_divideComplex, imaginary part

Average Error: 25.8 → 14.3

Time: 52.0s

Precision: binary64

Cost: 2064

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := -\frac{x.re}{y.im}\\ t_2 := \frac{y.re \cdot x.im}{t_0} + x.re \cdot \frac{-y.im}{t_0}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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.re y.im)))
        (t_2 (+ (/ (* y.re x.im) t_0) (* x.re (/ (- y.im) t_0)))))
   (if (<= y.im -1.25e+118)
     t_1
     (if (<= y.im -2.5e-179)
       t_2
       (if (<= y.im 3.9e-168)
         (/ x.im y.re)
         (if (<= y.im 8.2e+145) t_2 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_re / y_46_im);
	double t_2 = ((y_46_re * x_46_im) / t_0) + (x_46_re * (-y_46_im / t_0));
	double tmp;
	if (y_46_im <= -1.25e+118) {
		tmp = t_1;
	} else if (y_46_im <= -2.5e-179) {
		tmp = t_2;
	} else if (y_46_im <= 3.9e-168) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 8.2e+145) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = -(x_46re / y_46im)
    t_2 = ((y_46re * x_46im) / t_0) + (x_46re * (-y_46im / t_0))
    if (y_46im <= (-1.25d+118)) then
        tmp = t_1
    else if (y_46im <= (-2.5d-179)) then
        tmp = t_2
    else if (y_46im <= 3.9d-168) then
        tmp = x_46im / y_46re
    else if (y_46im <= 8.2d+145) then
        tmp = t_2
    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_re / y_46_im);
	double t_2 = ((y_46_re * x_46_im) / t_0) + (x_46_re * (-y_46_im / t_0));
	double tmp;
	if (y_46_im <= -1.25e+118) {
		tmp = t_1;
	} else if (y_46_im <= -2.5e-179) {
		tmp = t_2;
	} else if (y_46_im <= 3.9e-168) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 8.2e+145) {
		tmp = t_2;
	} 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_re / y_46_im)
	t_2 = ((y_46_re * x_46_im) / t_0) + (x_46_re * (-y_46_im / t_0))
	tmp = 0
	if y_46_im <= -1.25e+118:
		tmp = t_1
	elif y_46_im <= -2.5e-179:
		tmp = t_2
	elif y_46_im <= 3.9e-168:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 8.2e+145:
		tmp = t_2
	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_re / y_46_im))
	t_2 = Float64(Float64(Float64(y_46_re * x_46_im) / t_0) + Float64(x_46_re * Float64(Float64(-y_46_im) / t_0)))
	tmp = 0.0
	if (y_46_im <= -1.25e+118)
		tmp = t_1;
	elseif (y_46_im <= -2.5e-179)
		tmp = t_2;
	elseif (y_46_im <= 3.9e-168)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 8.2e+145)
		tmp = t_2;
	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_re / y_46_im);
	t_2 = ((y_46_re * x_46_im) / t_0) + (x_46_re * (-y_46_im / t_0));
	tmp = 0.0;
	if (y_46_im <= -1.25e+118)
		tmp = t_1;
	elseif (y_46_im <= -2.5e-179)
		tmp = t_2;
	elseif (y_46_im <= 3.9e-168)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 8.2e+145)
		tmp = t_2;
	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[(x$46$re / y$46$im), $MachinePrecision])}, Block[{t$95$2 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(x$46$re * N[((-y$46$im) / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.25e+118], t$95$1, If[LessEqual[y$46$im, -2.5e-179], t$95$2, If[LessEqual[y$46$im, 3.9e-168], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+145], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.24999999999999993e118 or 8.2000000000000003e145 < y.im

   1. Initial program 42.9

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

   2. Taylor expanded in y.re around 0 14.7

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

   3. Simplified 14.7

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

      [Start] 14.7	\[ -1 \cdot \frac{x.re}{y.im} \]
      rational_best-simplify-1 [=>] 14.7	\[ \color{blue}{\frac{x.re}{y.im} \cdot -1} \]
      rational_best-simplify-10 [=>] 14.7	\[ \color{blue}{-\frac{x.re}{y.im}} \]

   if -1.24999999999999993e118 < y.im < -2.4999999999999999e-179 or 3.90000000000000012e-168 < y.im < 8.2000000000000003e145

   1. Initial program 16.8

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

   2. Applied egg-rr 33.2

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

   3. Applied egg-rr 17.3

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

   4. Simplified 14.6

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

      [Start] 17.3	\[ \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} + \left(-y.im\right) \cdot \frac{x.re}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 17.3	\[ \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} + \left(-y.im\right) \cdot \frac{x.re}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 14.6	\[ \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} + \color{blue}{x.re \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

   if -2.4999999999999999e-179 < y.im < 3.90000000000000012e-168

   1. Initial program 23.5

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

   2. Taylor expanded in y.re around inf 12.7

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

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+118}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-179}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} + x.re \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} + x.re \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.5
Cost	1488

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ t_1 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-167}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	22.1
Cost	1164

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+129}:\\ \;\;\;\;\frac{y.im}{y.re \cdot \left(-y.re\right) - y.im \cdot y.im} \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	22.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -400000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2023099 
(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))))