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

Average Accuracy: 99.9% → 100.0%

Time: 8.2s

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 99.9%

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

2. Simplified 100.0%

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

   [Start] 99.9	\[ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   +-commutative [=>] 99.9	\[ \color{blue}{x + \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right)} \]
   +-commutative [=>] 99.9	\[ x + \left(\color{blue}{\left(x + \left(\left(x + y\right) + y\right)\right)} + z\right) \]
   associate-+l+ [=>] 99.9	\[ x + \left(\left(x + \color{blue}{\left(x + \left(y + y\right)\right)}\right) + z\right) \]
   associate-+r+ [=>] 99.9	\[ x + \left(\color{blue}{\left(\left(x + x\right) + \left(y + y\right)\right)} + z\right) \]
   associate-+l+ [=>] 99.9	\[ x + \color{blue}{\left(\left(x + x\right) + \left(\left(y + y\right) + z\right)\right)} \]
   associate-+r+ [=>] 99.9	\[ \color{blue}{\left(x + \left(x + x\right)\right) + \left(\left(y + y\right) + z\right)} \]
   count-2 [=>] 99.9	\[ \left(x + \color{blue}{2 \cdot x}\right) + \left(\left(y + y\right) + z\right) \]
   distribute-rgt1-in [=>] 99.9	\[ \color{blue}{\left(2 + 1\right) \cdot x} + \left(\left(y + y\right) + z\right) \]
   *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \left(2 + 1\right)} + \left(\left(y + y\right) + z\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, 2 + 1, \left(y + y\right) + z\right)} \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, \color{blue}{3}, \left(y + y\right) + z\right) \]
   count-2 [=>] 100.0	\[ \mathsf{fma}\left(x, 3, \color{blue}{2 \cdot y} + z\right) \]
   *-commutative [=>] 100.0	\[ \mathsf{fma}\left(x, 3, \color{blue}{y \cdot 2} + z\right) \]
   fma-def [=>] 100.0	\[ \mathsf{fma}\left(x, 3, \color{blue}{\mathsf{fma}\left(y, 2, z\right)}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	52.3%
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-20}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-178}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-232}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-160}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+80}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Accuracy	86.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+15} \lor \neg \left(z \leq 4.1 \cdot 10^{+45}\right):\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 3
Accuracy	85.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+18} \lor \neg \left(y \leq 0.00096\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 4
Accuracy	79.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+121}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+119}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	576

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

Alternative 6
Accuracy	52.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7
Accuracy	34.4%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023141 
(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))