Average Error: 31.6 → 0.1
Time: 5.9s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x} \]
↓
\[\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(\frac{x}{2}\right)}} \]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x}}{\frac{x}{\tan \left(\frac{x}{2}\right)}}

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (/ (/ (sin x) x) (/ x (tan (/ x 2.0)))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) / x) / (x / tan(x / 2.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  2. Applied flip--_binary6431.8

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

  3. Simplified16.2

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

  4. Applied associate-/r*_binary6415.5

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

  5. Simplified0.1

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

  6. Applied associate-/l*_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2021280 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))