Average Error: 31.3 → 0.2
Time: 4.4s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\frac{\sin \left(x \cdot 0.5\right)}{x}}{\cos \left(x \cdot 0.5\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\frac{\sin \left(x \cdot 0.5\right)}{x}}{\cos \left(x \cdot 0.5\right)}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (* (/ (sin x) x) (/ (/ (sin (* x 0.5)) x) (cos (* x 0.5)))))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) / x) * ((sin(x * 0.5) / x) / cos(x * 0.5));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied flip--_binary6431.4

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

  4. Simplified15.4

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

  5. Taylor expanded around inf 15.3

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

  6. Simplified0.1

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

  7. Taylor expanded around inf 0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right) \cdot x}}\]

  8. Simplified0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{x \cdot \cos \left(x \cdot 0.5\right)}}\]
  9. Using strategy rm

  10. Applied associate-/r*_binary640.2

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\frac{\sin \left(x \cdot 0.5\right)}{x}}{\cos \left(x \cdot 0.5\right)}}\]

  11. Simplified0.2

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{x}}}{\cos \left(x \cdot 0.5\right)}\]

  12. Final simplification0.2

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\frac{\sin \left(x \cdot 0.5\right)}{x}}{\cos \left(x \cdot 0.5\right)}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))