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

Average Error: 0.1 → 0.1

Time: 8.2s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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(x, x, fma(y, y, Float64(Float64(y * y) * 2.0)))
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[(x * x + N[(y * y + N[(N[(y * y), $MachinePrecision] * 2.0), $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. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	6848

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

Alternative 2
Error	13.6
Cost	2140

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{-238}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 5 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 5 \cdot 10^{-114}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 10^{-48}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	12.8
Cost	1748

\[\begin{array}{l} t_0 := y \cdot y + x \cdot x\\ t_1 := y \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 5 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot y \leq 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot y \leq 5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.1
Cost	576

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

Alternative 5
Error	25.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.508733983896255 \cdot 10^{-151}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.04611013677248 \cdot 10^{-145}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	50.6
Cost	192

\[y \cdot y \]

Reproduce

herbie shell --seed 2022316 
(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)))