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

Average Error: 0.1% → 0.05%

Time: 4.6s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))

↓

(FPCore (x y z) :precision binary64 (fma y 2.0 (fma 3.0 x 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(y, 2.0, fma(3.0, x, 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(y, 2.0, fma(3.0, x, 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[(y * 2.0 + N[(3.0 * x + 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 0.05

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

   [Start] 0.1	\[ \left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
   associate-+l+ [=>] 0.11	\[ \color{blue}{\left(\left(\left(x + y\right) + y\right) + \left(x + z\right)\right)} + x \]
   associate-+l+ [=>] 0.12	\[ \left(\color{blue}{\left(x + \left(y + y\right)\right)} + \left(x + z\right)\right) + x \]
   +-commutative [=>] 0.12	\[ \left(\color{blue}{\left(\left(y + y\right) + x\right)} + \left(x + z\right)\right) + x \]
   associate-+l+ [=>] 0.11	\[ \color{blue}{\left(\left(y + y\right) + \left(x + \left(x + z\right)\right)\right)} + x \]
   associate-+l+ [=>] 0.11	\[ \color{blue}{\left(y + y\right) + \left(\left(x + \left(x + z\right)\right) + x\right)} \]
   count-2 [=>] 0.11	\[ \color{blue}{2 \cdot y} + \left(\left(x + \left(x + z\right)\right) + x\right) \]
   *-commutative [=>] 0.11	\[ \color{blue}{y \cdot 2} + \left(\left(x + \left(x + z\right)\right) + x\right) \]
   +-commutative [=>] 0.11	\[ y \cdot 2 + \left(\color{blue}{\left(\left(x + z\right) + x\right)} + x\right) \]
   associate-+r+ [<=] 0.1	\[ y \cdot 2 + \color{blue}{\left(\left(x + z\right) + \left(x + x\right)\right)} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(y, 2, \left(x + z\right) + \left(x + x\right)\right)} \]
   +-commutative [=>] 0.1	\[ \mathsf{fma}\left(y, 2, \color{blue}{\left(x + x\right) + \left(x + z\right)}\right) \]
   associate-+r+ [=>] 0.1	\[ \mathsf{fma}\left(y, 2, \color{blue}{\left(\left(x + x\right) + x\right) + z}\right) \]
   +-commutative [<=] 0.1	\[ \mathsf{fma}\left(y, 2, \color{blue}{\left(x + \left(x + x\right)\right)} + z\right) \]
   count-2 [=>] 0.1	\[ \mathsf{fma}\left(y, 2, \left(x + \color{blue}{2 \cdot x}\right) + z\right) \]
   distribute-rgt1-in [=>] 0.1	\[ \mathsf{fma}\left(y, 2, \color{blue}{\left(2 + 1\right) \cdot x} + z\right) \]
   fma-def [=>] 0.05	\[ \mathsf{fma}\left(y, 2, \color{blue}{\mathsf{fma}\left(2 + 1, x, z\right)}\right) \]
   metadata-eval [=>] 0.05	\[ \mathsf{fma}\left(y, 2, \mathsf{fma}\left(\color{blue}{3}, x, z\right)\right) \]

3. Final simplification 0.05

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

Alternatives

Alternative 1
Error	47.26%
Cost	852

\[\begin{array}{l} \mathbf{if}\;y \leq -0.36:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-247}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-190}:\\ \;\;\;\;3 \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-63}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+70}:\\ \;\;\;\;3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 2
Error	14.38%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+17} \lor \neg \left(y \leq 3.6 \cdot 10^{+33}\right):\\ \;\;\;\;x + 2 \cdot \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + 3 \cdot x\\ \end{array} \]

Alternative 3
Error	14.25%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+14} \lor \neg \left(y \leq 1.35 \cdot 10^{+68}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + 3 \cdot x\\ \end{array} \]

Alternative 4
Error	22.18%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+23}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+69}:\\ \;\;\;\;z + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	0.1%
Cost	576

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

Alternative 6
Error	46.86%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -16:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+38}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 7
Error	64.61%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023090 
(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))