Expanding a square

Average Error: 39.0 → 0.0

Time: 7.5s

Precision: binary64

Cost: 6784

Math

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

↓

\[2 \cdot x + {x}^{2} \]

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return (2.0 * x) + pow(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 = (2.0d0 * 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 (2.0 * x) + Math.pow(x, 2.0);
}

Python

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

↓

def code(x):
	return (2.0 * x) + math.pow(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(2.0 * x) + (x ^ 2.0))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (2.0 * 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[(2.0 * x), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0

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

2. Taylor expanded in x around 0 0.0

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

3. Final simplification 0.0

   \[\leadsto 2 \cdot x + {x}^{2} \]

Alternatives

Alternative 1
Error	0.5
Cost	1800

\[\begin{array}{l} \mathbf{if}\;x + 1 \leq 0.9996:\\ \;\;\;\;\left(x + 1\right) \cdot \left(x + 1\right) - 1\\ \mathbf{elif}\;x + 1 \leq 1.0000002:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-1 - x\right) \cdot \left(-1 - x\right)\right) \cdot \left(1 + \left(-\frac{\frac{1}{-1 - x}}{-1 - x}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1096

\[\begin{array}{l} t_0 := \left(x + 1\right) \cdot \left(x + 1\right) - 1\\ \mathbf{if}\;x + 1 \leq 0.9996:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x + 1 \leq 1.0000002:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.5
Cost	192

\[2 \cdot x \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))