Average Error: 31.5 → 0.1
Time: 5.5s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\right)\right) \]
\frac{1 - \cos x}{x \cdot x}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\right)\right)

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (expm1 (log1p (* (/ (sin x) x) (/ (sin (/ x 2.0)) (* x (cos (/ x 2.0))))))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return expm1(log1p((sin(x) / x) * (sin(x / 2.0) / (x * cos(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 31.5

    \[\frac{1 - \cos x}{x \cdot x} \]

  2. Applied flip--_binary6431.7

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

  3. Simplified16.3

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

  4. Applied expm1-log1p-u_binary6416.3

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x \cdot x}\right)\right)} \]

  5. Simplified0.1

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\right)}\right) \]

  6. Applied tan-quot_binary640.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \frac{\color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}}{x}\right)\right) \]

  7. Applied associate-/l/_binary640.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}}\right)\right) \]

  8. Applied pow1_binary640.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\frac{\sin x}{x}\right)}^{1}} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\right)\right) \]

  9. Final simplification0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022095 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))