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))): 14 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)))): 1 points increase in error, 2 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)): 1 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, 0 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	10.0
Cost	1112

\[\begin{array}{l} t_0 := z + y \cdot 2\\ t_1 := z + x \cdot 3\\ \mathbf{if}\;z \leq -3.858146655908538 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.128170569936288 \cdot 10^{-41}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;z \leq 1.5307179277189962 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.357648115137136 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8578760292416322 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.0028531950651006 \cdot 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.0
Cost	1112

\[\begin{array}{l} t_0 := z + y \cdot 2\\ t_1 := z + x \cdot 3\\ \mathbf{if}\;z \leq -3.858146655908538 \cdot 10^{+22}:\\ \;\;\;\;x \cdot 2 + \left(x + z\right)\\ \mathbf{elif}\;z \leq 1.128170569936288 \cdot 10^{-41}:\\ \;\;\;\;x + 2 \cdot \left(x + y\right)\\ \mathbf{elif}\;z \leq 1.5307179277189962 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.357648115137136 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8578760292416322 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.0028531950651006 \cdot 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	31.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7746813156148457 \cdot 10^{+110}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.128170569936288 \cdot 10^{-41}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;z \leq 10075837077497612000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.357648115137136 \cdot 10^{+99}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Error	30.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9924937162573 \cdot 10^{+107}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq -3.035545945824555 \cdot 10^{-39}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;y \leq 6.394692703165846 \cdot 10^{-262}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5.620275277971621 \cdot 10^{+73}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 5
Error	13.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9924937162573 \cdot 10^{+107}:\\ \;\;\;\;y \cdot 2\\ \mathbf{elif}\;y \leq 5.620275277971621 \cdot 10^{+73}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 2\\ \end{array} \]

Alternative 6
Error	9.6
Cost	584

\[\begin{array}{l} t_0 := z + y \cdot 2\\ \mathbf{if}\;y \leq -2.9924937162573 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.620275277971621 \cdot 10^{+73}:\\ \;\;\;\;z + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.1
Cost	576

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

Alternative 8
Error	41.4
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022317 
(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))