Expanding a square

Average Error: 38.9 → 0.0

Time: 1.7s

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 38.9

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

2. Simplified 0.0

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

   [Start] 38.9	\[ \left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   difference-of-sqr-1 [=>] 38.9	\[ \color{blue}{\left(\left(x + 1\right) + 1\right) \cdot \left(\left(x + 1\right) - 1\right)} \]
   *-commutative [=>] 38.9	\[ \color{blue}{\left(\left(x + 1\right) - 1\right) \cdot \left(\left(x + 1\right) + 1\right)} \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\left(x + \left(1 - 1\right)\right)} \cdot \left(\left(x + 1\right) + 1\right) \]
   metadata-eval [=>] 0.0	\[ \left(x + \color{blue}{0}\right) \cdot \left(\left(x + 1\right) + 1\right) \]
   +-rgt-identity [=>] 0.0	\[ \color{blue}{x} \cdot \left(\left(x + 1\right) + 1\right) \]
   associate-+l+ [=>] 0.0	\[ x \cdot \color{blue}{\left(x + \left(1 + 1\right)\right)} \]
   metadata-eval [=>] 0.0	\[ x \cdot \left(x + \color{blue}{2}\right) \]

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	1.9
Cost	456

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

Alternative 2
Error	0.0
Cost	320

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

Alternative 3
Error	21.6
Cost	192

\[x \cdot 2 \]

Reproduce

herbie shell --seed 2023055 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))