Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3

Average Error: 0.0 → 0.3

Time: 1.4s

Precision: binary64

Cost: 448

Math

\[\sqrt{1 - x \cdot x} \]

↓

\[1 + -0.5 \cdot \left(x \cdot x\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (+ 1.0 (* -0.5 (* x x))))

C

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

↓

double code(double x) {
	return 1.0 + (-0.5 * (x * x));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-0.5d0) * (x * x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 - (x * x)));
}

↓

public static double code(double x) {
	return 1.0 + (-0.5 * (x * x));
}

Python

def code(x):
	return math.sqrt((1.0 - (x * x)))

↓

def code(x):
	return 1.0 + (-0.5 * (x * x))

Julia

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

↓

function code(x)
	return Float64(1.0 + Float64(-0.5 * Float64(x * x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 1.0 + (-0.5 * (x * x));
end

Wolfram

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

↓

code[x_] := N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\sqrt{1 - x \cdot x} \]

2. Taylor expanded in x around 0 0.3

   \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]

3. Simplified 0.3

   \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]
   Proof

   [Start] 0.3	\[ 1 + -0.5 \cdot {x}^{2} \]
   unpow2 [=>] 0.3	\[ 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

4. Final simplification 0.3

   \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]

Alternatives

Alternative 1
Error	0.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))