Average Error: 31.0 → 0.1
Time: 3.4s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (/ (* (/ (sin x) x) (tan (/ x 2.0))) x))
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}
↓
double code(double x) {
	return (((double) ((((double) sin(x)) / x) * ((double) tan((x / 2.0))))) / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  3. Applied flip--_binary6431.1

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

  4. Simplified15.1

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

  6. Applied associate-/r*_binary6414.6

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))