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

Average Error: 0.21% → 0.21%

Time: 7.0s

Precision: binary64

Cost: 6784

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

Math

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

↓

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

FPCore

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

↓

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

C

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);
}

Julia

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

Wolfram

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]

Target

Original	0.21%
Target	0.25%
Herbie	0.21%

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

Derivation

1. Initial program 0.21

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

2. Simplified 0.21

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

   [Start] 0.21	\[ \left(x \cdot 3\right) \cdot y - z \]
   fma-neg [=>] 0.21	\[ \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -z\right)} \]

3. Final simplification 0.21

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

Alternatives

Alternative 1
Error	25.21%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-54}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	25.24%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-42}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	0.25%
Cost	448

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

Alternative 4
Error	0.21%
Cost	448

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

Alternative 5
Error	42.51%
Cost	128

\[-z \]

Alternative 6
Error	97.76%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023090 
(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))