Average Error: 31.3 → 0.2
Time: 1.0m
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}
double f(double x) {
        double r3500350 = 1.0;
        double r3500351 = x;
        double r3500352 = cos(r3500351);
        double r3500353 = r3500350 - r3500352;
        double r3500354 = r3500351 * r3500351;
        double r3500355 = r3500353 / r3500354;
        return r3500355;
}
↓
double f(double x) {
        double r3500356 = x;
        double r3500357 = 0.5;
        double r3500358 = r3500356 * r3500357;
        double r3500359 = sin(r3500358);
        double r3500360 = r3500359 / r3500356;
        double r3500361 = cos(r3500358);
        double r3500362 = sin(r3500356);
        double r3500363 = r3500362 / r3500356;
        double r3500364 = r3500361 / r3500363;
        double r3500365 = r3500360 / r3500364;
        return r3500365;
}

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. Simplified15.7

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

  5. Taylor expanded around inf 15.7

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

  6. Simplified15.6

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

  7. Taylor expanded around inf 15.6

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))