Average Error: 31.7 → 0.1
Time: 5.1s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x} \]
↓
\[\frac{\sin x}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x} \]
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (* (/ (sin x) x) (/ (tan (* x 0.5)) x)))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) / x) * (tan((x * 0.5)) / x);
}

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) / x) * (tan((x * 0.5d0)) / x)
end function

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}
↓
public static double code(double x) {
	return (Math.sin(x) / x) * (Math.tan((x * 0.5)) / x);
}

def code(x):
	return (1.0 - math.cos(x)) / (x * x)
↓
def code(x):
	return (math.sin(x) / x) * (math.tan((x * 0.5)) / x)

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end
↓
function code(x)
	return Float64(Float64(sin(x) / x) * Float64(tan(Float64(x * 0.5)) / x))
end

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end
↓
function tmp = code(x)
	tmp = (sin(x) / x) * (tan((x * 0.5)) / x);
end

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  2. Applied egg-rr16.1

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

  3. Applied egg-rr16.1

    \[\leadsto \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot e^{-\mathsf{log1p}\left(\cos x\right)}\right)}}{x \cdot x} \]

  4. Taylor expanded in x around inf 16.1

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

  5. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(0.5 \cdot x\right)}{x}} \]

  6. Final simplification0.1

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))