Difference of squares

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Percentage Accurate: 93.8% → 96.9%

Time: 2.2s

Precision: binary64

Cost: 6784

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

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\[a \cdot a - b \cdot b \]

↓

\[\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right) \]

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

↓

(FPCore (a b) :precision binary64 (fma a a (* b (- b))))

double code(double a, double b) {
	return (a * a) - (b * b);
}

↓

double code(double a, double b) {
	return fma(a, a, (b * -b));
}

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

↓

function code(a, b)
	return fma(a, a, Float64(b * Float64(-b)))
end

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]

a \cdot a - b \cdot b

↓

\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)

Local Percentage Accuracy?

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.

Target

Original	93.8%
Target	100.0%
Herbie	96.9%

\[\left(a + b\right) \cdot \left(a - b\right) \]

Derivation?

Initial program 92.6%
\[a \cdot a - b \cdot b \]

Simplified96.9%

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

Step-by-step derivation

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

Final simplification96.9%
\[\leadsto \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right) \]

Alternatives

Alternative 1
Accuracy	76.7%
Cost	786

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

Alternative 2
Accuracy	96.8%
Cost	708

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

Alternative 3
Accuracy	54.1%
Cost	192

\[a \cdot a \]

Error

Reproduce?

herbie shell --seed 2023159 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))

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