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

Average Error: 0.1 → 0.2

Time: 2.1s

Precision: binary64

Cost: 6784

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

Math

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

↓

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

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(x, Float64(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[(x * N[(3.0 * y), $MachinePrecision] + (-z)), $MachinePrecision]

Target

Original	0.1
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.2

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

   [Start] 0.1	\[ \left(x \cdot 3\right) \cdot y - z \]
   associate-*l* [=>] 0.2	\[ \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]
   fma-neg [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	15.2
Cost	1096

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Error	17.7
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0008:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-24} \lor \neg \left(z \leq -6 \cdot 10^{-87}\right) \land z \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	0.2
Cost	448

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

Alternative 5
Error	27.0
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2022364 
(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))