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

Average Error: 0.1 → 0.0

Time: 54.0s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma 3.0 x (fma 2.0 y 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(3.0, x, fma(2.0, y, 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(3.0, x, fma(2.0, y, 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[(3.0 * x + N[(2.0 * y + 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 0.1

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

3. Taylor expanded in x around 0 0.1

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

4. Simplified 0.0

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

Alternatives

Alternative 1
Error	9.4
Cost	716

\[\begin{array}{l} t_0 := 2 \cdot y + z\\ \mathbf{if}\;z \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 24000000000:\\ \;\;\;\;2 \cdot \left(y + x\right) + x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot x + z\\ \end{array} \]

Alternative 2
Error	9.9
Cost	712

\[\begin{array}{l} t_0 := 2 \cdot y + z\\ \mathbf{if}\;y \leq -8 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.02:\\ \;\;\;\;2 \cdot x + \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	30.5
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+93}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{-46}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Error	12.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+87}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 5
Error	9.9
Cost	584

\[\begin{array}{l} t_0 := 2 \cdot y + z\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;3 \cdot x + z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.1
Cost	576

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

Alternative 7
Error	29.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+94}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 10^{+69}:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 8
Error	42.2
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023033 
(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))