\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]

↓

\[\mathsf{fma}\left(x \cdot 3, x, -2 \cdot {x}^{3}\right) \]

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

↓

(FPCore (x) :precision binary64 (fma (* x 3.0) x (* -2.0 (pow x 3.0))))

double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}

↓

double code(double x) {
	return fma((x * 3.0), x, (-2.0 * pow(x, 3.0)));
}

function code(x)
	return Float64(Float64(x * x) * Float64(3.0 - Float64(x * 2.0)))
end

↓

function code(x)
	return fma(Float64(x * 3.0), x, Float64(-2.0 * (x ^ 3.0)))
end

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(3.0 - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(x * 3.0), $MachinePrecision] * x + N[(-2.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)

↓

\mathsf{fma}\left(x \cdot 3, x, -2 \cdot {x}^{3}\right)

Original	0.2
Target	0.2
Herbie	0.1

Derivation

Initial program 0.2
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
Simplified0.2
\[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \]
Proof
(*.f64 x (*.f64 x (fma.f64 x -2 3))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (fma.f64 x (Rewrite<= metadata-eval (neg.f64 2)) 3))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (neg.f64 2)) 3)))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 2))) 3))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x) 2)) 3))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 3 (*.f64 (neg.f64 x) 2))))): 0 points increase in error, 0 points decrease in error
(*.f64 x (*.f64 x (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 3 (*.f64 x 2))))): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) (-.f64 3 (*.f64 x 2)))): 24 points increase in error, 23 points decrease in error
Applied egg-rr0.2
\[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -2\right) + \left(x \cdot x\right) \cdot 3} \]
Applied egg-rr0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 3, x, -2 \cdot {x}^{3}\right)} \]
Final simplification0.1
\[\leadsto \mathsf{fma}\left(x \cdot 3, x, -2 \cdot {x}^{3}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	7040

\[-2 \cdot {x}^{3} + x \cdot \left(x \cdot 3\right) \]

Alternative 2
Error	2.3
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{if}\;x \leq -3.3109288446661744:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.989449587946897 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	704

\[x \cdot \left(x \cdot 3 + x \cdot \left(x \cdot -2\right)\right) \]

Alternative 4
Error	0.2
Cost	576

\[x \cdot \left(x \cdot \left(3 + x \cdot -2\right)\right) \]

Alternative 5
Error	16.4
Cost	320

\[x \cdot \left(x \cdot 3\right) \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

  (* (* x x) (- 3.0 (* x 2.0))))

Error

Target

Derivation

Alternatives

Error

Reproduce