Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Average Accuracy: 99.9% → 99.9%

Time: 10.9s

Precision: binary64

Cost: 7104

Math

\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

↓

\[\mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot 2 + t\right)\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

↓

(FPCore (x y z t) :precision binary64 (fma y 5.0 (* x (+ (* (+ y z) 2.0) t))))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

↓

double code(double x, double y, double z, double t) {
	return fma(y, 5.0, (x * (((y + z) * 2.0) + t)));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

↓

function code(x, y, z, t)
	return fma(y, 5.0, Float64(x * Float64(Float64(Float64(y + z) * 2.0) + t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(x * N[(N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]

2. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot 2 + t\right)\right)} \]
   Proof

   [Start] 99.9	\[ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   +-commutative [=>] 99.9	\[ \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]
   associate-+l+ [=>] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]
   *-un-lft-identity [=>] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(\color{blue}{1 \cdot \left(y + z\right)} + \left(z + y\right)\right) + t\right)\right) \]
   +-commutative [<=] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(1 \cdot \left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]
   *-un-lft-identity [=>] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(1 \cdot \left(y + z\right) + \color{blue}{1 \cdot \left(y + z\right)}\right) + t\right)\right) \]
   distribute-rgt-out [=>] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(y + z\right) \cdot \left(1 + 1\right)} + t\right)\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot \color{blue}{2} + t\right)\right) \]

3. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot 2 + t\right)\right) \]

Alternatives

Alternative 1
Accuracy	77.6%
Cost	1108

\[\begin{array}{l} t_1 := y \cdot 5 + x \cdot t\\ t_2 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{elif}\;y \leq 0.0075:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	77.6%
Cost	1108

\[\begin{array}{l} t_1 := y \cdot 5 + x \cdot t\\ t_2 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.7 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(z + z\right) + x \cdot t\\ \mathbf{elif}\;y \leq 0.055:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	92.3%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y \leq -36000000000000 \lor \neg \left(y \leq 3\right):\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + \left(x \cdot \left(z + z\right) + x \cdot t\right)\\ \end{array} \]

Alternative 4
Accuracy	58.7%
Cost	976

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ t_2 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	60.7%
Cost	976

\[\begin{array}{l} t_1 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\left(y + z\right) \cdot \left(x + x\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-113}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-108}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	83.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-168} \lor \neg \left(y \leq 1.28 \cdot 10^{-110}\right):\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right) + x \cdot t\\ \end{array} \]

Alternative 7
Accuracy	92.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -33000000000000 \lor \neg \left(y \leq 15\right):\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]

Alternative 8
Accuracy	99.9%
Cost	960

\[y \cdot 5 + x \cdot \left(t + \left(y + \left(z + \left(y + z\right)\right)\right)\right) \]

Alternative 9
Accuracy	99.9%
Cost	960

\[\left(x \cdot \left(\left(y + z\right) \cdot 2\right) + x \cdot t\right) + y \cdot 5 \]

Alternative 10
Accuracy	49.5%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-40} \lor \neg \left(x \leq -9.5 \cdot 10^{-58}\right) \land x \leq 3.35 \cdot 10^{-28}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]

Alternative 11
Accuracy	49.5%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-150}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-298}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-135}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]

Alternative 12
Accuracy	77.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-42} \lor \neg \left(y \leq 1.1 \cdot 10^{-38}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]

Alternative 13
Accuracy	26.3%
Cost	192

\[x \cdot t \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))