Linear.Quaternion:$c/ from linear-1.19.1.3, E

Average Error: 0.22% → 0.09%

Time: 8.4s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.22%
Target	0.2%
Herbie	0.09%

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

Derivation

1. Initial program 0.22

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

2. Simplified 0.09

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

   [Start] 0.22	\[ \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
   +-commutative [=>] 0.22	\[ \color{blue}{y \cdot y + \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right)} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(y, y, \left(x \cdot x + y \cdot y\right) + y \cdot y\right)} \]
   associate-+l+ [=>] 0.09	\[ \mathsf{fma}\left(y, y, \color{blue}{x \cdot x + \left(y \cdot y + y \cdot y\right)}\right) \]
   fma-def [=>] 0.09	\[ \mathsf{fma}\left(y, y, \color{blue}{\mathsf{fma}\left(x, x, y \cdot y + y \cdot y\right)}\right) \]
   distribute-lft-out [=>] 0.09	\[ \mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y + y\right)}\right)\right) \]

3. Final simplification 0.09

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

Alternatives

Alternative 1
Error	0.21%
Cost	6848

\[\mathsf{fma}\left(x, x, 3 \cdot \left(y \cdot y\right)\right) \]

Alternative 2
Error	19.94%
Cost	1241

\[\begin{array}{l} t_0 := 3 \cdot \left(y \cdot y\right)\\ t_1 := y \cdot y + x \cdot x\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-114} \lor \neg \left(y \leq 2.3 \cdot 10^{-75}\right) \land y \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 3
Error	0.22%
Cost	960

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

Alternative 4
Error	20.63%
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-12} \lor \neg \left(y \leq -9.2 \cdot 10^{-48} \lor \neg \left(y \leq -7.6 \cdot 10^{-91}\right) \land y \leq 5 \cdot 10^{-114}\right):\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Error	20.67%
Cost	848

\[\begin{array}{l} t_0 := 3 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-115}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 6
Error	0.22%
Cost	576

\[3 \cdot \left(y \cdot y\right) + x \cdot x \]

Alternative 7
Error	41.31%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-118}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-160}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	43.46%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))