Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4

Average Error: 0.1 → 0.0

Time: 7.0s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

(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)))

C

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));
}

Julia

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

Wolfram

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]

Derivation

1. Initial program 0.1

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]
   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 simplification 0.0

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

Alternatives

Alternative 1
Error	30.8
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-43}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-127}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-250}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-242}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-116}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+55}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	14.2
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+224}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+168} \lor \neg \left(x \leq -7.2 \cdot 10^{+124}\right) \land x \leq 1.4 \cdot 10^{+161}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 3
Error	9.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+27}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(x + z\right)\\ \end{array} \]

Alternative 4
Error	9.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -45000 \lor \neg \left(x \leq 11500000000000\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]

Alternative 5
Error	0.1
Cost	576

\[x + \left(z + 2 \cdot \left(x + y\right)\right) \]

Alternative 6
Error	30.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7
Error	42.0
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023016 
(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))