Average Error: 0.2 → 0.1
Time: 4.4s
Precision: binary64
Cost: 6848
\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
↓
\[\mathsf{fma}\left(x, 6, -9 \cdot \left(x \cdot x\right)\right) \]
(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))
↓
(FPCore (x) :precision binary64 (fma x 6.0 (* -9.0 (* x x))))
double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}
↓
double code(double x) {
	return fma(x, 6.0, (-9.0 * (x * x)));
}

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end
↓
function code(x)
	return fma(x, 6.0, Float64(-9.0 * Float64(x * x)))
end

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]
↓
code[x_] := N[(x * 6.0 + N[(-9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x
↓
\mathsf{fma}\left(x, 6, -9 \cdot \left(x \cdot x\right)\right)

Error

Target

Original0.2
Target0.2
Herbie0.1
\[6 \cdot x - 9 \cdot \left(x \cdot x\right) \]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
    Proof
    (*.f64 x (+.f64 6 (*.f64 x -9))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (+.f64 (Rewrite<= metadata-eval (*.f64 2 3)) (*.f64 x -9))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (+.f64 (*.f64 2 3) (*.f64 x (Rewrite<= metadata-eval (*.f64 -3 3))))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (+.f64 (*.f64 2 3) (*.f64 x (*.f64 (Rewrite<= metadata-eval (neg.f64 3)) 3)))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (+.f64 (*.f64 2 3) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (neg.f64 3)) 3)))): 12 points increase in error, 5 points decrease in error
    (*.f64 x (+.f64 (*.f64 2 3) (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 3))) 3))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (+.f64 (*.f64 2 3) (*.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x) 3)) 3))): 0 points increase in error, 0 points decrease in error
    (*.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 3 (+.f64 2 (*.f64 (neg.f64 x) 3))))): 6 points increase in error, 1 points decrease in error
    (*.f64 x (*.f64 3 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 2 (*.f64 x 3))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 3 (-.f64 2 (*.f64 x 3))) x)): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, -9 \cdot \left(x \cdot x\right)\right)} \]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot \left(x \cdot x\right)\right) \]

Alternatives

Alternative 1
Error0.2
Cost6720
\[x \cdot \mathsf{fma}\left(x, -9, 6\right) \]
Alternative 2
Error2.1
Cost584
\[\begin{array}{l} t_0 := -9 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error2.1
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \end{array} \]
Alternative 4
Error0.2
Cost448
\[x \cdot \left(6 + x \cdot -9\right) \]
Alternative 5
Error21.5
Cost192
\[x \cdot 6 \]
Alternative 6
Error62.1
Cost64
\[4 \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

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