Average Error: 31.4 → 0.1
Time: 6.3s
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. Applied flip--_binary6431.6

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

  3. Simplified15.8

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

  4. Applied *-un-lft-identity_binary6415.8

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

  5. Applied times-frac_binary6415.8

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

  6. Applied associate-/l*_binary6415.9

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

  7. Simplified0.4

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

  8. Applied associate-/r/_binary640.4

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

  9. Applied associate-/r*_binary640.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2021220 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))