_divideComplex, imaginary part

Average Error: 26.0 → 0.9

Time: 16.0s

Precision: binary64

Cost: 20352

Math

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

↓

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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))
  (- (* y.re (/ x.im (hypot y.re y.im))) (* (/ y.im (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_im * y_46_re) - (x_46_re * 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)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - ((y_46_im / 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_im * y_46_re) - (x_46_re * 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)) * ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - ((y_46_im / 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_im * y_46_re) - (x_46_re * 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)) * ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - ((y_46_im / 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_im * y_46_re) - Float64(x_46_re * 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(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(Float64(y_46_im / 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_im * y_46_re) - (x_46_re * 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)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - ((y_46_im / 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.0

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

2. Applied egg-rr 16.9

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

3. Applied egg-rr 0.9

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

4. Final simplification 0.9

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

Alternatives

Alternative 1
Error	10.6
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+121}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(-1, \frac{y.re}{\frac{y.im}{x.im}}, x.re\right)\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2
Error	12.3
Cost	13700

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(-1, \frac{y.re}{\frac{y.im}{x.im}}, x.re\right)\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3
Error	12.3
Cost	7236

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+120}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	12.4
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.3
Cost	1100

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.26 \cdot 10^{-70}:\\ \;\;\;\;\frac{-x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1100

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{-x.re}}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	16.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{-70} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Error	16.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.26 \cdot 10^{-70} \lor \neg \left(y.im \leq 4.9 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Error	19.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-13} \lor \neg \left(y.im \leq 1.6 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 10
Error	23.3
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-13} \lor \neg \left(y.im \leq 4 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	34.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+174}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+211}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 12
Error	36.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq 1.85 \cdot 10^{+209}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13
Error	58.8
Cost	192

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

Reproduce

herbie shell --seed 2023083 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))