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

Average Error: 0.22% → 0.21%

Time: 9.8s

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 0.22

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

2. Simplified 0.21

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

   [Start] 0.22	\[ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   metadata-eval [=>] 0.22	\[ \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   +-commutative [=>] 0.22	\[ 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z} + x\right)} \]
   fma-def [=>] 0.21	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

3. Final simplification 0.21

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

Alternatives

Alternative 1
Error	27.47%
Cost	20104

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \end{array} \]

Alternative 2
Error	27.47%
Cost	20041

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-102} \lor \neg \left(t_0 \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3
Error	0.22%
Cost	6848

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

Alternative 4
Error	46.77%
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023090 
(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)))))