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

Average Accuracy: 99.9% → 100.0%

Time: 10.8s

Precision: binary64

Cost: 7104

Math

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

↓

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

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 (+ t (* (+ y z) 2.0)))))

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 * (t + ((y + z) * 2.0))));
}

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(t + Float64(Float64(y + z) * 2.0))))
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[(t + N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision]), $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 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot 2 + t\right)\right)} \]
   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 [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]
   *-un-lft-identity [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(1 \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)}\right) \]
   *-un-lft-identity [<=] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]
   associate-+l+ [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]
   *-un-lft-identity [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(\color{blue}{1 \cdot \left(y + z\right)} + \left(z + y\right)\right) + t\right)\right) \]
   +-commutative [<=] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(1 \cdot \left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]
   *-un-lft-identity [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(1 \cdot \left(y + z\right) + \color{blue}{1 \cdot \left(y + z\right)}\right) + t\right)\right) \]
   distribute-rgt-out [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(y + z\right) \cdot \left(1 + 1\right)} + t\right)\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(y, 5, x \cdot \left(\left(y + z\right) \cdot \color{blue}{2} + t\right)\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	7104

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

Alternative 2
Accuracy	85.7%
Cost	1233

\[\begin{array}{l} t_1 := y \cdot 5 + x \cdot t\\ \mathbf{if}\;t \leq -1.86 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-23} \lor \neg \left(t \leq 2 \cdot 10^{+66}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	60.0%
Cost	1108

\[\begin{array}{l} t_1 := \left(y + z\right) \cdot \left(x + x\right)\\ t_2 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-256}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-23}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	60.8%
Cost	1108

\[\begin{array}{l} t_1 := y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-302}:\\ \;\;\;\;\left(y + z\right) \cdot \left(x + x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-253}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-186}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(t + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	972

\[\begin{array}{l} t_1 := x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{if}\;x \leq -7.1 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-150}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	88.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1350000000 \lor \neg \left(z \leq 2.6 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot 5 + 2 \cdot \left(x \cdot \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + y \cdot 2\right)\\ \end{array} \]

Alternative 7
Accuracy	99.9%
Cost	960

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

Alternative 8
Accuracy	50.3%
Cost	852

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-39}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-253}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-23}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]

Alternative 9
Accuracy	49.9%
Cost	848

\[\begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;x \leq -7.1 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	59.7%
Cost	844

\[\begin{array}{l} t_1 := \left(y + z\right) \cdot \left(x + x\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	84.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-24} \lor \neg \left(x \leq 1.45 \cdot 10^{-58}\right):\\ \;\;\;\;x \cdot \left(t + \left(y + z\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]

Alternative 12
Accuracy	77.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-36} \lor \neg \left(y \leq 5.3 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]

Alternative 13
Accuracy	49.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-47}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]

Alternative 14
Accuracy	26.1%
Cost	192

\[x \cdot t \]

Alternative 15
Accuracy	2.6%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023152 
(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)))