Average Error: 30.9 → 0.3
Time: 48.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \sin x}{\cos x \cdot x + x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \sin x}{\cos x \cdot x + x}
double f(double x) {
        double r3707554 = 1.0;
        double r3707555 = x;
        double r3707556 = cos(r3707555);
        double r3707557 = r3707554 - r3707556;
        double r3707558 = r3707555 * r3707555;
        double r3707559 = r3707557 / r3707558;
        return r3707559;
}
↓
double f(double x) {
        double r3707560 = x;
        double r3707561 = sin(r3707560);
        double r3707562 = r3707561 / r3707560;
        double r3707563 = r3707562 * r3707561;
        double r3707564 = cos(r3707560);
        double r3707565 = r3707564 * r3707560;
        double r3707566 = r3707565 + r3707560;
        double r3707567 = r3707563 / r3707566;
        return r3707567;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied flip--31.0

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

  4. Applied associate-/l/31.0

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

  5. Simplified14.7

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

  6. Taylor expanded around inf 14.7

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

  7. Simplified0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-/l/0.3

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

  11. Taylor expanded around inf 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))