Average Error: 26.9 → 1.7
Time: 23.8s
Precision: binary64
Cost: 33152
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right) \]
(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)) (/ (hypot y.im y.re) x.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)) / (hypot(y_46_im, y_46_re) / x_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(Float64(-y_46_im) / hypot(y_46_im, y_46_re)) / Float64(hypot(y_46_im, y_46_re) / x_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[(N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision] / x$46$re), $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{\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right)

Error

Derivation

  1. Initial program 26.9

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

  2. Applied egg-rr15.7

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

  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. Applied egg-rr1.7

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

  5. Final simplification1.7

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

Alternatives

Alternative 1
Error9.0
Cost33552
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot \left(-y.im\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{if}\;y.im \leq -4.794042462619568 \cdot 10^{+124}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.697590635772782 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-215}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.9808424949256643 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 2
Error1.5
Cost33152
\[\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
Alternative 3
Error9.2
Cost27216
\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.833171361393942 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-y.im}{y.re}\right)\\ \end{array} \]
Alternative 4
Error9.2
Cost27216
\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.833171361393942 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{-y.im}{y.re}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\right)\\ \end{array} \]
Alternative 5
Error8.3
Cost27216
\[\begin{array}{l} t_0 := \frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}\\ t_1 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_3 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -2.9912388202249646 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(t_2, t_3, \frac{\frac{y.im}{y.re}}{t_0}\right)\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, t_3, \frac{\frac{-y.im}{y.re}}{t_0}\right)\\ \end{array} \]
Alternative 6
Error10.1
Cost14160
\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1.833171361393942 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-152}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.5045501848422174 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error16.3
Cost1892
\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{-x.re}{y.im}\\ t_2 := \frac{x.im}{y.re} - x.re \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ t_3 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_4 := \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.7913582216935604 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq -1.9396320210000686 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 1.1134830631313484 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.0791024974413246 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 5.150325609861833 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6.8713236446597505 \cdot 10^{-40}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 6.197458397330996 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 8
Error16.3
Cost1892
\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{y.re \cdot x.im}{t_1}\\ t_3 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_4 := \frac{x.im}{y.re} - x.re \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.7913582216935604 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq -2.2922731553287687 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.re \cdot \left(-y.im\right)}{t_1}\\ \mathbf{elif}\;y.re \leq -2.5357738530320827 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.1134830631313484 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.0791024974413246 \cdot 10^{-73}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 5.150325609861833 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8713236446597505 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 6.197458397330996 \cdot 10^{-22}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 9
Error11.8
Cost1488
\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1.337754889392654 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -9.47708778455609 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-150}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0079564568981761 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error18.7
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.571001699659394 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 11
Error18.8
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.571001699659394 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 12
Error15.1
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -49309.53180018831:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 13
Error15.1
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -49309.53180018831:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
Alternative 14
Error18.9
Cost840
\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.571001699659394 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 15
Error18.8
Cost840
\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.571001699659394 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
Alternative 16
Error23.0
Cost520
\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.571001699659394 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 827300406763.5386:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 17
Error37.3
Cost192
\[\frac{x.im}{y.re} \]

Error

Reproduce

herbie shell --seed 2022321 
(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))))