Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

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Percentage Accurate: 99.8% → 99.8%

Time: 4.1s

Precision: binary64

Cost: 6784

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

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\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

\[\left(x \cdot 3\right) \cdot y - z \]

↓

\[\mathsf{fma}\left(x \cdot 3, y, -z\right) \]

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

↓

(FPCore (x y z) :precision binary64 (fma (* x 3.0) y (- z)))

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

↓

double code(double x, double y, double z) {
	return fma((x * 3.0), y, -z);
}

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

↓

function code(x, y, z)
	return fma(Float64(x * 3.0), y, Float64(-z))
end

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x * 3.0), $MachinePrecision] * y + (-z)), $MachinePrecision]

\left(x \cdot 3\right) \cdot y - z

↓

\mathsf{fma}\left(x \cdot 3, y, -z\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	99.8%
Target	99.8%
Herbie	99.8%

\[x \cdot \left(3 \cdot y\right) - z \]

Derivation?

Initial program 99.8%
\[\left(x \cdot 3\right) \cdot y - z \]
Simplified99.8%
\[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -z\right)} \]
Step-by-step derivation
[Start]99.8
\[ \left(x \cdot 3\right) \cdot y - z \]
fma-neg [=>]99.8
\[ \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -z\right)} \]
Final simplification99.8%
\[\leadsto \mathsf{fma}\left(x \cdot 3, y, -z\right) \]

[Start]99.8	\[ \left(x \cdot 3\right) \cdot y - z \]
fma-neg [=>]99.8	\[ \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -z\right)} \]

Alternatives

Alternative 1
Accuracy	70.9%
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+16} \lor \neg \left(x \leq -2.4 \cdot 10^{-32} \lor \neg \left(x \leq -6.2 \cdot 10^{-74}\right) \land x \leq 1.55 \cdot 10^{-75}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Accuracy	70.7%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+16}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-26} \lor \neg \left(x \leq -3.1 \cdot 10^{-75}\right) \land x \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	448

\[x \cdot \left(3 \cdot y\right) - z \]

Alternative 4
Accuracy	99.8%
Cost	448

\[\left(x \cdot 3\right) \cdot y - z \]

Alternative 5
Accuracy	51.2%
Cost	128

\[-z \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))

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