math.square on complex, real part

Percentage Accurate: 93.7% → 96.2%

Time: 7.6s

Precision: binary64

Cost: 708

• 43.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \cdot re \leq 4 \cdot 10^{+294}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

↓

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* re re) 4e+294) (- (* re re) (* im im)) (* re re)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

↓

double re_sqr(double re, double im) {
	double tmp;
	if ((re * re) <= 4e+294) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

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 ((re * re) <= 4d+294) then
        tmp = (re * re) - (im * im)
    else
        tmp = re * re
    end if
    re_sqr = tmp
end function

Java

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 ((re * re) <= 4e+294) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

↓

def re_sqr(re, im):
	tmp = 0
	if (re * re) <= 4e+294:
		tmp = (re * re) - (im * im)
	else:
		tmp = re * re
	return tmp

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

↓

function re_sqr(re, im)
	tmp = 0.0
	if (Float64(re * re) <= 4e+294)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

↓

function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((re * re) <= 4e+294)
		tmp = (re * re) - (im * im);
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

↓

re$95$sqr[re_, im_] := If[LessEqual[N[(re * re), $MachinePrecision], 4e+294], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 4.00000000000000027e294

   1. Initial program 100.0%

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

   if 4.00000000000000027e294 < (*.f64 re re)

   1. Initial program 73.9%

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

   2. Taylor expanded in re around inf 92.8%

      \[\leadsto \color{blue}{{re}^{2}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{re \cdot re} \]
      Step-by-step derivation

      [Start] 92.8%	\[ {re}^{2} \]
      unpow2 [=>] 92.8%	\[ \color{blue}{re \cdot re} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 4 \cdot 10^{+294}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \cdot re \leq 4 \cdot 10^{+294}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Accuracy	76.4%
Cost	520

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+63}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-72}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3
Accuracy	54.0%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023277 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))