\[x \cdot \left(1 - x \cdot 0.5\right) \]

↓

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

(FPCore (x) :precision binary64 (* x (- 1.0 (* x 0.5))))

↓

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

double code(double x) {
	return x * (1.0 - (x * 0.5));
}

↓

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

function code(x)
	return Float64(x * Float64(1.0 - Float64(x * 0.5)))
end

↓

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

code[x_] := N[(x * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

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

x \cdot \left(1 - x \cdot 0.5\right)

↓

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

Derivation

Initial program 0.0
\[x \cdot \left(1 - x \cdot 0.5\right) \]
Simplified0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
Proof
(fma.f64 x (*.f64 x -1/2) x): 0 points increase in error, 0 points decrease in error
(fma.f64 x (*.f64 x (Rewrite<= metadata-eval (neg.f64 1/2))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 x (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 1/2))) x): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (neg.f64 (*.f64 x 1/2))) x)): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 x (neg.f64 (*.f64 x 1/2))) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))): 2 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x 1) (*.f64 x (neg.f64 (*.f64 x 1/2))))): 0 points increase in error, 2 points decrease in error
(Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 1 (neg.f64 (*.f64 x 1/2))))): 0 points increase in error, 0 points decrease in error
(*.f64 x (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 x 1/2)))): 0 points increase in error, 0 points decrease in error
Final simplification0
\[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, x\right) \]

Alternatives

Alternative 1
Error	1.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

\[x \cdot \left(1 + x \cdot -0.5\right) \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	20.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (- 1.0 (* x 0.5))))

Error

Derivation

Alternatives

Error

Reproduce