_divideComplex, imaginary part

Average Error: 26.3 → 1.6

Time: 18.5s

Precision: binary64

Cost: 20352

Math

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

↓

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

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(y_46_im / Float64(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[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.3

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

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 1.1

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

4. Applied egg-rr 19.5

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)} - 1\right)}\right) \]

5. Simplified 1.6

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

   [Start] 19.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re - \left(e^{\mathsf{log1p}\left(\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)} - 1\right)\right) \]
   expm1-def [=>] 12.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\right)}\right) \]
   expm1-log1p [=>] 1.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re - \color{blue}{\frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right) \]

6. Final simplification 1.6

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

Alternatives

Alternative 1
Error	1.1
Cost	20352

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

Alternative 2
Error	8.6
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\\ \end{array} \]

Alternative 3
Error	14.8
Cost	14168

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := y.im \cdot y.im + y.re \cdot y.re\\ t_2 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\frac{y.im \cdot y.im}{x.im}}\right)\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_2 + 2 \cdot \left(2 \cdot \mathsf{fma}\left(-y.im, x.re, y.im \cdot x.re\right)\right)}{t_1}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	14.6
Cost	14168

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := y.im \cdot y.im + y.re \cdot y.re\\ t_2 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\frac{y.im \cdot y.im}{x.im}}\right)\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -2.65 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_2 + 2 \cdot \left(2 \cdot \mathsf{fma}\left(-y.im, x.re, y.im \cdot x.re\right)\right)}{t_1}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-54}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	12.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6
Error	16.4
Cost	1428

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.66 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -7.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re \cdot \left(-y.im\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.9
Cost	1369

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2800000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -4.9 \cdot 10^{-40} \lor \neg \left(y.im \leq 2.25 \cdot 10^{+33}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	17.1
Cost	1369

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3300000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -6.7 \cdot 10^{-40} \lor \neg \left(y.im \leq 4 \cdot 10^{+23}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -7.7 \cdot 10^{+15} \lor \neg \left(y.re \leq 2.4 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 10
Error	23.3
Cost	786

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+97} \lor \neg \left(y.im \leq -2400000000000\right) \land \left(y.im \leq -7.7 \cdot 10^{-25} \lor \neg \left(y.im \leq 3 \cdot 10^{+23}\right)\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2023237 
(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))))