Average Error: 30.2 → 0.1
Time: 37.2s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}
double f(double x) {
        double r993484 = 1.0;
        double r993485 = x;
        double r993486 = cos(r993485);
        double r993487 = r993484 - r993486;
        double r993488 = r993485 * r993485;
        double r993489 = r993487 / r993488;
        return r993489;
}
↓
double f(double x) {
        double r993490 = x;
        double r993491 = 2.0;
        double r993492 = r993490 / r993491;
        double r993493 = tan(r993492);
        double r993494 = r993493 / r993490;
        double r993495 = sin(r993490);
        double r993496 = r993494 * r993495;
        double r993497 = r993496 / r993490;
        return r993497;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.3

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

  4. Simplified14.8

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

  6. Applied *-un-lft-identity14.8

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

  7. Applied *-un-lft-identity14.8

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

  8. Applied distribute-lft-out14.8

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

  9. Applied times-frac14.8

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

  10. Applied times-frac0.3

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

  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. Using strategy rm

  14. Applied associate-*l/0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))