Average Error: 26.2 → 1.5
Time: 23.8s
Precision: binary64
Cost: 33152
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
(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))
  (* (/ x.re (hypot y.im y.re)) (/ (- y.im) (hypot y.im y.re)))))
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)), ((x_46_re / hypot(y_46_im, y_46_re)) * (-y_46_im / hypot(y_46_im, y_46_re))));
}

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(x_46_re / hypot(y_46_im, y_46_re)) * Float64(Float64(-y_46_im) / hypot(y_46_im, y_46_re))))
end

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

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

Error

Derivation

  1. Initial program 26.2

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

  2. Applied egg-rr15.3

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

  3. Applied egg-rr1.5

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

  4. Final simplification1.5

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

Alternatives

Alternative 1
Error4.4
Cost26952
\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \mathsf{fma}\left(t_0, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -3.6973860497430285 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.im, t_0, \frac{y.im \cdot \left(-x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error6.6
Cost22024
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ t_2 := \frac{t_1}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+284}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\frac{\frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error10.3
Cost14160
\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.4786926448593535 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-198}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 4
Error13.9
Cost7956
\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -41.12221041906736:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-136}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t_1}\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 5
Error13.9
Cost7300
\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -41.12221041906736:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 6
Error13.9
Cost7172
\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, t_2\right)\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -41.12221041906736:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 7
Error13.9
Cost1884
\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -41.12221041906736:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.3301041980390175 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 8
Error16.7
Cost1232
\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.087015133523303 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error16.7
Cost1232
\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.2165722800599223 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.282901720871127 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.087015133523303 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error19.2
Cost968
\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.087015133523303 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error23.1
Cost520
\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.091015625630344 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.087015133523303 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error58.8
Cost192
\[\frac{x.im}{y.im} \]
Alternative 13
Error37.8
Cost192
\[\frac{x.im}{y.re} \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))