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

Average Error: 0.1 → 0.0

Time: 7.3s

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
   (fma.f64 x 3 (fma.f64 y 2 z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= metadata-eval (+.f64 2 1)) (fma.f64 y 2 z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 2 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y 2) z))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 2 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 y)) z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 2 1) (+.f64 (Rewrite<= count-2_binary64 (+.f64 y y)) z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 2 1)) (+.f64 (+.f64 y y) z))): 19 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 2 1) x)) (+.f64 (+.f64 y y) z)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 x (*.f64 2 x))) (+.f64 (+.f64 y y) z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 (*.f64 2 x) (+.f64 (+.f64 y y) z)))): 2 points increase in error, 3 points decrease in error
   (+.f64 x (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 x) (+.f64 y y)) z))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (+.f64 (Rewrite<= count-2_binary64 (+.f64 x x)) (+.f64 y y)) z)): 0 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 x (+.f64 y y)))) z)): 3 points increase in error, 0 points decrease in error
   (+.f64 x (+.f64 (+.f64 x (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x y) y))) z)): 0 points increase in error, 1 points decrease in error
   (+.f64 x (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 x y) y) x)) z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 x y) y) x) z) x)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	30.3
Cost	1512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6546878232628167 \cdot 10^{+51}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq -10326716484.632036:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq -5.2531677840937536 \cdot 10^{-27}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -2.4806960435952307 \cdot 10^{-92}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq -2.2381988435619657 \cdot 10^{-289}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.1228670593086178 \cdot 10^{-218}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 3.0382935784584405 \cdot 10^{-136}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.832333123914727 \cdot 10^{+29}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;x \leq 7.998429987993812 \cdot 10^{+71}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.002723479378877 \cdot 10^{+108}:\\ \;\;\;\;y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \]

Alternative 2
Error	9.7
Cost	976

\[\begin{array}{l} t_0 := x + 2 \cdot \left(x + y\right)\\ t_1 := z + y \cdot 2\\ \mathbf{if}\;x \leq -1.6546878232628167 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9205791044564139 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.832333123914727 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.998429987993812 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.7
Cost	976

\[\begin{array}{l} t_0 := x + 2 \cdot \left(x + y\right)\\ t_1 := z + y \cdot 2\\ \mathbf{if}\;x \leq -1.6546878232628167 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9205791044564139 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.832333123914727 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.998429987993812 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2 + x \cdot 3\\ \end{array} \]

Alternative 4
Error	12.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5428507742474736 \cdot 10^{+128}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 5.531758936158507 \cdot 10^{+131}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	9.0
Cost	584

\[\begin{array}{l} t_0 := z + x \cdot 3\\ \mathbf{if}\;x \leq -1.6546878232628167 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.579941344673226 \cdot 10^{+20}:\\ \;\;\;\;z + y \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.1
Cost	576

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

Alternative 7
Error	30.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -5.7838754150937935 \cdot 10^{+109}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 2.3890750917486223 \cdot 10^{+128}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 8
Error	42.1
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022308 
(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))