Examples.Basics.ProofTests:f4 from sbv-4.4

Average Error: 0.0 → 0.0

Time: 10.2s

Precision: binary64

Cost: 704

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

Math

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

↓

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

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))))

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));
}

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))
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));
}

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))

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 * x) + Float64(y * Float64(Float64(x * 2.0) + 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));
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 * x), $MachinePrecision] + N[(y * N[(N[(x * 2.0), $MachinePrecision] + 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. Simplified 0.0

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

   [Start] 0.0	\[ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{y \cdot y + \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right)} \]
   rational.json-simplify-41 [=>] 0.0	\[ \color{blue}{x \cdot x + \left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]
   rational.json-simplify-1 [=>] 0.0	\[ x \cdot x + \color{blue}{\left(y \cdot y + \left(x \cdot 2\right) \cdot y\right)} \]
   rational.json-simplify-51 [=>] 0.0	\[ x \cdot x + \color{blue}{y \cdot \left(x \cdot 2 + y\right)} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	8.6
Cost	972

\[\begin{array}{l} t_0 := \left(x + \left(y + y\right)\right) \cdot x\\ \mathbf{if}\;y \leq 7.7 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\left(y + \left(x + x\right)\right) \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right) + y \cdot y\\ \end{array} \]

Alternative 2
Error	8.6
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{-149}:\\ \;\;\;\;x \cdot x + y \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\left(y + \left(x + x\right)\right) \cdot y\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-11}:\\ \;\;\;\;\left(x + \left(y + y\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(y \cdot x\right) + y \cdot y\\ \end{array} \]

Alternative 3
Error	8.6
Cost	844

\[\begin{array}{l} t_0 := \left(x + \left(y + y\right)\right) \cdot x\\ t_1 := \left(y + \left(x + x\right)\right) \cdot y\\ \mathbf{if}\;y \leq 7.7 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	28.1
Cost	448

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

Alternative 5
Error	55.2
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(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)))