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

Average Accuracy: 99.8% → 99.9%

Time: 5.4s

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	99.8%
Target	99.8%
Herbie	99.9%

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

Derivation

1. Initial program 99.8%

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

2. Simplified 99.9%

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

   [Start] 99.8	\[ \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
   +-commutative [=>] 99.8	\[ \color{blue}{y \cdot y + \left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right)} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(y, y, \left(x \cdot x + y \cdot y\right) + y \cdot y\right)} \]
   associate-+l+ [=>] 99.9	\[ \mathsf{fma}\left(y, y, \color{blue}{x \cdot x + \left(y \cdot y + y \cdot y\right)}\right) \]
   fma-def [=>] 99.9	\[ \mathsf{fma}\left(y, y, \color{blue}{\mathsf{fma}\left(x, x, y \cdot y + y \cdot y\right)}\right) \]
   distribute-lft-out [=>] 99.9	\[ \mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y + y\right)}\right)\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	80.5%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.95 \cdot 10^{-180} \lor \neg \left(y \cdot y \leq 1.05 \cdot 10^{-133}\right) \land y \cdot y \leq 2.2 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	80.6%
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-181} \lor \neg \left(y \cdot y \leq 4 \cdot 10^{-134}\right) \land y \cdot y \leq 2 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	977

\[\begin{array}{l} t_0 := x \cdot x + y \cdot y\\ \mathbf{if}\;x \leq -350000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.05 \cdot 10^{-27}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-141} \lor \neg \left(x \leq 5.8 \cdot 10^{-62}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	576

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

Alternative 5
Accuracy	57.8%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023137 
(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)))