math.square on complex, real part

Percentage Accurate: 93.8% → 97.1%

Time: 1.9s

Precision: binary64

Cost: 6784

• 43.6% 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 96.5%

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

2. Simplified 99.2%

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

   [Start] 96.5	\[ re \cdot re - im \cdot im \]
   fma-neg [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(re, re, -im \cdot im\right)} \]
   distribute-rgt-neg-in [=>] 99.2	\[ \mathsf{fma}\left(re, re, \color{blue}{im \cdot \left(-im\right)}\right) \]

3. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	75.4%
Cost	785

\[\begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-145} \lor \neg \left(re \leq 2.4 \cdot 10^{-77}\right) \land re \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Accuracy	97.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 2 \cdot 10^{+303}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 3
Accuracy	54.4%
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2023160 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))