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

Average Error: 0.1 → 0.0

Time: 7.4s

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	30.4
Cost	1116

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+119}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-66}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-264}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-107}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+40}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+110}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	10.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+108}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot 2\\ \end{array} \]

Alternative 3
Error	10.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+135} \lor \neg \left(y \leq 8 \cdot 10^{-21}\right):\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 3\\ \end{array} \]

Alternative 4
Error	13.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+92}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+99}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 5
Error	0.1
Cost	576

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

Alternative 6
Error	31.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+107}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7
Error	41.4
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023031 
(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))