Average Error: 31.3 → 0.3
Time: 17.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02309310542475334626644212221435736864805:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(x + x\right)\right)}} \cdot \frac{1}{\left(1 \cdot \cos x + \cos x \cdot \cos x\right) + 1 \cdot 1}\\ \mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02309310542475334626644212221435736864805:\\
\;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(x + x\right)\right)}} \cdot \frac{1}{\left(1 \cdot \cos x + \cos x \cdot \cos x\right) + 1 \cdot 1}\\

\mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\

\end{array}
double f(double x) {
        double r727649 = 1.0;
        double r727650 = x;
        double r727651 = cos(r727650);
        double r727652 = r727649 - r727651;
        double r727653 = r727650 * r727650;
        double r727654 = r727652 / r727653;
        return r727654;
}
↓
double f(double x) {
        double r727655 = x;
        double r727656 = -0.023093105424753346;
        bool r727657 = r727655 <= r727656;
        double r727658 = 1.0;
        double r727659 = r727658 / r727655;
        double r727660 = 1.0;
        double r727661 = r727660 * r727660;
        double r727662 = r727660 * r727661;
        double r727663 = cos(r727655);
        double r727664 = 0.5;
        double r727665 = r727655 + r727655;
        double r727666 = cos(r727665);
        double r727667 = r727664 * r727666;
        double r727668 = r727664 + r727667;
        double r727669 = r727663 * r727668;
        double r727670 = r727662 - r727669;
        double r727671 = r727655 / r727670;
        double r727672 = r727659 / r727671;
        double r727673 = r727660 * r727663;
        double r727674 = r727663 * r727663;
        double r727675 = r727673 + r727674;
        double r727676 = r727675 + r727661;
        double r727677 = r727658 / r727676;
        double r727678 = r727672 * r727677;
        double r727679 = 0.02891102314655495;
        bool r727680 = r727655 <= r727679;
        double r727681 = r727655 * r727655;
        double r727682 = -0.041666666666666664;
        double r727683 = 0.001388888888888889;
        double r727684 = r727683 * r727681;
        double r727685 = fma(r727681, r727684, r727664);
        double r727686 = fma(r727681, r727682, r727685);
        double r727687 = r727660 - r727663;
        double r727688 = r727655 / r727687;
        double r727689 = r727658 / r727688;
        double r727690 = r727659 * r727689;
        double r727691 = r727680 ? r727686 : r727690;
        double r727692 = r727657 ? r727678 : r727691;
        return r727692;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.023093105424753346

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied clear-num0.5

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

    8. Applied flip3--0.6

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

    9. Applied associate-/r/0.5

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

    10. Applied *-un-lft-identity0.5

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

    11. Applied times-frac0.6

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

    12. Applied associate-*r*0.6

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

    13. Simplified0.6

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{x}{\left(1 \cdot 1\right) \cdot 1 - \left(\cos x \cdot \cos x\right) \cdot \cos x}}} \cdot \frac{1}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\]
    14. Using strategy rm

    15. Applied sqr-cos0.5

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

    16. Simplified0.5

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

    if -0.023093105424753346 < x < 0.02891102314655495

    1. Initial program 62.2

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

    3. Applied *-un-lft-identity62.2

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

    4. Applied times-frac61.2

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

    5. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \frac{1}{2}\right)\right)}\]

    if 0.02891102314655495 < x

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.5

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

    6. Applied clear-num0.5

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02309310542475334626644212221435736864805:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 \cdot \left(1 \cdot 1\right) - \cos x \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(x + x\right)\right)}} \cdot \frac{1}{\left(1 \cdot \cos x + \cos x \cdot \cos x\right) + 1 \cdot 1}\\ \mathbf{elif}\;x \le 0.02891102314655494970319082881360372994095:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))