Average Error: 31.2 → 0.1
Time: 4.5s
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 31.2

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

  3. Applied flip--_binary64_5431.3

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

  4. Simplified15.7

    \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x}\]
  5. Using strategy rm

  6. Applied associate-/r*_binary64_2315.1

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

  7. Simplified0.1

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

  9. Applied div-inv_binary64_760.2

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

  10. Applied associate-*l*_binary64_200.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020280 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))