_divideComplex, imaginary part

Average Error: 26.4 → 15.3

Time: 23.2s

Precision: binary64

Cost: 7632

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} + y.im \cdot \left(-\frac{x.re}{{y.re}^{2}}\right)\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-160}:\\ \;\;\;\;\left(-\frac{x.re}{y.im}\right) + y.re \cdot \frac{x.im}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.im y.re) (* y.im (- (/ x.re (pow y.re 2.0)))))))
   (if (<= y.re -9.5e+95)
     t_1
     (if (<= y.re -7e-135)
       t_0
       (if (<= y.re 4.5e-160)
         (+ (- (/ x.re y.im)) (* y.re (/ x.im (pow y.im 2.0))))
         (if (<= y.re 1.42e+141) t_0 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 = ((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 t_1 = (x_46_im / y_46_re) + (y_46_im * -(x_46_re / pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -9.5e+95) {
		tmp = t_1;
	} else if (y_46_re <= -7e-135) {
		tmp = t_0;
	} else if (y_46_re <= 4.5e-160) {
		tmp = -(x_46_re / y_46_im) + (y_46_re * (x_46_im / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.42e+141) {
		tmp = t_0;
	} 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 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) + (y_46im * -(x_46re / (y_46re ** 2.0d0)))
    if (y_46re <= (-9.5d+95)) then
        tmp = t_1
    else if (y_46re <= (-7d-135)) then
        tmp = t_0
    else if (y_46re <= 4.5d-160) then
        tmp = -(x_46re / y_46im) + (y_46re * (x_46im / (y_46im ** 2.0d0)))
    else if (y_46re <= 1.42d+141) then
        tmp = t_0
    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 = ((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 t_1 = (x_46_im / y_46_re) + (y_46_im * -(x_46_re / Math.pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -9.5e+95) {
		tmp = t_1;
	} else if (y_46_re <= -7e-135) {
		tmp = t_0;
	} else if (y_46_re <= 4.5e-160) {
		tmp = -(x_46_re / y_46_im) + (y_46_re * (x_46_im / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.42e+141) {
		tmp = t_0;
	} 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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((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 / math.pow(y_46_re, 2.0)))
	tmp = 0
	if y_46_re <= -9.5e+95:
		tmp = t_1
	elif y_46_re <= -7e-135:
		tmp = t_0
	elif y_46_re <= 4.5e-160:
		tmp = -(x_46_re / y_46_im) + (y_46_re * (x_46_im / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 1.42e+141:
		tmp = t_0
	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(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)))
	t_1 = Float64(Float64(x_46_im / y_46_re) + Float64(y_46_im * Float64(-Float64(x_46_re / (y_46_re ^ 2.0)))))
	tmp = 0.0
	if (y_46_re <= -9.5e+95)
		tmp = t_1;
	elseif (y_46_re <= -7e-135)
		tmp = t_0;
	elseif (y_46_re <= 4.5e-160)
		tmp = Float64(Float64(-Float64(x_46_re / y_46_im)) + Float64(y_46_re * Float64(x_46_im / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 1.42e+141)
		tmp = t_0;
	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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((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 / (y_46_re ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -9.5e+95)
		tmp = t_1;
	elseif (y_46_re <= -7e-135)
		tmp = t_0;
	elseif (y_46_re <= 4.5e-160)
		tmp = -(x_46_re / y_46_im) + (y_46_re * (x_46_im / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 1.42e+141)
		tmp = t_0;
	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[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(y$46$im * (-N[(x$46$re / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e+95], t$95$1, If[LessEqual[y$46$re, -7e-135], t$95$0, If[LessEqual[y$46$re, 4.5e-160], N[((-N[(x$46$re / y$46$im), $MachinePrecision]) + N[(y$46$re * N[(x$46$im / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.42e+141], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.5000000000000004e95 or 1.42000000000000005e141 < y.re

   1. Initial program 41.5

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

   2. Taylor expanded in y.re around inf 16.7

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

   3. Simplified 15.2

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

      [Start] 16.7	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      rational.json-simplify-49 [=>] 15.2	\[ \frac{x.im}{y.re} + -1 \cdot \color{blue}{\left(y.im \cdot \frac{x.re}{{y.re}^{2}}\right)} \]
      rational.json-simplify-43 [=>] 15.2	\[ \frac{x.im}{y.re} + \color{blue}{y.im \cdot \left(\frac{x.re}{{y.re}^{2}} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 15.2	\[ \frac{x.im}{y.re} + y.im \cdot \color{blue}{\left(-\frac{x.re}{{y.re}^{2}}\right)} \]

   if -9.5000000000000004e95 < y.re < -6.9999999999999997e-135 or 4.50000000000000026e-160 < y.re < 1.42000000000000005e141

   1. Initial program 16.4

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

   if -6.9999999999999997e-135 < y.re < 4.50000000000000026e-160

   1. Initial program 23.9

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

   2. Taylor expanded in y.re around 0 9.5

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

   3. Simplified 13.6

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

      [Start] 9.5	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-2 [=>] 9.5	\[ \color{blue}{\frac{x.re}{y.im} \cdot -1} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-9 [=>] 9.5	\[ \color{blue}{\left(-\frac{x.re}{y.im}\right)} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      rational.json-simplify-2 [=>] 9.5	\[ \left(-\frac{x.re}{y.im}\right) + \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} \]
      rational.json-simplify-49 [=>] 13.6	\[ \left(-\frac{x.re}{y.im}\right) + \color{blue}{y.re \cdot \frac{x.im}{{y.im}^{2}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(-\frac{x.re}{{y.re}^{2}}\right)\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-160}:\\ \;\;\;\;\left(-\frac{x.re}{y.im}\right) + y.re \cdot \frac{x.im}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(-\frac{x.re}{{y.re}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.9
Cost	7500

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -6.5 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\left(-\frac{x.re}{y.im}\right) + y.re \cdot \frac{x.im}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 2
Error	16.0
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-161}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 3
Error	21.5
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-116}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.22 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Error	22.7
Cost	1232

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{-x.re}{\frac{t_0}{y.im}}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-115}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 5
Error	23.5
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 6
Error	24.0
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 7
Error	37.7
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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))))