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

Average Error: 0.19% → 0.08%

Time: 8.5s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.19%
Target	0.16%
Herbie	0.08%

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

Derivation

1. Initial program 0.19

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

2. Simplified 0.08

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

   [Start] 0.19	\[ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   +-commutative [=>] 0.19	\[ \color{blue}{z \cdot z + \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   fma-def [=>] 0.09	\[ \color{blue}{\mathsf{fma}\left(z, z, \left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   associate-+l+ [=>] 0.09	\[ \mathsf{fma}\left(z, z, \color{blue}{x \cdot y + \left(z \cdot z + z \cdot z\right)}\right) \]
   fma-def [=>] 0.08	\[ \mathsf{fma}\left(z, z, \color{blue}{\mathsf{fma}\left(x, y, z \cdot z + z \cdot z\right)}\right) \]
   count-2 [=>] 0.08	\[ \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)}\right)\right) \]

3. Final simplification 0.08

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

Alternatives

Alternative 1
Error	0.09%
Cost	7104

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

Alternative 2
Error	0.19%
Cost	6848

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

Alternative 3
Error	18.19%
Cost	976

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot 3\\ t_1 := z \cdot z + x \cdot y\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4
Error	18.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-80} \lor \neg \left(z \leq 4.8 \cdot 10^{-63}\right):\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5
Error	18.68%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-80}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-63}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 6
Error	0.16%
Cost	576

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

Alternative 7
Error	37.54%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023104 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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