Examples.Basics.ProofTests:f4 from sbv-4.4

Average Error: 0.03% → 0.02%

Time: 3.6s

Precision: binary64

Cost: 704

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * (x + (2.0d0 * y))) + (y * y)
end function

Java

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

↓

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

Python

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

↓

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

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

↓

function tmp = code(x, y)
	tmp = (x * (x + (2.0 * y))) + (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[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.03%
Target	0.03%
Herbie	0.02%

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

Derivation

1. Initial program 0.03

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

2. Taylor expanded in x around 0 0.03

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

3. Simplified 0.02

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

   [Start] 0.03	\[ \left(2 \cdot \left(y \cdot x\right) + {x}^{2}\right) + y \cdot y \]
   associate-*r* [=>] 0.03	\[ \left(\color{blue}{\left(2 \cdot y\right) \cdot x} + {x}^{2}\right) + y \cdot y \]
   unpow2 [=>] 0.03	\[ \left(\left(2 \cdot y\right) \cdot x + \color{blue}{x \cdot x}\right) + y \cdot y \]
   distribute-rgt-in [<=] 0.02	\[ \color{blue}{x \cdot \left(2 \cdot y + x\right)} + y \cdot y \]
   +-commutative [<=] 0.02	\[ x \cdot \color{blue}{\left(x + 2 \cdot y\right)} + y \cdot y \]

4. Final simplification 0.02

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

Alternatives

Alternative 1
Error	12.67%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-125}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 2
Error	1.96%
Cost	448

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

Alternative 3
Error	12.74%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-126}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Error	43.52%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023088 
(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)))