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

Average Accuracy: 99.9% → 100.0%

Time: 7.4s

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	99.9%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 100.0%

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

   [Start] 99.9	\[ x \cdot \left(y + z\right) + z \cdot 5 \]
   +-commutative [=>] 99.9	\[ \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	6848

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

Alternative 2
Accuracy	62.1%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;x \leq -860000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-41}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-100}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+28}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+89}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 3
Accuracy	81.1%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-82}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 39:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot x\\ \end{array} \]

Alternative 4
Accuracy	80.9%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{z \cdot 25}{5 - x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 22.5:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot x\\ \end{array} \]

Alternative 5
Accuracy	81.0%
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-82}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 25.5:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	62.3%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.26 \cdot 10^{-82} \lor \neg \left(x \leq -7.5 \cdot 10^{-99}\right) \land x \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Accuracy	74.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-167} \lor \neg \left(z \leq 6.5 \cdot 10^{-38}\right):\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8
Accuracy	99.9%
Cost	576

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

Alternative 9
Accuracy	45.4%
Cost	192

\[z \cdot 5 \]

Reproduce

herbie shell --seed 2023147 
(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)))