Difference of squares

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Percentage Accurate: 93.8% → 97.1%

Time: 2.5s

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

↓

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

(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.

Herbie found 4 alternatives:

Alternative	Accuracy	Speedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Target

Original	93.8%
Target	100.0%
Herbie	97.1%

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

Derivation?

Initial program 96.5%
\[a \cdot a - b \cdot b \]

Simplified99.2%

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

Step-by-step derivation

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

Final simplification99.2%
\[\leadsto \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right) \]

Alternatives

Alternative 1
Accuracy	75.4%
Cost	785

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

Alternative 2
Accuracy	97.1%
Cost	708

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

Alternative 3
Accuracy	54.4%
Cost	192

\[a \cdot a \]

Reproduce?

herbie shell --seed 2023160 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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

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