Average Error: 31.0 → 0.3
Time: 36.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\log_* (1 + (e^{\frac{\sin x}{x}} - 1)^*) \cdot \frac{\sin x}{x}}{1 + \cos x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\log_* (1 + (e^{\frac{\sin x}{x}} - 1)^*) \cdot \frac{\sin x}{x}}{1 + \cos x}
double f(double x) {
        double r2086587 = 1.0;
        double r2086588 = x;
        double r2086589 = cos(r2086588);
        double r2086590 = r2086587 - r2086589;
        double r2086591 = r2086588 * r2086588;
        double r2086592 = r2086590 / r2086591;
        return r2086592;
}
↓
double f(double x) {
        double r2086593 = x;
        double r2086594 = sin(r2086593);
        double r2086595 = r2086594 / r2086593;
        double r2086596 = expm1(r2086595);
        double r2086597 = log1p(r2086596);
        double r2086598 = r2086597 * r2086595;
        double r2086599 = 1.0;
        double r2086600 = cos(r2086593);
        double r2086601 = r2086599 + r2086600;
        double r2086602 = r2086598 / r2086601;
        return r2086602;
}

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. Simplified14.9

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

  6. Taylor expanded around -inf 14.9

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

  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 log1p-expm1-u0.3

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \color{blue}{\log_* (1 + (e^{\frac{\sin x}{x}} - 1)^*)}}{\cos x + 1}\]

  10. Final simplification0.3

    \[\leadsto \frac{\log_* (1 + (e^{\frac{\sin x}{x}} - 1)^*) \cdot \frac{\sin x}{x}}{1 + \cos x}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))