math.square on complex, real part

Percentage Accurate: 93.6% → 97.2%

Time: 3.3s

Precision: binary64

Cost: 6916

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

FPCore

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

↓

(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 3.2e+170) (fma 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 <= 3.2e+170) {
		tmp = fma(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 (re <= 3.2e+170)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(re * re);
	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, 3.2e+170], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.19999999999999979e170

   1. Initial program 97.0%

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

   2. Simplified 99.6%

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

      [Start] 97.0%	\[ re \cdot re - im \cdot im \]
      fma-neg [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(re, re, -im \cdot im\right)} \]
      distribute-rgt-neg-in [=>] 99.6%	\[ \mathsf{fma}\left(re, re, \color{blue}{im \cdot \left(-im\right)}\right) \]

   if 3.19999999999999979e170 < re

   1. Initial program 70.8%

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

   2. Taylor expanded in re around inf 95.8%

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

   3. Simplified 95.8%

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

      [Start] 95.8%	\[ {re}^{2} \]
      unpow2 [=>] 95.8%	\[ \color{blue}{re \cdot re} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Accuracy	93.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+204}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3
Accuracy	76.6%
Cost	521

\[\begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{-64} \lor \neg \left(im \leq 1.8 \cdot 10^{+65}\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 4
Accuracy	54.0%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023263 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))