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

Average Error: 0.1 → 0.1

Time: 8.0s

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	12.2
Cost	848

\[\begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 3\\ \mathbf{if}\;x \leq -5.047766529709451 \cdot 10^{-23}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -8.947712136562253 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.766440654394941 \cdot 10^{-90}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.24787262127744 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	12.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -5.047766529709451 \cdot 10^{-23}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -8.947712136562253 \cdot 10^{-39}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{elif}\;x \leq -7.766440654394941 \cdot 10^{-90}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.24787262127744 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Error	0.1
Cost	576

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

Alternative 4
Error	27.6
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022317 
(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)))