?

Average Error: 0.1 → 0.0
Time: 8.8s
Precision: binary64
Cost: 13120

?

\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
↓
\[\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right) \]
(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))
↓
(FPCore (x y z) :precision binary64 (fma x 3.0 (fma y 2.0 z)))
double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}
↓
double code(double x, double y, double z) {
	return fma(x, 3.0, fma(y, 2.0, z));
}

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end
↓
function code(x, y, z)
	return fma(x, 3.0, fma(y, 2.0, z))
end

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]
↓
code[x_, y_, z_] := N[(x * 3.0 + N[(y * 2.0 + z), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)

Error?

Derivation?

  1. Initial program 0.1

    \[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]
    Proof

    [Start]0.1

    \[ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]

    +-commutative [=>]0.1

    \[ \color{blue}{x + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right)} \]

    +-commutative [=>]0.1

    \[ x + \left(\color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)} + z\right) \]

    associate-+l+ [=>]0.1

    \[ x + \left(\left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right) + z\right) \]

    associate-+r+ [=>]0.1

    \[ x + \left(\color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)} + z\right) \]

    associate-+l+ [=>]0.1

    \[ x + \color{blue}{\left(\left(x + x\right) + \left(\left(y + y\right) + z\right)\right)} \]

    associate-+r+ [=>]0.1

    \[ \color{blue}{\left(x + \left(x + x\right)\right) + \left(\left(y + y\right) + z\right)} \]

    count-2 [=>]0.1

    \[ \left(x + \color{blue}{2 \cdot x}\right) + \left(\left(y + y\right) + z\right) \]

    distribute-rgt1-in [=>]0.1

    \[ \color{blue}{\left(2 + 1\right) \cdot x} + \left(\left(y + y\right) + z\right) \]

    *-commutative [=>]0.1

    \[ \color{blue}{x \cdot \left(2 + 1\right)} + \left(\left(y + y\right) + z\right) \]

    fma-def [=>]0.0

    \[ \color{blue}{\mathsf{fma}\left(x, 2 + 1, \left(y + y\right) + z\right)} \]

    metadata-eval [=>]0.0

    \[ \mathsf{fma}\left(x, \color{blue}{3}, \left(y + y\right) + z\right) \]

    count-2 [=>]0.0

    \[ \mathsf{fma}\left(x, 3, \color{blue}{2 \cdot y} + z\right) \]

    *-commutative [=>]0.0

    \[ \mathsf{fma}\left(x, 3, \color{blue}{y \cdot 2} + z\right) \]

    fma-def [=>]0.0

    \[ \mathsf{fma}\left(x, 3, \color{blue}{\mathsf{fma}\left(y, 2, z\right)}\right) \]

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error33.2
Cost1905
\[\begin{array}{l} t_0 := x + \left(y + y\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+206}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-209}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-236}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-201}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-140}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-91}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+76} \lor \neg \left(x \leq 6.8 \cdot 10^{+158}\right) \land x \leq 3.6 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 2
Error33.2
Cost1777
\[\begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+115}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-209}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-290}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-236}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-204}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+75} \lor \neg \left(x \leq 7 \cdot 10^{+159}\right) \land x \leq 3.6 \cdot 10^{+229}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 3
Error28.0
Cost1117
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-55}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq -2.95 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-85}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-66} \lor \neg \left(y \leq 2.8 \cdot 10^{-20}\right) \land y \leq 5 \cdot 10^{+56}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 4
Error10.5
Cost976
\[\begin{array}{l} t_0 := z + y \cdot 2\\ t_1 := y \cdot 2 + x \cdot 3\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.95 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-66}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error11.4
Cost850
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+115} \lor \neg \left(x \leq 2.05 \cdot 10^{+73} \lor \neg \left(x \leq 5.9 \cdot 10^{+159}\right) \land x \leq 3.6 \cdot 10^{+229}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
Alternative 6
Error14.4
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+192}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+229}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 7
Error0.1
Cost576
\[x + \left(z + 2 \cdot \left(x + y\right)\right) \]
Alternative 8
Error29.8
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+107}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+51}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
Alternative 9
Error41.4
Cost64
\[z \]

Error

Reproduce?

herbie shell --seed 2023039 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))