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

Average Error: 0.2 → 0.2

Time: 10.7s

Precision: binary64

Cost: 13120

Math

\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

↓

\[0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

↓

(FPCore (x y z) :precision binary64 (* 0.5 (fma y (sqrt z) x)))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

↓

double code(double x, double y, double z) {
	return 0.5 * fma(y, sqrt(z), x);
}

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

↓

function code(x, y, z)
	return Float64(0.5 * fma(y, sqrt(z), x))
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

2. Simplified 0.2

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   Proof

   [Start] 0.2	\[ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   metadata-eval [=>] 0.2	\[ \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   +-commutative [=>] 0.2	\[ 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z} + x\right)} \]
   fma-def [=>] 0.2	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

3. Final simplification 0.2

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

Alternatives

Alternative 1
Error	17.1
Cost	33362

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-23} \lor \neg \left(t_0 \leq 5 \cdot 10^{-117}\right) \land \left(t_0 \leq 5 \cdot 10^{-96} \lor \neg \left(t_0 \leq 2 \cdot 10^{+45}\right)\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 2
Error	17.0
Cost	33361

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \frac{0.5}{{z}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-117} \lor \neg \left(t_0 \leq 5 \cdot 10^{-96}\right) \land t_0 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \]

Alternative 3
Error	17.0
Cost	33361

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{\frac{{z}^{-0.5}}{y}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-117} \lor \neg \left(t_0 \leq 5 \cdot 10^{-96}\right) \land t_0 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \]

Alternative 4
Error	17.0
Cost	33361

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-117} \lor \neg \left(t_0 \leq 5 \cdot 10^{-96}\right) \land t_0 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \]

Alternative 5
Error	0.2
Cost	6848

\[0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]

Alternative 6
Error	29.3
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))