Commute and associate

Average Error: 0.0 → 0

Time: 3.7s

Precision: binary64

Cost: 64

Math

\[\left(\left(x + y\right) + z\right) - \left(x + \left(y + z\right)\right) \]

↓

\[0 \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (+ x y) z) (+ x (+ y z))))

↓

(FPCore (x y z) :precision binary64 0.0)

C

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

↓

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

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

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

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

↓

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return ((x + y) + z) - (x + (y + z))

↓

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + y) + z) - Float64(x + Float64(y + z)))
end

↓

function code(x, y, z)
	return 0.0
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

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

↓

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 0.0

   \[\left(\left(x + y\right) + z\right) - \left(x + \left(y + z\right)\right) \]

2. Simplified 0

   \[\leadsto \color{blue}{0} \]
   Proof
   0: 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-inverses_binary64 (-.f64 (+.f64 (+.f64 x y) z) (+.f64 (+.f64 x y) z))): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (+.f64 x y) z) (Rewrite=> associate-+l+_binary64 (+.f64 x (+.f64 y z)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z)
  :name "Commute and associate"
  :precision binary64
  (- (+ (+ x y) z) (+ x (+ y z))))