Average Error: 31.3 → 0.1
Time: 30.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{x}}{\cos \left(\frac{x}{2}\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{x}}{\cos \left(\frac{x}{2}\right)}
double f(double x) {
        double r934303 = 1.0;
        double r934304 = x;
        double r934305 = cos(r934304);
        double r934306 = r934303 - r934305;
        double r934307 = r934304 * r934304;
        double r934308 = r934306 / r934307;
        return r934308;
}
↓
double f(double x) {
        double r934309 = x;
        double r934310 = sin(r934309);
        double r934311 = r934310 / r934309;
        double r934312 = 2.0;
        double r934313 = r934309 / r934312;
        double r934314 = sin(r934313);
        double r934315 = r934311 * r934314;
        double r934316 = r934315 / r934309;
        double r934317 = cos(r934313);
        double r934318 = r934316 / r934317;
        return r934318;
}

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--31.4

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

  4. Simplified15.4

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

  6. Applied associate-/r*15.3

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

  7. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}}{x}\]
  8. Using strategy rm

  9. Applied tan-quot0.1

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

  10. Applied associate-*r/0.1

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

  11. Applied associate-/l/0.1

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{x \cdot \cos \left(\frac{x}{2}\right)}}\]
  12. Using strategy rm

  13. Applied associate-/r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))