Average Error: 31.3 → 0.1
Time: 1.4m
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{x \cdot \cos \left(\frac{1}{2} \cdot x\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{x \cdot \cos \left(\frac{1}{2} \cdot x\right)}
double f(double x) {
        double r1414145 = 1.0;
        double r1414146 = x;
        double r1414147 = cos(r1414146);
        double r1414148 = r1414145 - r1414147;
        double r1414149 = r1414146 * r1414146;
        double r1414150 = r1414148 / r1414149;
        return r1414150;
}
↓
double f(double x) {
        double r1414151 = x;
        double r1414152 = sin(r1414151);
        double r1414153 = r1414152 / r1414151;
        double r1414154 = 0.5;
        double r1414155 = r1414154 * r1414151;
        double r1414156 = sin(r1414155);
        double r1414157 = cos(r1414155);
        double r1414158 = r1414151 * r1414157;
        double r1414159 = r1414156 / r1414158;
        double r1414160 = r1414153 * r1414159;
        return r1414160;
}

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. Simplified15.1

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

  6. Applied *-un-lft-identity15.1

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

  7. Applied *-un-lft-identity15.1

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

  8. Applied distribute-lft-out15.1

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

  9. Applied times-frac15.1

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

  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. Taylor expanded around -inf 0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))