Examples.Basics.ProofTests:f4 from sbv-4.4

Average Accuracy: 100.0% → 100.0%

Time: 4.5s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* y (* x 2.0))) (* 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) + (y * (x * 2.0))) + (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) + (y * (x * 2.0d0))) + (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) + (y * (x * 2.0))) + (y * y);
}

Python

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

↓

def code(x, y):
	return ((x * x) + (y * (x * 2.0))) + (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(Float64(x * x) + Float64(y * Float64(x * 2.0))) + 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) + (y * (x * 2.0))) + (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[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	88.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot x + x \cdot \left(2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	704

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

Alternative 3
Accuracy	87.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Accuracy	88.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 5
Accuracy	87.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-157}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6
Accuracy	55.8%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023138 
(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)))