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

Average Error: 0.2 → 0.1

Time: 3.0s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right) + -1 \cdot z} \]

3. Simplified 0.1

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

   [Start] 0.1	\[ 3 \cdot \left(y \cdot x\right) + -1 \cdot z \]
   mul-1-neg [=>] 0.1	\[ 3 \cdot \left(y \cdot x\right) + \color{blue}{\left(-z\right)} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(3, y \cdot x, -z\right)} \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	16.7
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-87} \lor \neg \left(z \leq -5.6 \cdot 10^{-112}\right) \land z \leq 5000:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	16.7
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-86}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-112}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 36:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	0.2
Cost	448

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

Alternative 5
Error	27.8
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023054 
(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))