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

Average Error: 0.2 → 0.1

Time: 9.4s

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 0.1

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   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 [=>] 0.1	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	16.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-81}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - \frac{y}{\frac{\frac{x}{z}}{y}}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	6848

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

Alternative 3
Error	29.9
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023023 
(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)))))