Average Error: 31.4 → 0.4
Time: 16.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin x}{\cos x + 1}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin x}{\cos x + 1}
double f(double x) {
        double r836421 = 1.0;
        double r836422 = x;
        double r836423 = cos(r836422);
        double r836424 = r836421 - r836423;
        double r836425 = r836422 * r836422;
        double r836426 = r836424 / r836425;
        return r836426;
}
↓
double f(double x) {
        double r836427 = x;
        double r836428 = sin(r836427);
        double r836429 = r836428 / r836427;
        double r836430 = r836429 / r836427;
        double r836431 = r836430 * r836428;
        double r836432 = cos(r836427);
        double r836433 = 1.0;
        double r836434 = r836432 + r836433;
        double r836435 = r836431 / r836434;
        return r836435;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  3. Applied flip--31.5

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

  4. Simplified15.7

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

  5. Taylor expanded around inf 15.5

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

  6. Simplified0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied add-sqr-sqrt31.4

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

  10. Applied times-frac31.4

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

  11. Applied *-un-lft-identity31.4

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

  12. Applied add-sqr-sqrt31.5

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

  13. Applied times-frac31.5

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

  14. Applied swap-sqr31.5

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

  15. Simplified31.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))