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

Average Accuracy: 98.2% → 99.4%

Time: 7.7s

Precision: binary64

Cost: 13376

• Unsound rule application detected in e-graph. Results may not be sound.

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 (* z (+ 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, (z * (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(z * 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[(z * N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	98.2%
Target	98.2%
Herbie	99.4%

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

Derivation

1. Initial program 99.1%

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

2. Simplified 100.0%

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

   [Start] 99.1	\[ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   +-commutative [=>] 99.1	\[ \color{blue}{z \cdot z + \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   fma-def [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(z, z, \left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   associate-+l+ [=>] 99.2	\[ \mathsf{fma}\left(z, z, \color{blue}{x \cdot y + \left(z \cdot z + z \cdot z\right)}\right) \]
   fma-def [=>] 100.0	\[ \mathsf{fma}\left(z, z, \color{blue}{\mathsf{fma}\left(x, y, z \cdot z + z \cdot z\right)}\right) \]
   distribute-lft-out [=>] 100.0	\[ \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(z + z\right)}\right)\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6848

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

Alternative 2
Accuracy	81.6%
Cost	1115

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+133} \lor \neg \left(z \leq -3.6 \cdot 10^{+49}\right) \land \left(z \leq -2.2 \cdot 10^{-18} \lor \neg \left(z \leq 5.8 \cdot 10^{-38}\right) \land \left(z \leq 2.1 \cdot 10^{-17} \lor \neg \left(z \leq 1.25 \cdot 10^{+21}\right)\right)\right):\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	81.7%
Cost	1112

\[\begin{array}{l} t_0 := z \cdot \left(z \cdot 3\right)\\ t_1 := 3 \cdot \left(z \cdot z\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+52}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	98.2%
Cost	960

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

Alternative 5
Accuracy	98.2%
Cost	576

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

Alternative 6
Accuracy	52.5%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023160 
(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)))