_divideComplex, imaginary part

Average Error: 26.9 → 16.4

Time: 18.0s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{t_0}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y.re \cdot x.im}{t_0} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (- (/ x.im y.re) (/ (* y.im x.re) t_0))))
   (if (<= y.re -1.8e+57)
     t_1
     (if (<= y.re 3.4e-17) (- (/ (* y.re x.im) t_0) (/ x.re y.im)) t_1))))

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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / t_0);
	double tmp;
	if (y_46_re <= -1.8e+57) {
		tmp = t_1;
	} else if (y_46_re <= 3.4e-17) {
		tmp = ((y_46_re * x_46_im) / t_0) - (x_46_re / y_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

↓

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46im / y_46re) - ((y_46im * x_46re) / t_0)
    if (y_46re <= (-1.8d+57)) then
        tmp = t_1
    else if (y_46re <= 3.4d-17) then
        tmp = ((y_46re * x_46im) / t_0) - (x_46re / y_46im)
    else
        tmp = t_1
    end if
    code = tmp
end function

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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / t_0);
	double tmp;
	if (y_46_re <= -1.8e+57) {
		tmp = t_1;
	} else if (y_46_re <= 3.4e-17) {
		tmp = ((y_46_re * x_46_im) / t_0) - (x_46_re / y_46_im);
	} else {
		tmp = t_1;
	}
	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 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / t_0)
	tmp = 0
	if y_46_re <= -1.8e+57:
		tmp = t_1
	elif y_46_re <= 3.4e-17:
		tmp = ((y_46_re * x_46_im) / t_0) - (x_46_re / y_46_im)
	else:
		tmp = t_1
	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(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * x_46_re) / t_0))
	tmp = 0.0
	if (y_46_re <= -1.8e+57)
		tmp = t_1;
	elseif (y_46_re <= 3.4e-17)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) / t_0) - Float64(x_46_re / y_46_im));
	else
		tmp = t_1;
	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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / t_0);
	tmp = 0.0;
	if (y_46_re <= -1.8e+57)
		tmp = t_1;
	elseif (y_46_re <= 3.4e-17)
		tmp = ((y_46_re * x_46_im) / t_0) - (x_46_re / y_46_im);
	else
		tmp = t_1;
	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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.8e+57], t$95$1, If[LessEqual[y$46$re, 3.4e-17], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.8000000000000001e57 or 3.3999999999999998e-17 < y.re

   1. Initial program 34.4

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

   2. Applied egg-rr 34.5

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

   3. Applied egg-rr 34.4

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

   4. Taylor expanded in y.re around inf 18.9

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

   if -1.8000000000000001e57 < y.re < 3.3999999999999998e-17

   1. Initial program 19.6

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

   2. Applied egg-rr 19.9

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

   3. Applied egg-rr 19.6

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

   4. Taylor expanded in y.im around inf 14.0

      \[\leadsto \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{\frac{x.re}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.9
Cost	1556

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+21}:\\ \;\;\;\;t_1 \cdot \left(y.re \cdot x.im\right)\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;t_1 \cdot \left(y.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 2
Error	21.8
Cost	1556

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-68}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-94}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(y.re \cdot x.im\right)\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(-y.im\right) \cdot \frac{x.re}{y.im \cdot y.im + y.re \cdot y.re}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 3
Error	21.8
Cost	1428

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\frac{y.re \cdot x.im}{t_0}\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{y.im}{y.re \cdot \left(-y.re\right) - y.im \cdot y.im} \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Error	21.8
Cost	1428

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-95}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{y.re \cdot x.im}{t_0}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 5
Error	21.8
Cost	1428

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-88}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(y.re \cdot x.im\right)\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 6
Error	18.1
Cost	1224

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	23.8
Cost	1100

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	24.0
Cost	1100

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.7
Cost	520

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	38.1
Cost	192

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

Reproduce

herbie shell --seed 2023100 
(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))))