_divideComplex, real part

Average Error: 26.4 → 0.8

Time: 17.1s

Precision: binary64

Cost: 20352

Math

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

↓

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

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)) (* (/ y.re (hypot y.re y.im)) x.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)) + ((y_46_re / hypot(y_46_re, y_46_im)) * x_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)) + ((y_46_re / Math.hypot(y_46_re, y_46_im)) * x_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)) + ((y_46_re / math.hypot(y_46_re, y_46_im)) * x_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(Float64(y_46_re / hypot(y_46_re, y_46_im)) * x_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)) + ((y_46_re / hypot(y_46_re, y_46_im)) * x_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[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.4

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

2. Applied egg-rr 17.3

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

3. Applied egg-rr 17.3

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

4. Simplified 9.3

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

   [Start] 17.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   +-commutative [=>] 17.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   associate-/l* [=>] 9.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   *-commutative [=>] 9.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

5. Applied egg-rr 0.8

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

6. Final simplification 0.8

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

Alternatives

Alternative 1
Error	10.7
Cost	20824

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -850000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.7
Cost	20824

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ t_2 := \mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)\\ t_3 := t_0 \cdot \frac{t_2}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -850000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{t_0}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_2}}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	12.7
Cost	13896

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \end{array} \]

Alternative 4
Error	12.4
Cost	1752

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -900000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.48 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.1
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+105} \lor \neg \left(y.re \leq -1.25 \cdot 10^{+57} \lor \neg \left(y.re \leq -9.5 \cdot 10^{+14}\right) \land y.re \leq 1.88 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 6
Error	15.8
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+101} \lor \neg \left(y.re \leq -1.25 \cdot 10^{+57} \lor \neg \left(y.re \leq -2.9 \cdot 10^{+14}\right) \land y.re \leq 2.3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 7
Error	15.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ t_1 := \frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	19.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+105} \lor \neg \left(y.re \leq 2.45 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 10
Error	23.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3400000000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	36.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq 3.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(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))))