Average Error: 31.6 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right)\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{1}{x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right)
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (* (/ 1.0 x) (* (/ (sin x) x) (tan (/ x 2.0)))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (1.0 / x) * ((sin(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. Using strategy rm

  3. Applied flip--_binary6431.7

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

  4. Simplified16.1

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

  6. Applied *-un-lft-identity_binary6416.1

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

  7. Applied times-frac_binary6415.5

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))