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

Average Error: 0.1 → 0.0

Time: 7.6s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.1
Target	0.1
Herbie	0.0

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ x \cdot \left(y + z\right) + z \cdot 5 \]
   distribute-rgt-in [=>] 0.1	\[ \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]
   associate-+l+ [=>] 0.1	\[ \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]
   *-commutative [=>] 0.1	\[ \color{blue}{x \cdot y} + \left(z \cdot x + z \cdot 5\right) \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot x + z \cdot 5\right)} \]
   distribute-lft-out [=>] 0.1	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x + 5\right)}\right) \]

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{\left(y + z\right) \cdot x + 5 \cdot z} \]

4. Simplified 0.0

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

   [Start] 0.1	\[ \left(y + z\right) \cdot x + 5 \cdot z \]
   +-commutative [=>] 0.1	\[ \color{blue}{5 \cdot z + \left(y + z\right) \cdot x} \]
   *-commutative [=>] 0.1	\[ \color{blue}{z \cdot 5} + \left(y + z\right) \cdot x \]
   *-commutative [<=] 0.1	\[ z \cdot 5 + \color{blue}{x \cdot \left(y + z\right)} \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(z, 5, x \cdot \color{blue}{\left(z + y\right)}\right) \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	6848

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

Alternative 2
Error	12.9
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ t_1 := z \cdot \left(5 + x\right)\\ \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.4 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	24.6
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-16}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-54} \lor \neg \left(x \leq -5 \cdot 10^{-115}\right) \land x \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Error	24.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-16}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-54}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 5
Error	1.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -550000000000 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot 5\\ \end{array} \]

Alternative 6
Error	16.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-79} \lor \neg \left(z \leq 8.5 \cdot 10^{-147}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Error	0.1
Cost	576

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

Alternative 8
Error	35.0
Cost	192

\[z \cdot 5 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))