Average Error: 0.2 → 0.2
Time: 6.9s
Precision: binary64
Cost: 6976
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
↓
\[x \cdot \mathsf{fma}\left(x, 3, x \cdot \left(x \cdot -2\right)\right) \]
(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))
↓
(FPCore (x) :precision binary64 (* x (fma x 3.0 (* x (* x -2.0)))))
double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}
↓
double code(double x) {
	return x * fma(x, 3.0, (x * (x * -2.0)));
}

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

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

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

Error

Target

Original0.2
Target0.2
Herbie0.2
\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Derivation

  1. Initial program 0.2

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

  2. 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)))): 26 points increase in error, 15 points decrease in error

  3. Applied egg-rr0.2

    \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot -2\right) \cdot x + 3 \cdot x\right)} \]

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost6848
\[x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right) \]
Alternative 2
Error2.1
Cost712
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot -2\right)\right)\\ \mathbf{if}\;x \leq -792.8649358483495:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2942125103405895:\\ \;\;\;\;x \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error0.2
Cost576
\[x \cdot \left(x \cdot \left(3 + x \cdot -2\right)\right) \]
Alternative 4
Error61.1
Cost192
\[x \cdot 4.5 \]
Alternative 5
Error61.4
Cost64
\[6.75 \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))