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

Average Error: 0.1 → 0.1

Time: 6.7s

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 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]
   Proof
   (*.f64 1/2 (fma.f64 y (sqrt.f64 z) x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (fma.f64 y (sqrt.f64 z) x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 1 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (sqrt.f64 z)) x))): 3 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 1 2) (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 y (sqrt.f64 z))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	15.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-48}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	6912

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

Alternative 3
Error	0.1
Cost	6848

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

Alternative 4
Error	29.2
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2022329 
(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)))))