_divideComplex, imaginary part

Average Error: 26.2 → 0.7

Time: 18.3s

Precision: binary64

Cost: 22409

Math

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

↓

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+297}\right):\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

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 (- (* x.im y.re) (* x.re y.im)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+297)))
     (*
      (/ 1.0 (hypot y.re y.im))
      (-
       (* y.re (/ x.im (hypot y.re y.im)))
       (* x.re (/ y.im (hypot y.re y.im)))))
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im)))))

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 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+297)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re * (y_46_im / hypot(y_46_re, y_46_im))));
	} else {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e+297)) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - (x_46_re * (y_46_im / Math.hypot(y_46_re, y_46_im))));
	} else {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e+297):
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - (x_46_re * (y_46_im / math.hypot(y_46_re, y_46_im))))
	else:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	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 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+297))
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(x_46_re * Float64(y_46_im / hypot(y_46_re, y_46_im)))));
	else
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e+297)))
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re * (y_46_im / hypot(y_46_re, y_46_im))));
	else
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = 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[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+297]], $MachinePrecision]], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

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))) < -inf.0 or 4.9999999999999998e297 < (/.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 63.4

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

   2. Applied egg-rr 58.7

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 0.8

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 4.9999999999999998e297

   1. Initial program 11.8

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

   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. Applied egg-rr 0.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty \lor \neg \left(\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}\right):\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.4
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.1 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \end{array} \]

Alternative 2
Error	11.1
Cost	14028

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \end{array} \]

Alternative 3
Error	11.2
Cost	14028

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-75}:\\ \;\;\;\;\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \end{array} \]

Alternative 4
Error	11.7
Cost	2000

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re \cdot y.im + \left(t_0 - x.re \cdot y.im\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \end{array} \]

Alternative 5
Error	11.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -7.6 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \end{array} \]

Alternative 6
Error	15.6
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+49} \lor \neg \left(y.re \leq -3.4 \cdot 10^{+18}\right) \land \left(y.re \leq -5.6 \cdot 10^{-39} \lor \neg \left(y.re \leq 5.4 \cdot 10^{-7}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	15.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{+18}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	15.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{y.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	19.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+44} \lor \neg \left(y.re \leq 6.1 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 11
Error	22.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12
Error	58.8
Cost	192

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

Alternative 13
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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))))