Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A

?

Percentage Accurate: 99.9% → 100.0%

Time: 1.6s

Precision: binary64

Cost: 6720

Unsound rule application detected in e-graph. Results may not be sound. (more)

?

\[x - \left(y \cdot 4\right) \cdot z \]

↓

\[\begin{array}{l} \\ \mathsf{fma}\left(z, y \cdot -4, x\right) \end{array} \]

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma z (* y -4.0) x))

double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}

↓

double code(double x, double y, double z) {
	return fma(z, (y * -4.0), x);
}

function code(x, y, z)
	return Float64(x - Float64(Float64(y * 4.0) * z))
end

↓

function code(x, y, z)
	return fma(z, Float64(y * -4.0), x)
end

code[x_, y_, z_] := N[(x - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(z * N[(y * -4.0), $MachinePrecision] + x), $MachinePrecision]

x - \left(y \cdot 4\right) \cdot z

↓

\begin{array}{l}

\\
\mathsf{fma}\left(z, y \cdot -4, x\right)
\end{array}

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 100.0%
\[x - \left(y \cdot 4\right) \cdot z \]

Simplified100.0%

\[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot -4, x\right)} \]

Step-by-step derivation

[Start]100.0%	\[ x - \left(y \cdot 4\right) \cdot z \]
sub-neg [=>]100.0%	\[ \color{blue}{x + \left(-\left(y \cdot 4\right) \cdot z\right)} \]
+-commutative [=>]100.0%	\[ \color{blue}{\left(-\left(y \cdot 4\right) \cdot z\right) + x} \]
*-commutative [=>]100.0%	\[ \left(-\color{blue}{z \cdot \left(y \cdot 4\right)}\right) + x \]
distribute-rgt-neg-in [=>]100.0%	\[ \color{blue}{z \cdot \left(-y \cdot 4\right)} + x \]
fma-def [=>]100.0%	\[ \color{blue}{\mathsf{fma}\left(z, -y \cdot 4, x\right)} \]
distribute-rgt-neg-in [=>]100.0%	\[ \mathsf{fma}\left(z, \color{blue}{y \cdot \left(-4\right)}, x\right) \]
metadata-eval [=>]100.0%	\[ \mathsf{fma}\left(z, y \cdot \color{blue}{-4}, x\right) \]

Final simplification100.0%
\[\leadsto \mathsf{fma}\left(z, y \cdot -4, x\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6720

\[\mathsf{fma}\left(z, y \cdot -4, x\right) \]

Alternative 2
Accuracy	67.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-159} \lor \neg \left(z \leq 7 \cdot 10^{+45}\right):\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	448

\[x - z \cdot \left(y \cdot 4\right) \]

Alternative 4
Accuracy	50.6%
Cost	64

\[x \]

Reproduce?

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))

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