cos2 (problem 3.4.1)

Average Error: 31.4 → 0.1

Time: 6.8s

Precision: binary64

Math

\[\frac{1 - \cos x}{x \cdot x} \]

↓

\[\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

↓

(FPCore (x) :precision binary64 (/ (* (sin x) (/ (tan (/ x 2.0)) x)) x))

C

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

↓

double code(double x) {
	return (sin(x) * (tan((x / 2.0)) / x)) / x;
}

Fortran

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

↓

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

Java

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

↓

public static double code(double x) {
	return (Math.sin(x) * (Math.tan((x / 2.0)) / x)) / x;
}

Python

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

↓

def code(x):
	return (math.sin(x) * (math.tan((x / 2.0)) / x)) / x

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(sin(x) * Float64(tan(Float64(x / 2.0)) / x)) / x)
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (sin(x) * (tan((x / 2.0)) / x)) / x;
end

Wolfram

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

↓

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 31.4

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Applied flip--_binary64 31.5

   \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

3. Simplified 16.1

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x} \]

4. Taylor expanded in x around inf 15.9

   \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{\left(1 + \cos x\right) \cdot {x}^{2}}} \]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}} \]

6. Applied associate-*l/_binary64 0.1

   \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}} \]

7. Final simplification 0.1

   \[\leadsto \frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))