math.square on complex, real part

Percentage Accurate: 93.8% → 96.9%

Time: 2.0s

Precision: binary64

Cost: 6784

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right) \]

FPCore

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

↓

(FPCore re_sqr (re im) :precision binary64 (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) {
	return fma(re, re, (im * -im));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 92.6%

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

2. Simplified 96.9%

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

   [Start] 92.6	\[ re \cdot re - im \cdot im \]
   fma-neg [=>] 96.9	\[ \color{blue}{\mathsf{fma}\left(re, re, -im \cdot im\right)} \]
   distribute-rgt-neg-in [=>] 96.9	\[ \mathsf{fma}\left(re, re, \color{blue}{im \cdot \left(-im\right)}\right) \]

3. Final simplification 96.9%

   \[\leadsto \mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right) \]

Alternatives

Alternative 1
Accuracy	76.7%
Cost	786

\[\begin{array}{l} \mathbf{if}\;im \leq -3.8 \cdot 10^{+81} \lor \neg \left(im \leq -2.6 \cdot 10^{-21}\right) \land \left(im \leq -5.3 \cdot 10^{-89} \lor \neg \left(im \leq 2500000000000\right)\right):\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Accuracy	96.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 4 \cdot 10^{+305}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3
Accuracy	54.1%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023159 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))