Examples.Basics.BasicTests:f3 from sbv-4.4

Average Error: 0.0 → 0.0

Time: 14.5s

Precision: binary64

Cost: 832

Math

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

↓

\[\left(\left(y + x\right) \cdot 0.5\right) \cdot \left(y + \left(y + \left(x + x\right)\right)\right) \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (* (* (+ y x) 0.5) (+ y (+ y (+ x x)))))

C

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

↓

double code(double x, double y) {
	return ((y + x) * 0.5) * (y + (y + (x + x)));
}

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 + x) * 0.5d0) * (y + (y + (x + x)))
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 + x) * 0.5) * (y + (y + (x + x)));
}

Python

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

↓

def code(x, y):
	return ((y + x) * 0.5) * (y + (y + (x + x)))

Julia

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

↓

function code(x, y)
	return Float64(Float64(Float64(y + x) * 0.5) * Float64(y + Float64(y + Float64(x + x))))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = ((y + x) * 0.5) * (y + (y + (x + x)));
end

Wolfram

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

↓

code[x_, y_] := N[(N[(N[(y + x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(y + N[(y + N[(x + x), $MachinePrecision]), $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. Applied egg-rr 18.5

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

3. Applied egg-rr 18.6

   \[\leadsto \frac{\color{blue}{\frac{x + y}{\frac{1}{\left(x + y\right) \cdot \left(x + y\right)}}}}{x + y} \]

4. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot 0.5\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot 0.5\right)} \]

5. Simplified 0.0

   \[\leadsto \color{blue}{\left(\left(y + x\right) \cdot 0.5\right) \cdot \left(y + \left(y + \left(x + x\right)\right)\right)} \]
   Proof

   [Start] 0.0	\[ \left(x + y\right) \cdot \left(\left(x + y\right) \cdot 0.5\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot 0.5\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{\left(\left(x + y\right) \cdot 0.5\right) \cdot \left(x + y\right)} + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot 0.5\right) \]
   rational.json-simplify-51 [=>] 0.0	\[ \color{blue}{\left(\left(x + y\right) \cdot 0.5\right) \cdot \left(\left(x + y\right) + \left(x + y\right)\right)} \]
   rational.json-simplify-1 [=>] 0.0	\[ \left(\color{blue}{\left(y + x\right)} \cdot 0.5\right) \cdot \left(\left(x + y\right) + \left(x + y\right)\right) \]
   rational.json-simplify-1 [=>] 0.0	\[ \left(\left(y + x\right) \cdot 0.5\right) \cdot \left(\left(x + y\right) + \color{blue}{\left(y + x\right)}\right) \]
   rational.json-simplify-41 [=>] 0.0	\[ \left(\left(y + x\right) \cdot 0.5\right) \cdot \color{blue}{\left(y + \left(x + \left(x + y\right)\right)\right)} \]
   rational.json-simplify-1 [=>] 0.0	\[ \left(\left(y + x\right) \cdot 0.5\right) \cdot \left(y + \left(x + \color{blue}{\left(y + x\right)}\right)\right) \]
   rational.json-simplify-41 [=>] 0.0	\[ \left(\left(y + x\right) \cdot 0.5\right) \cdot \left(y + \color{blue}{\left(y + \left(x + x\right)\right)}\right) \]

6. Final simplification 0.0

   \[\leadsto \left(\left(y + x\right) \cdot 0.5\right) \cdot \left(y + \left(y + \left(x + x\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	448

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

Alternative 2
Error	55.0
Cost	320

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

Reproduce

herbie shell --seed 2023074 
(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)))