Average Error: 30.9 → 0.3
Time: 40.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\frac{1 + {\left(\cos x\right)}^{3}}{\left(1 - \cos x\right) + \cos x \cdot \cos x}}\]
double f(double x) {
        double r2406393 = 1.0;
        double r2406394 = x;
        double r2406395 = cos(r2406394);
        double r2406396 = r2406393 - r2406395;
        double r2406397 = r2406394 * r2406394;
        double r2406398 = r2406396 / r2406397;
        return r2406398;
}
↓
double f(double x) {
        double r2406399 = x;
        double r2406400 = sin(r2406399);
        double r2406401 = r2406400 / r2406399;
        double r2406402 = r2406401 * r2406401;
        double r2406403 = 1.0;
        double r2406404 = cos(r2406399);
        double r2406405 = 3.0;
        double r2406406 = pow(r2406404, r2406405);
        double r2406407 = r2406403 + r2406406;
        double r2406408 = r2406403 - r2406404;
        double r2406409 = r2406404 * r2406404;
        double r2406410 = r2406408 + r2406409;
        double r2406411 = r2406407 / r2406410;
        double r2406412 = r2406402 / r2406411;
        return r2406412;
}


\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\frac{1 + {\left(\cos x\right)}^{3}}{\left(1 - \cos x\right) + \cos x \cdot \cos x}}

Error

Bits error versus x

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 flip3-+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))