Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A

Average Error: 0.05% → 0.04%

Time: 2.1s

Precision: binary64

Cost: 6720

Math

\[x - \left(y \cdot 4\right) \cdot z \]

↓

\[\mathsf{fma}\left(z, y \cdot -4, x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma z (* y -4.0) x))

C

double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}

↓

double code(double x, double y, double z) {
	return fma(z, (y * -4.0), x);
}

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(y * 4.0) * z))
end

↓

function code(x, y, z)
	return fma(z, Float64(y * -4.0), x)
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(z * N[(y * -4.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 0.05

   \[x - \left(y \cdot 4\right) \cdot z \]

2. Simplified 0.04

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot -4, x\right)} \]
   Proof

   [Start] 0.05	\[ x - \left(y \cdot 4\right) \cdot z \]
   cancel-sign-sub-inv [=>] 0.05	\[ \color{blue}{x + \left(-y \cdot 4\right) \cdot z} \]
   +-commutative [=>] 0.05	\[ \color{blue}{\left(-y \cdot 4\right) \cdot z + x} \]
   *-commutative [=>] 0.05	\[ \color{blue}{z \cdot \left(-y \cdot 4\right)} + x \]
   fma-def [=>] 0.04	\[ \color{blue}{\mathsf{fma}\left(z, -y \cdot 4, x\right)} \]
   distribute-rgt-neg-in [=>] 0.04	\[ \mathsf{fma}\left(z, \color{blue}{y \cdot \left(-4\right)}, x\right) \]
   metadata-eval [=>] 0.04	\[ \mathsf{fma}\left(z, y \cdot \color{blue}{-4}, x\right) \]

3. Final simplification 0.04

   \[\leadsto \mathsf{fma}\left(z, y \cdot -4, x\right) \]

Alternatives

Alternative 1
Error	27.14%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-32} \lor \neg \left(x \leq 10^{+22}\right) \land x \leq 6.8 \cdot 10^{+45}:\\ \;\;\;\;-4 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.05%
Cost	448

\[x - z \cdot \left(y \cdot 4\right) \]

Alternative 3
Error	42.99%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))