Average Error: 31.4 → 0.1
Time: 6.2s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (/ (* (sin x) (/ (tan (/ x 2.0)) x)) x))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) * (tan(x / 2.0) / x)) / x;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

    \[\frac{1 - \cos x}{x \cdot x}\]
  2. Using strategy rm

  3. Applied flip--_binary64_5331.5

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

  4. Simplified16.0

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

  5. Taylor expanded around inf 15.8

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

  6. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}\]
  7. Using strategy rm

  8. Applied associate-*l/_binary64_210.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021077 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))