Examples.Basics.BasicTests:f2 from sbv-4.4

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Percentage Accurate: 93.8% → 96.8%

Time: 3.7s

Precision: binary64

Cost: 6784

44.2% of points produce a very large (infinite) output. You may want to add a precondition. (more)

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\[x \cdot x - y \cdot y \]

↓

\[\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right) \]

(FPCore (x y) :precision binary64 (- (* x x) (* y y)))

↓

(FPCore (x y) :precision binary64 (fma x x (* y (- y))))

double code(double x, double y) {
	return (x * x) - (y * y);
}

↓

double code(double x, double y) {
	return fma(x, x, (y * -y));
}

function code(x, y)
	return Float64(Float64(x * x) - Float64(y * y))
end

↓

function code(x, y)
	return fma(x, x, Float64(y * Float64(-y)))
end

code[x_, y_] := N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision]

x \cdot x - y \cdot y

↓

\mathsf{fma}\left(x, x, y \cdot \left(-y\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%
\[x \cdot x - y \cdot y \]

Simplified98.0%

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

Step-by-step derivation

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

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

Alternatives

Alternative 1
Accuracy	96.8%
Cost	6784

\[\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right) \]

Alternative 2
Accuracy	77.1%
Cost	786

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+104} \lor \neg \left(y \leq -1.25 \cdot 10^{+52}\right) \land \left(y \leq -0.08 \lor \neg \left(y \leq 1.65 \cdot 10^{+47}\right)\right):\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Accuracy	96.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+296}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]

Alternative 4
Accuracy	54.3%
Cost	192

\[x \cdot x \]

Reproduce?

herbie shell --seed 2023258 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))

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