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

Average Error: 0.2% → 0.2%

Time: 5.0s

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	0.2%
Target	0.22%
Herbie	0.2%

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	36.55%
Cost	1380

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+117} \lor \neg \left(x \leq -6.6 \cdot 10^{+100} \lor \neg \left(x \leq -1.36 \cdot 10^{+75}\right) \land \left(x \leq -1.45 \cdot 10^{+35} \lor \neg \left(x \leq -1.4 \cdot 10^{-60}\right) \land \left(x \leq -4.8 \cdot 10^{-199} \lor \neg \left(x \leq -2.05 \cdot 10^{-225}\right) \land x \leq 4.5 \cdot 10^{-46}\right)\right)\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	36.43%
Cost	1376

\[\begin{array}{l} t_0 := 3 \cdot \left(x \cdot y\right)\\ t_1 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+101}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+35}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-199}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-42}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.22%
Cost	448

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

Alternative 4
Error	42.87%
Cost	128

\[-z \]

Alternative 5
Error	97.76%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023121 
(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))