System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2

Average Accuracy: 99.9% → 99.9%

Time: 1.6s

Precision: binary64

Cost: 6784

Math

\[0.5 \cdot \left(x \cdot x - y\right) \]

↓

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

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (- (* x x) y)))

↓

(FPCore (x y) :precision binary64 (* 0.5 (fma x x (- y))))

C

double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}

↓

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

Julia

function code(x, y)
	return Float64(0.5 * Float64(Float64(x * x) - y))
end

↓

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

Wolfram

code[x_, y_] := N[(0.5 * N[(N[(x * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

↓

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

Derivation

1. Initial program 99.9%

   \[0.5 \cdot \left(x \cdot x - y\right) \]

2. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ 0.5 \cdot \left(x \cdot x - y\right) \]
   fma-neg [=>] 99.9	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, x, -y\right)} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	82.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-24} \lor \neg \left(x \leq 0.00048\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	448

\[0.5 \cdot \left(x \cdot x - y\right) \]

Alternative 3
Accuracy	66.3%
Cost	192

\[y \cdot -0.5 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))