_divideComplex, imaginary part

Average Error: 26.0 → 1.1

Time: 19.8s

Precision: binary64

Cost: 35208

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ t_2 := \frac{t_1}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{+248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
 (let* ((t_0
         (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)))))
        (t_1 (- (* x.im y.re) (* x.re y.im)))
        (t_2 (/ t_1 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_2 -4e+248)
     t_0
     (if (<= t_2 5e+271)
       (* (/ 1.0 (hypot y.re y.im)) (/ t_1 (hypot y.re y.im)))
       t_0))))

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) {
	double t_0 = 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))));
	double t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_2 = t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_2 <= -4e+248) {
		tmp = t_0;
	} else if (t_2 <= 5e+271) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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)
	t_0 = 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))))
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_2 = Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_2 <= -4e+248)
		tmp = t_0;
	elseif (t_2 <= 5e+271)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+248], t$95$0, If[LessEqual[t$95$2, 5e+271], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -4.00000000000000018e248 or 5.0000000000000003e271 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 59.7

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

   2. Applied egg-rr 35.7

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

   3. Applied egg-rr 1.8

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

   if -4.00000000000000018e248 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000003e271

   1. Initial program 11.7

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

   2. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -4 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\\ \mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.9
Cost	33552

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot \left(-y.im\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{if}\;y.im \leq -7.146207147472019 \cdot 10^{+125}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re}{\frac{y.re}{y.im}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.im \leq 7.44935694106442 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 2
Error	1.7
Cost	33152

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

Alternative 3
Error	9.8
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5.755640880864757 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-195}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re}{\frac{y.re}{y.im}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.im \leq 1.917726867369815 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 4
Error	12.3
Cost	7696

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -89842096553.5778:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-195}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 3.9508936352397686 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 5
Error	11.5
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.8553213279974144 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.897018463432072 \cdot 10^{-118}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re}{\frac{y.re}{y.im}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8799572103187346 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.4
Cost	1360

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.755640880864757 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.449980886049424 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1694614927.6689146:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.873608231729959:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re}{\frac{y.re}{y.im}} \cdot \frac{-1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.755640880864757 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.449980886049424 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1694614927.6689146:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.873608231729959:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.755640880864757 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.449980886049424 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1694614927.6689146:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.873608231729959:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	16.0
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.755640880864757 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.449980886049424 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1694614927.6689146:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.873608231729959:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	19.2
Cost	840

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -9.197655707976395:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.484607801108024 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -9.197655707976395:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.484607801108024 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12
Error	57.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.275363659072502 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.0653595180822158 \cdot 10^{+167}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 13
Error	36.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.15 \cdot 10^{-294}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 14
Error	58.8
Cost	192

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

Reproduce

herbie shell --seed 2022317 
(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))))