_divideComplex, imaginary part

Average Error: 25.8 → 16.3

Time: 2.4s

Precision: binary64

Cost: 1488

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.9 \cdot 10^{-159}:\\ \;\;\;\;-1 \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

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)))))
   (if (<= y.re -3.3e+101)
     (/ x.im y.re)
     (if (<= y.re -9.2e-137)
       t_0
       (if (<= y.re 7.9e-159)
         (* -1.0 (/ x.re y.im))
         (if (<= y.re 1.65e+59) t_0 (/ x.im y.re)))))))

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 tmp;
	if (y_46_re <= -3.3e+101) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.2e-137) {
		tmp = t_0;
	} else if (y_46_re <= 7.9e-159) {
		tmp = -1.0 * (x_46_re / y_46_im);
	} else if (y_46_re <= 1.65e+59) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	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) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-3.3d+101)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-9.2d-137)) then
        tmp = t_0
    else if (y_46re <= 7.9d-159) then
        tmp = (-1.0d0) * (x_46re / y_46im)
    else if (y_46re <= 1.65d+59) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    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 tmp;
	if (y_46_re <= -3.3e+101) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.2e-137) {
		tmp = t_0;
	} else if (y_46_re <= 7.9e-159) {
		tmp = -1.0 * (x_46_re / y_46_im);
	} else if (y_46_re <= 1.65e+59) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	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))
	tmp = 0
	if y_46_re <= -3.3e+101:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -9.2e-137:
		tmp = t_0
	elif y_46_re <= 7.9e-159:
		tmp = -1.0 * (x_46_re / y_46_im)
	elif y_46_re <= 1.65e+59:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	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)))
	tmp = 0.0
	if (y_46_re <= -3.3e+101)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -9.2e-137)
		tmp = t_0;
	elseif (y_46_re <= 7.9e-159)
		tmp = Float64(-1.0 * Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.65e+59)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	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));
	tmp = 0.0;
	if (y_46_re <= -3.3e+101)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -9.2e-137)
		tmp = t_0;
	elseif (y_46_re <= 7.9e-159)
		tmp = -1.0 * (x_46_re / y_46_im);
	elseif (y_46_re <= 1.65e+59)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	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]}, If[LessEqual[y$46$re, -3.3e+101], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -9.2e-137], t$95$0, If[LessEqual[y$46$re, 7.9e-159], N[(-1.0 * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+59], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.30000000000000011e101 or 1.65e59 < y.re

   1. Initial program 37.0

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

   2. Taylor expanded in y.re around inf 17.9

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

   if -3.30000000000000011e101 < y.re < -9.20000000000000032e-137 or 7.89999999999999972e-159 < y.re < 1.65e59

   1. Initial program 15.8

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

   if -9.20000000000000032e-137 < y.re < 7.89999999999999972e-159

   1. Initial program 22.9

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

   2. Taylor expanded in y.re around 0 14.5

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.9 \cdot 10^{-159}:\\ \;\;\;\;-1 \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.8
Cost	1100

\[\begin{array}{l} t_0 := -1 \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 2
Error	24.2
Cost	1100

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.14 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-x.re \cdot y.im}{t_0}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{y.re \cdot x.im}{t_0}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+20}:\\ \;\;\;\;-1 \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 3
Error	23.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;-1 \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2023053 
(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))))