Examples.Basics.ProofTests:f4 from sbv-4.4

Average Error: 0.0 → 0.0

Time: 3.7s

Precision: binary64

Cost: 7104

Math

\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

↓

function code(x, y)
	return fma(x, x, Float64(Float64(Float64(x * 2.0) * y) + Float64(y * y)))
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x * x + N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \]

Derivation

1. Initial program 0.0

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

2. Applied egg-rr 0.0

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

3. Taylor expanded in y around 0 0.0

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

4. Simplified 0.0

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

5. Applied egg-rr 0.0

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

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	20.5
Cost	844

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot 2 + y\right)\\ \mathbf{if}\;x \leq -1.5106622313505857 \cdot 10^{-38}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -6.423523404190331 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6409460070423962 \cdot 10^{-151}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	704

\[y \cdot y + x \cdot \left(x + 2 \cdot y\right) \]

Alternative 3
Error	20.6
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5106622313505857 \cdot 10^{-38}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -6.423523404190331 \cdot 10^{-57}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -2.6409460070423962 \cdot 10^{-151}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Error	1.2
Cost	448

\[y \cdot y + x \cdot x \]

Alternative 5
Error	28.1
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))