Average Error: 31.0 → 0.3
Time: 45.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\left(1 + \left(\cos x \cdot \cos x - \cos x\right)\right) \cdot \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x \cdot \left(\cos x \cdot \cos x\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\left(1 + \left(\cos x \cdot \cos x - \cos x\right)\right) \cdot \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{1 + \cos x \cdot \left(\cos x \cdot \cos x\right)}
double f(double x) {
        double r4168588 = 1.0;
        double r4168589 = x;
        double r4168590 = cos(r4168589);
        double r4168591 = r4168588 - r4168590;
        double r4168592 = r4168589 * r4168589;
        double r4168593 = r4168591 / r4168592;
        return r4168593;
}
↓
double f(double x) {
        double r4168594 = 1.0;
        double r4168595 = x;
        double r4168596 = cos(r4168595);
        double r4168597 = r4168596 * r4168596;
        double r4168598 = r4168597 - r4168596;
        double r4168599 = r4168594 + r4168598;
        double r4168600 = sin(r4168595);
        double r4168601 = r4168600 / r4168595;
        double r4168602 = r4168601 * r4168601;
        double r4168603 = r4168596 * r4168597;
        double r4168604 = r4168594 + r4168603;
        double r4168605 = r4168602 / r4168604;
        double r4168606 = r4168599 * r4168605;
        return r4168606;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  3. Applied flip--31.1

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

  4. Applied associate-/l/31.1

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

  5. Simplified15.6

    \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
  6. Using strategy rm

  7. Applied flip3-+15.6

    \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

  8. Taylor expanded around -inf 15.6

    \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2} \cdot \left(\left({\left(\cos x\right)}^{2} + 1\right) - \cos x\right)}{{x}^{2} \cdot \left({\left(\cos x\right)}^{3} + 1\right)}}\]

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))