Expanding a square

Average Error: 39.5 → 0.0

Time: 2.2s

Precision: binary64

Cost: 448

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

↓

\[x \cdot x + x \cdot 2 \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

↓

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

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) * (x + 1.0d0)) - 1.0d0
end function

↓

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

Java

public static double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

public static double code(double x) {
	return (x * x) + (x * 2.0);
}

Python

def code(x):
	return ((x + 1.0) * (x + 1.0)) - 1.0

↓

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

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

↓

function code(x)
	return Float64(Float64(x * x) + Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) * (x + 1.0)) - 1.0;
end

↓

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

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

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

Derivation

1. Initial program 39.5

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]
   Proof
   (*.f64 x (+.f64 x 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 x 0)) (+.f64 x 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (+.f64 x (Rewrite<= metadata-eval (-.f64 1 1))) (+.f64 x 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 x 1) 1)) (+.f64 x 2)): 181 points increase in error, 2 points decrease in error
   (*.f64 (-.f64 (+.f64 x 1) 1) (+.f64 x (Rewrite<= metadata-eval (+.f64 1 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (+.f64 x 1) 1) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x 1) 1))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (+.f64 x 1) 1) (-.f64 (+.f64 x 1) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= difference-of-sqr-1_binary64 (-.f64 (*.f64 (+.f64 x 1) (+.f64 x 1)) 1)): 7 points increase in error, 2 points decrease in error

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{x \cdot x + x \cdot 2} \]

4. Final simplification 0.0

   \[\leadsto x \cdot x + x \cdot 2 \]

Alternatives

Alternative 1
Error	1.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -336.42660365535187:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 0.10266196555053207:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

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

Alternative 3
Error	41.5
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))