_divideComplex, imaginary part

Average Error: 25.6 → 4.6

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -7.579958549150986 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im}{y.re} - t_0\\ \mathbf{elif}\;y.re \leq -3.9696921097903 \cdot 10^{-311}:\\ \;\;\;\;\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot {\left(\frac{\sqrt{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{2} - t_0\\ \end{array} \]

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 (/ x.re (+ y.im (* y.re (/ y.re y.im))))))
   (if (<= y.re -7.579958549150986e+153)
     (- (/ x.im y.re) t_0)
     (if (<= y.re -3.9696921097903e-311)
       (- (/ x.im (/ (pow (hypot y.re y.im) 2.0) y.re)) t_0)
       (- (* x.im (pow (/ (sqrt y.re) (hypot y.re y.im)) 2.0)) 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 = x_46_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_re <= -7.579958549150986e+153) {
		tmp = (x_46_im / y_46_re) - t_0;
	} else if (y_46_re <= -3.9696921097903e-311) {
		tmp = (x_46_im / (pow(hypot(y_46_re, y_46_im), 2.0) / y_46_re)) - t_0;
	} else {
		tmp = (x_46_im * pow((sqrt(y_46_re) / hypot(y_46_re, y_46_im)), 2.0)) - t_0;
	}
	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_re / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_re <= -7.579958549150986e+153) {
		tmp = (x_46_im / y_46_re) - t_0;
	} else if (y_46_re <= -3.9696921097903e-311) {
		tmp = (x_46_im / (Math.pow(Math.hypot(y_46_re, y_46_im), 2.0) / y_46_re)) - t_0;
	} else {
		tmp = (x_46_im * Math.pow((Math.sqrt(y_46_re) / Math.hypot(y_46_re, y_46_im)), 2.0)) - t_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 + (y_46_re * (y_46_re / y_46_im)))
	tmp = 0
	if y_46_re <= -7.579958549150986e+153:
		tmp = (x_46_im / y_46_re) - t_0
	elif y_46_re <= -3.9696921097903e-311:
		tmp = (x_46_im / (math.pow(math.hypot(y_46_re, y_46_im), 2.0) / y_46_re)) - t_0
	else:
		tmp = (x_46_im * math.pow((math.sqrt(y_46_re) / math.hypot(y_46_re, y_46_im)), 2.0)) - 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 = Float64(x_46_re / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re / y_46_im))))
	tmp = 0.0
	if (y_46_re <= -7.579958549150986e+153)
		tmp = Float64(Float64(x_46_im / y_46_re) - t_0);
	elseif (y_46_re <= -3.9696921097903e-311)
		tmp = Float64(Float64(x_46_im / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_re)) - t_0);
	else
		tmp = Float64(Float64(x_46_im * (Float64(sqrt(y_46_re) / hypot(y_46_re, y_46_im)) ^ 2.0)) - t_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 + (y_46_re * (y_46_re / y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -7.579958549150986e+153)
		tmp = (x_46_im / y_46_re) - t_0;
	elseif (y_46_re <= -3.9696921097903e-311)
		tmp = (x_46_im / ((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_re)) - t_0;
	else
		tmp = (x_46_im * ((sqrt(y_46_re) / hypot(y_46_re, y_46_im)) ^ 2.0)) - t_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 / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.579958549150986e+153], N[(N[(x$46$im / y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[y$46$re, -3.9696921097903e-311], N[(N[(x$46$im / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x$46$im * N[Power[N[(N[Sqrt[y$46$re], $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 3 regimes

2. if y.re < -7.5799585491509855e153

   1. Initial program 46.4

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

   2. Simplified 46.4

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

   3. Applied egg-rr 46.4

      \[\leadsto \color{blue}{\left(y.im \cdot x.re - x.im \cdot y.re\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{2}}} \]

   4. Taylor expanded in x.re around 0 46.4

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

   5. Simplified 45.3

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

   6. Taylor expanded in y.re around 0 45.3

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

   7. Simplified 38.8

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

   8. Taylor expanded in y.re around inf 7.8

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

   if -7.5799585491509855e153 < y.re < -3.96969210979026e-311

   1. Initial program 18.9

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

   2. Simplified 18.9

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

   3. Applied egg-rr 19.1

      \[\leadsto \color{blue}{\left(y.im \cdot x.re - x.im \cdot y.re\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{2}}} \]

   4. Taylor expanded in x.re around 0 18.9

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

   5. Simplified 15.0

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

   6. Taylor expanded in y.re around 0 5.3

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

   7. Simplified 5.3

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

   8. Applied egg-rr 5.3

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

   if -3.96969210979026e-311 < y.re

   1. Initial program 24.9

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

   2. Simplified 24.9

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

   3. Applied egg-rr 25.1

      \[\leadsto \color{blue}{\left(y.im \cdot x.re - x.im \cdot y.re\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{2}}} \]

   4. Taylor expanded in x.re around 0 24.9

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

   5. Simplified 21.9

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

   6. Taylor expanded in y.re around 0 15.1

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

   7. Simplified 13.6

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

   8. Applied egg-rr 3.3

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.579958549150986 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -3.9696921097903 \cdot 10^{-311}:\\ \;\;\;\;\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot {\left(\frac{\sqrt{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{2} - \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))