Average Error: 26.6 → 1.6
Time: 24.2s
Precision: binary64
Cost: 33152
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
(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))
  (* (/ y.im (hypot y.im y.re)) (/ (- x.re) (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)), ((y_46_im / hypot(y_46_im, y_46_re)) * (-x_46_re / 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(y_46_im / hypot(y_46_im, y_46_re)) * Float64(Float64(-x_46_re) / 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[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-x$46$re) / 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{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)

Error

Derivation

  1. Initial program 26.6

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

  2. Applied egg-rr15.6

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

  3. Applied egg-rr1.6

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

  4. Final simplification1.6

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

Alternatives

Alternative 1
Error7.4
Cost33288
\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.3786115264042156 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.0953591642983293 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{x.im}{y.im}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\\ \end{array} \]
Alternative 2
Error10.0
Cost33156
\[\begin{array}{l} t_0 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -5.734033481894504 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, t_0, y.im \cdot \left(x.re \cdot \left(-{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2}\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 9.399082510003664 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t_0, \frac{\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right)\\ \end{array} \]
Alternative 3
Error1.7
Cost33152
\[\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right) \]
Alternative 4
Error8.1
Cost27588
\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right)\\ \end{array} \]
Alternative 5
Error10.0
Cost14288
\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.4589299905809306 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-175}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.377624204014503 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \left(x.im \cdot \frac{1}{y.im}\right) - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 6
Error16.1
Cost1496
\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ t_2 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.2360995689060646 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error15.8
Cost1496
\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ t_2 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.2360995689060646 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error15.8
Cost1496
\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.2360995689060646 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 9
Error16.1
Cost1492
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.8764357443912225 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.9428857570054914 \cdot 10^{+88}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \left(x.im \cdot \frac{1}{y.im}\right) - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 10
Error11.5
Cost1488
\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7.69449897482202 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-129}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.9428857570054914 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \left(x.im \cdot \frac{1}{y.im}\right) - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 11
Error16.2
Cost1428
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.8764357443912225 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.9428857570054914 \cdot 10^{+88}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \end{array} \]
Alternative 12
Error17.5
Cost1368
\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.2360995689060646 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error16.3
Cost1368
\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ t_2 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.0149644145814087 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.988174147505243 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.2360995689060646 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error21.0
Cost1104
\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 15
Error24.3
Cost784
\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.979558976022407 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7305573003182963 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1926.0100326518855:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 16
Error37.5
Cost192
\[\frac{x.im}{y.re} \]

Error

Reproduce

herbie shell --seed 2022310 
(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))))