math.square on complex, real part

Percentage Accurate: 93.0% → 97.6%

Time: 1.9s

Precision: binary64

Cost: 6916

• 43.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+239}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re -5e+239) (* re re) (fma re re (* im (- im)))))

C

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

↓

double re_sqr(double re, double im) {
	double tmp;
	if (re <= -5e+239) {
		tmp = re * re;
	} else {
		tmp = fma(re, re, (im * -im));
	}
	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 (re <= -5e+239)
		tmp = Float64(re * re);
	else
		tmp = fma(re, re, Float64(im * Float64(-im)));
	end
	return 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[re, -5e+239], N[(re * re), $MachinePrecision], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.00000000000000007e239

   1. Initial program 52.9%

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

   2. Taylor expanded in re around inf 100.0%

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

   3. Simplified 100.0%

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

      [Start] 100.0%	\[ {re}^{2} \]
      unpow2 [=>] 100.0%	\[ \color{blue}{re \cdot re} \]

   if -5.00000000000000007e239 < re

   1. Initial program 94.6%

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

   2. Simplified 97.9%

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

      [Start] 94.6%	\[ re \cdot re - im \cdot im \]
      fma-neg [=>] 97.9%	\[ \color{blue}{\mathsf{fma}\left(re, re, -im \cdot im\right)} \]
      distribute-rgt-neg-in [=>] 97.9%	\[ \mathsf{fma}\left(re, re, \color{blue}{im \cdot \left(-im\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+239}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+239}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	95.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \cdot re \leq 5 \cdot 10^{+273}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3
Accuracy	76.9%
Cost	516

\[\begin{array}{l} \mathbf{if}\;re \cdot re \leq 2.5 \cdot 10^{+66}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 4
Accuracy	54.2%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023229 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))