_divideComplex, real part

Average Error: 25.7 → 15.2

Time: 18.1s

Precision: binary64

Cost: 7436

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -4.7e+123)
     (/ x.im y.im)
     (if (<= y.im -4e-28)
       t_0
       (if (<= y.im 3.4e-101)
         (+ (/ x.re y.re) (/ (* y.im x.im) (pow y.re 2.0)))
         (if (<= y.im 1.7e+109) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.7e+123) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4e-28) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e-101) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 1.7e+109) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_im;
	}
	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_46re * y_46re) + (x_46im * 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) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-4.7d+123)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-4d-28)) then
        tmp = t_0
    else if (y_46im <= 3.4d-101) then
        tmp = (x_46re / y_46re) + ((y_46im * x_46im) / (y_46re ** 2.0d0))
    else if (y_46im <= 1.7d+109) then
        tmp = t_0
    else
        tmp = x_46im / y_46im
    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_re * y_46_re) + (x_46_im * 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.7e+123) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4e-28) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e-101) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / Math.pow(y_46_re, 2.0));
	} else if (y_46_im <= 1.7e+109) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -4.7e+123:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -4e-28:
		tmp = t_0
	elif y_46_im <= 3.4e-101:
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / math.pow(y_46_re, 2.0))
	elif y_46_im <= 1.7e+109:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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 <= -4.7e+123)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -4e-28)
		tmp = t_0;
	elseif (y_46_im <= 3.4e-101)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * x_46_im) / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 1.7e+109)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -4.7e+123)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -4e-28)
		tmp = t_0;
	elseif (y_46_im <= 3.4e-101)
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / (y_46_re ^ 2.0));
	elseif (y_46_im <= 1.7e+109)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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, -4.7e+123], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4e-28], t$95$0, If[LessEqual[y$46$im, 3.4e-101], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * x$46$im), $MachinePrecision] / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+109], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.69999999999999979e123 or 1.70000000000000003e109 < y.im

   1. Initial program 40.5

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

   2. Taylor expanded in y.re around 0 15.5

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

   if -4.69999999999999979e123 < y.im < -3.99999999999999988e-28 or 3.39999999999999989e-101 < y.im < 1.70000000000000003e109

   1. Initial program 17.4

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

   if -3.99999999999999988e-28 < y.im < 3.39999999999999989e-101

   1. Initial program 18.5

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

   2. Taylor expanded in y.re around inf 13.0

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

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.7
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 2
Error	22.9
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 3
Error	23.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 4
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2023077 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))