math.square on complex, real part

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Percentage Accurate: 93.3% → 96.6%

Time: 2.0s

Precision: binary64

Cost: 6784

43.6% of points produce a very large (infinite) output. You may want to add a precondition. (more)

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

↓

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

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

↓

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

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));
}

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

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]

re \cdot re - im \cdot im

↓

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

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 4 alternatives:

Alternative	Accuracy	Speedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

Initial program 96.1%
\[re \cdot re - im \cdot im \]

Simplified98.0%

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

Step-by-step derivation

[Start]96.1%	\[ re \cdot re - im \cdot im \]
fma-neg [=>]98.0%	\[ \color{blue}{\mathsf{fma}\left(re, re, -im \cdot im\right)} \]
distribute-rgt-neg-in [=>]98.0%	\[ \mathsf{fma}\left(re, re, \color{blue}{im \cdot \left(-im\right)}\right) \]

Final simplification98.0%
\[\leadsto \mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right) \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	6784

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

Alternative 2
Accuracy	74.8%
Cost	785

\[\begin{array}{l} \mathbf{if}\;re \leq -5.6 \cdot 10^{+167}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq -2.1 \cdot 10^{+140} \lor \neg \left(re \leq -1.1 \cdot 10^{+93}\right) \land re \leq 4.9 \cdot 10^{-83}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3
Accuracy	96.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{+293}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 4
Accuracy	53.4%
Cost	192

\[re \cdot re \]

Reproduce?

herbie shell --seed 2023165 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))

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