Average Error: 30.4 → 0.1
Time: 15.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}{x}
double f(double x) {
        double r827414 = 1.0;
        double r827415 = x;
        double r827416 = cos(r827415);
        double r827417 = r827414 - r827416;
        double r827418 = r827415 * r827415;
        double r827419 = r827417 / r827418;
        return r827419;
}
↓
double f(double x) {
        double r827420 = x;
        double r827421 = sin(r827420);
        double r827422 = r827421 / r827420;
        double r827423 = 2.0;
        double r827424 = r827420 / r827423;
        double r827425 = sin(r827424);
        double r827426 = r827422 * r827425;
        double r827427 = cos(r827424);
        double r827428 = r827426 / r827427;
        double r827429 = r827428 / r827420;
        return r827429;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.4

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

  3. Applied flip--30.5

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

  4. Simplified15.0

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

  6. Applied *-un-lft-identity15.0

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

  7. Applied times-frac15.0

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  10. Simplified0.1

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

  12. Applied associate-*r/0.1

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

  14. Applied tan-quot0.1

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

  15. Applied associate-*r/0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))