?

Average Error: 0.1% → 0%
Time: 6.6s
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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]
    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+ [=>]0.09

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

    associate-+l+ [=>]0.09

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

    associate-+r+ [=>]0.09

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

    count-2 [=>]0.09

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

    distribute-rgt1-in [=>]0.09

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

    *-commutative [=>]0.09

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

    fma-def [=>]0

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

    metadata-eval [=>]0

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

    count-2 [=>]0

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

    *-commutative [=>]0

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

    fma-def [=>]0

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

  3. Final simplification0

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

Alternatives

Alternative 1
Error48.86%
Cost1645
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+165}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+144}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+85}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-241}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-287}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-259}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{-104}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-54}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 29000000000000 \lor \neg \left(x \leq 2.2 \cdot 10^{+93}\right) \land x \leq 5.5 \cdot 10^{+122}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 2
Error15.74%
Cost1115
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+165} \lor \neg \left(x \leq -1.26 \cdot 10^{+144}\right) \land \left(x \leq -1.15 \cdot 10^{+101} \lor \neg \left(x \leq 21000000000 \lor \neg \left(x \leq 2.85 \cdot 10^{+81}\right) \land x \leq 7 \cdot 10^{+122}\right)\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]
Alternative 3
Error15.71%
Cost1112
\[\begin{array}{l} t_0 := z + x \cdot 3\\ t_1 := z + y \cdot 2\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1150000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;x \cdot 2 + \left(x + z\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error14.88%
Cost976
\[\begin{array}{l} t_0 := z + x \cdot 3\\ t_1 := z + y \cdot 2\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+92}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error20%
Cost849
\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+166}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{+144} \lor \neg \left(x \leq -1.46 \cdot 10^{+102}\right) \land x \leq 1.8 \cdot 10^{+159}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]
Alternative 6
Error0.09%
Cost576
\[x + \left(z + 2 \cdot \left(x + y\right)\right) \]
Alternative 7
Error47.64%
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+33}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 3800000000000:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]
Alternative 8
Error65.49%
Cost64
\[z \]

Error

Reproduce?

herbie shell --seed 2023121 
(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))