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

Average Error: 0.1 → 0.1

Time: 6.3s

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	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	12.2
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.2 \cdot 10^{-225} \lor \neg \left(y \cdot y \leq 1.45 \cdot 10^{-208}\right) \land y \cdot y \leq 4.75 \cdot 10^{-153}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 2
Error	12.2
Cost	1101

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-231} \lor \neg \left(y \cdot y \leq 5 \cdot 10^{-209}\right) \land y \cdot y \leq 4 \cdot 10^{-153}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 3
Error	12.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-12}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-79}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4
Error	0.1
Cost	576

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

Alternative 5
Error	26.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-160}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-151}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	28.0
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023018 
(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)))