_divideComplex, real part

Average Error: 26.3 → 0.6

Time: 16.8s

Precision: binary64

Cost: 20352

Math

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

↓

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

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
 (*
  (/ 1.0 (hypot y.re y.im))
  (+ (/ x.im (/ (hypot y.re y.im) y.im)) (/ x.re (/ (hypot y.re y.im) y.re)))))

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 (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (hypot(y_46_re, y_46_im) / y_46_im)) + (x_46_re / (hypot(y_46_re, y_46_im) / y_46_re)));
}

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_im / (Math.hypot(y_46_re, y_46_im) / y_46_im)) + (x_46_re / (Math.hypot(y_46_re, y_46_im) / y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_im / (math.hypot(y_46_re, y_46_im) / y_46_im)) + (x_46_re / (math.hypot(y_46_re, y_46_im) / y_46_re)))

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 Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) / y_46_im)) + Float64(x_46_re / Float64(hypot(y_46_re, y_46_im) / y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (hypot(y_46_re, y_46_im) / y_46_im)) + (x_46_re / (hypot(y_46_re, y_46_im) / y_46_re)));
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[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.3

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

2. Applied egg-rr 17.2

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

3. Applied egg-rr 17.2

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

4. Simplified 0.6

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

   [Start] 17.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   +-commutative [=>] 17.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   associate-/l* [=>] 9.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   associate-/l* [=>] 0.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}\right) \]

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	8.7
Cost	20560

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+157}:\\ \;\;\;\;\frac{x.im}{y.im} + {\left(\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}\right)}^{-1}\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.im + \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)\\ \end{array} \]

Alternative 2
Error	11.0
Cost	14160

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + {\left(\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}\right)}^{-1}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;x.im \cdot \frac{y.im}{t_0} + y.re \cdot \frac{x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)\\ \end{array} \]

Alternative 3
Error	11.6
Cost	7300

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + {\left(\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}\right)}^{-1}\\ \mathbf{elif}\;y.im \leq -1.02 \cdot 10^{-104}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.82 \cdot 10^{+127}:\\ \;\;\;\;x.im \cdot \frac{y.im}{t_0} + y.re \cdot \frac{x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 4
Error	11.6
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+127}:\\ \;\;\;\;x.im \cdot \frac{y.im}{t_0} + y.re \cdot \frac{x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 5
Error	12.0
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -11000000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq 1.36 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 7
Error	15.4
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im}{y.im} + \left(x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.25 \cdot 10^{+38}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8
Error	24.7
Cost	976

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{-51}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	18.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -310000000 \lor \neg \left(y.im \leq 3.1 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	18.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -38000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 11
Error	18.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 12
Error	15.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -560000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 13
Error	14.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -27000000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 14
Error	22.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -31000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 15
Error	37.8
Cost	192

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

Reproduce

herbie shell --seed 2023083 
(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))))