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

Average Error: 0.2% → 0.08%

Time: 6.9s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

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

↓

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

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 * 2.0), z, (z * z)) + (x * y);
}

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 Float64(fma(Float64(z * 2.0), z, Float64(z * z)) + Float64(x * y))
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[(N[(N[(z * 2.0), $MachinePrecision] * z + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.2%
Target	0.17%
Herbie	0.08%

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

Derivation

1. Initial program 0.2

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

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] 0.2	\[ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   +-commutative [=>] 0.2	\[ \color{blue}{z \cdot z + \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right)} \]
   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+ [=>] 0.08	\[ \mathsf{fma}\left(z, z, \color{blue}{x \cdot y + \left(z \cdot z + z \cdot z\right)}\right) \]
   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. Applied egg-rr 0.19

   \[\leadsto \color{blue}{\left(z \cdot z + 2 \cdot \left(z \cdot z\right)\right) + x \cdot y} \]

4. Applied egg-rr 0.08

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

5. Final simplification 0.08

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

Alternatives

Alternative 1
Error	18.88%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.7 \cdot 10^{-81} \lor \neg \left(z \cdot z \leq 3.8 \cdot 10^{-47}\right) \land z \cdot z \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]

Alternative 2
Error	18.74%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-47}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+17}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]

Alternative 3
Error	0.22%
Cost	576

\[x \cdot y + z \cdot \frac{z}{0.3333333333333333} \]

Alternative 4
Error	0.19%
Cost	576

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

Alternative 5
Error	37.63%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023121 
(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)))