math.square on complex, real part

?

Percentage Accurate: 93.4% → 96.6%
Time: 1.9s
Precision: binary64
Cost: 708

?

\[re \cdot re - im \cdot im \]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+304}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))
↓
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 1e+304) (- (* re re) (* im im)) (* im (- im))))
double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}
↓
double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+304) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function
↓
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 1d+304) then
        tmp = (re * re) - (im * im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}
↓
public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 1e+304) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

def re_sqr(re, im):
	return (re * re) - (im * im)
↓
def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 1e+304:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end
↓
function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e+304)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end
↓
function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e+304)
		tmp = (re * re) - (im * im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]
↓
re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e+304], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

re \cdot re - im \cdot im
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \leq 10^{+304}:\\
\;\;\;\;re \cdot re - im \cdot im\\

\mathbf{else}:\\
\;\;\;\;im \cdot \left(-im\right)\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 4 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (*.f64 im im) < 9.9999999999999994e303

    1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]

    if 9.9999999999999994e303 < (*.f64 im im)

    1. Initial program 79.0%

      \[re \cdot re - im \cdot im \]

    2. Taylor expanded in re around 0 90.3%

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]

    3. Simplified90.3%

      \[\leadsto \color{blue}{im \cdot \left(-im\right)} \]
      Step-by-step derivation

      [Start]90.3%

      \[ -1 \cdot {im}^{2} \]

      unpow2 [=>]90.3%

      \[ -1 \cdot \color{blue}{\left(im \cdot im\right)} \]

      mul-1-neg [=>]90.3%

      \[ \color{blue}{-im \cdot im} \]

      distribute-rgt-neg-in [=>]90.3%

      \[ \color{blue}{im \cdot \left(-im\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification97.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+304}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy96.6%
Cost708
\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+304}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]
Alternative 2
Accuracy96.7%
Cost6784
\[\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right) \]
Alternative 3
Accuracy75.4%
Cost521
\[\begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+109} \lor \neg \left(im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]
Alternative 4
Accuracy53.8%
Cost192
\[re \cdot re \]

Reproduce?

herbie shell --seed 2023178 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))