_divideComplex, imaginary part

Average Error: 26.7 → 1.4

Time: 28.8s

Precision: binary64

Cost: 33152

Math

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

↓

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

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
 (fma
  (/ y.re (hypot y.re y.im))
  (/ x.im (hypot y.re y.im))
  (/ (/ (- y.im) (/ (hypot y.im y.re) x.re)) (hypot 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_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 fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((-y_46_im / (hypot(y_46_im, y_46_re) / x_46_re)) / hypot(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_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 fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(Float64(-y_46_im) / Float64(hypot(y_46_im, y_46_re) / x_46_re)) / hypot(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$46$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[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[((-y$46$im) / N[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.7

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

2. Applied egg-rr 15.5

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)} \]

3. Applied egg-rr 1.5

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

4. Applied egg-rr 1.4

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

5. Final simplification 1.4

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

Alternatives

Alternative 1
Error	6.0
Cost	26952

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -6.566731613537308 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.8752470576642625 \cdot 10^{+115}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(x.im, \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 2
Error	10.4
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -4.348610485428097 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-132}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 3
Error	12.2
Cost	13832

\[\begin{array}{l} \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, -x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.798730089960014 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 4
Error	12.2
Cost	7824

\[\begin{array}{l} t_0 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-163}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.798730089960014 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 5
Error	12.2
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.798730089960014 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 6
Error	12.2
Cost	7172

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	16.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.1758521565422204 \cdot 10^{-48}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq -1.998814742675197 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.605581508364986 \cdot 10^{-74}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.4
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -122006031482371650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.5586786786398142 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	16.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	16.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 12
Error	16.5
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 13
Error	16.5
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.916160279112307 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 14
Error	20.4
Cost	1104

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	20.3
Cost	1104

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.9522731660807546 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	24.3
Cost	916

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.6689993466329684 \cdot 10^{-5}:\\ \;\;\;\;-\frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 17
Error	23.8
Cost	784

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.2088374622742729 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.151460632922877 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 18
Error	35.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.14191476856822 \cdot 10^{+205}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.070553457478154 \cdot 10^{+226}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 19
Error	58.7
Cost	192

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

Alternative 20
Error	57.0
Cost	192

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

Reproduce

herbie shell --seed 2022294 
(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))))