Average Error: 31.3 → 0.1
Time: 6.2s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\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 (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (sin(x) / x) * (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 31.3

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

  3. Applied flip--_binary64_60031.4

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

  4. Simplified15.7

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

  6. Applied *-un-lft-identity_binary64_57415.7

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

  7. Applied *-un-lft-identity_binary64_57415.7

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

  8. Applied distribute-lft-out_binary64_61815.7

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

  9. Applied times-frac_binary64_56915.7

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

  10. Applied times-frac_binary64_5690.3

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

  11. Simplified0.3

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020231 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))