Examples.Basics.BasicTests:f3 from sbv-4.4

Average Error: 0.0 → 0.0

Time: 1.1s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ 2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right) \]
   rational_best-simplify-44 [=>] 0.0	\[ \color{blue}{y \cdot \left(2 \cdot x\right)} + \left({y}^{2} + {x}^{2}\right) \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	448

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

Alternative 2
Error	54.9
Cost	320

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

Reproduce

herbie shell --seed 2023096 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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