_divideComplex, imaginary part

Average Error: 26.6 → 16.5

Time: 22.2s

Precision: binary64

Cost: 7632

Math

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

↓

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 + y.re \cdot \frac{x.im}{{y.im}^{2}}\\ \end{array} \]

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.re y.im)))
        (t_1
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.9e+144)
     t_0
     (if (<= y.im -2.9e-244)
       t_1
       (if (<= y.im 4.4e-121)
         (/ x.im y.re)
         (if (<= y.im 3.8e+130)
           t_1
           (+ t_0 (* y.re (/ x.im (pow y.im 2.0))))))))))

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 = -(x_46_re / y_46_im);
	double t_1 = ((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 tmp;
	if (y_46_im <= -1.9e+144) {
		tmp = t_0;
	} else if (y_46_im <= -2.9e-244) {
		tmp = t_1;
	} else if (y_46_im <= 4.4e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.8e+130) {
		tmp = t_1;
	} else {
		tmp = t_0 + (y_46_re * (x_46_im / pow(y_46_im, 2.0)));
	}
	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 = -(x_46re / y_46im)
    t_1 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.9d+144)) then
        tmp = t_0
    else if (y_46im <= (-2.9d-244)) then
        tmp = t_1
    else if (y_46im <= 4.4d-121) then
        tmp = x_46im / y_46re
    else if (y_46im <= 3.8d+130) then
        tmp = t_1
    else
        tmp = t_0 + (y_46re * (x_46im / (y_46im ** 2.0d0)))
    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 = -(x_46_re / y_46_im);
	double t_1 = ((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 tmp;
	if (y_46_im <= -1.9e+144) {
		tmp = t_0;
	} else if (y_46_im <= -2.9e-244) {
		tmp = t_1;
	} else if (y_46_im <= 4.4e-121) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.8e+130) {
		tmp = t_1;
	} else {
		tmp = t_0 + (y_46_re * (x_46_im / Math.pow(y_46_im, 2.0)));
	}
	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_re / y_46_im)
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.9e+144:
		tmp = t_0
	elif y_46_im <= -2.9e-244:
		tmp = t_1
	elif y_46_im <= 4.4e-121:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 3.8e+130:
		tmp = t_1
	else:
		tmp = t_0 + (y_46_re * (x_46_im / math.pow(y_46_im, 2.0)))
	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 = Float64(-Float64(x_46_re / y_46_im))
	t_1 = 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)))
	tmp = 0.0
	if (y_46_im <= -1.9e+144)
		tmp = t_0;
	elseif (y_46_im <= -2.9e-244)
		tmp = t_1;
	elseif (y_46_im <= 4.4e-121)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.8e+130)
		tmp = t_1;
	else
		tmp = Float64(t_0 + Float64(y_46_re * Float64(x_46_im / (y_46_im ^ 2.0))));
	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_re / y_46_im);
	t_1 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.9e+144)
		tmp = t_0;
	elseif (y_46_im <= -2.9e-244)
		tmp = t_1;
	elseif (y_46_im <= 4.4e-121)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 3.8e+130)
		tmp = t_1;
	else
		tmp = t_0 + (y_46_re * (x_46_im / (y_46_im ^ 2.0)));
	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[(x$46$re / y$46$im), $MachinePrecision])}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$im, -1.9e+144], t$95$0, If[LessEqual[y$46$im, -2.9e-244], t$95$1, If[LessEqual[y$46$im, 4.4e-121], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+130], t$95$1, N[(t$95$0 + N[(y$46$re * N[(x$46$im / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.90000000000000013e144

   1. Initial program 44.7

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

   2. Taylor expanded in y.re around 0 14.6

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

   3. Simplified 14.6

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

      [Start] 14.6	\[ -1 \cdot \frac{x.re}{y.im} \]
      rational.json-simplify-2 [=>] 14.6	\[ \color{blue}{\frac{x.re}{y.im} \cdot -1} \]
      rational.json-simplify-9 [=>] 14.6	\[ \color{blue}{-\frac{x.re}{y.im}} \]

   if -1.90000000000000013e144 < y.im < -2.89999999999999996e-244 or 4.40000000000000042e-121 < y.im < 3.8000000000000002e130

   1. Initial program 18.0

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

   if -2.89999999999999996e-244 < y.im < 4.40000000000000042e-121

   1. Initial program 24.3

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

   2. Taylor expanded in y.re around inf 16.2

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

   if 3.8000000000000002e130 < y.im

   1. Initial program 41.7

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

   2. Taylor expanded in y.re around 0 14.5

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   3. Simplified 13.1

      \[\leadsto \color{blue}{\left(-\frac{x.re}{y.im}\right) + y.re \cdot \frac{x.im}{{y.im}^{2}}} \]
      Proof

      [Start] 14.5	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-2 [=>] 14.5	\[ \color{blue}{\frac{x.re}{y.im} \cdot -1} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-9 [=>] 14.5	\[ \color{blue}{\left(-\frac{x.re}{y.im}\right)} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-2 [<=] 14.5	\[ \left(-\frac{x.re}{y.im}\right) + \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} \]
      rational.json-simplify-49 [=>] 13.1	\[ \left(-\frac{x.re}{y.im}\right) + \color{blue}{y.re \cdot \frac{x.im}{{y.im}^{2}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+144}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-244}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x.re}{y.im}\right) + y.re \cdot \frac{x.im}{{y.im}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	1488

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	24.0
Cost	1428

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;y.im \cdot \frac{-x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 3
Error	23.1
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-157}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+117}:\\ \;\;\;\;\frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Error	23.3
Cost	784

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 5
Error	37.6
Cost	192

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

Reproduce

herbie shell --seed 2023065 
(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))))