+287}\right):\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Error	93.4%
Cost	20616.00

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+71}:\\ \;\;\;\;t_0 \cdot \left(t_1 + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.re + t_1\right)\\ \end{array} \]

Alternative 3
Error	83.6%
Cost	14160.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -8.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.12 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-98}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 4
Error	79.3%
Cost	13904.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	81.9%
Cost	1488.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -7.2 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 6
Error	59.8%
Cost	1368.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 0.135:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 10^{+74}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+285}:\\ \;\;\;\;\frac{y.im}{\frac{y.re}{x.im}} \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	60.0%
Cost	1240.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 0.26:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+286}:\\ \;\;\;\;\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	73.4%
Cost	1234.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{-16} \lor \neg \left(y.re \leq 3.6 \cdot 10^{-92}\right) \land \left(y.re \leq 0.135 \lor \neg \left(y.re \leq 2.3 \cdot 10^{+69}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 9
Error	69.7%
Cost	1232.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ t_1 := \frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-135}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	72.5%
Cost	1232.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.9 \cdot 10^{-135}:\\ \;\;\;\;x.im \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	73.4%
Cost	1232.00

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -1.95 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 0.205:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	73.6%
Cost	1232.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 0.135:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	75.0%
Cost	1232.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 0.56:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	74.6%
Cost	1100.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 15
Error	74.2%
Cost	1100.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 16
Error	74.1%
Cost	1100.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 17
Error	63.6%
Cost	972.00

\[\begin{array}{l} t_0 := \frac{x.re}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\frac{y.re}{x.im}} \cdot \frac{1}{y.re}\\ \end{array} \]

Alternative 18
Error	62.1%
Cost	721.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92} \lor \neg \left(y.re \leq 0.72\right) \land y.re \leq 5.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 19
Error	41.4%
Cost	192.00

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

Reproduce

herbie shell --seed 2023093 
(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))))