Average Error: 31.5 → 0.2
Time: 28.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03465278425284570168463105233058740850538 \lor \neg \left(x \le 0.03299786672642589929749235011513519566506\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right) + \mathsf{fma}\left(1, \frac{\frac{1}{x}}{x}, \frac{\frac{-1}{x}}{x} \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03465278425284570168463105233058740850538 \lor \neg \left(x \le 0.03299786672642589929749235011513519566506\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{-1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right) + \mathsf{fma}\left(1, \frac{\frac{1}{x}}{x}, \frac{\frac{-1}{x}}{x} \cdot \cos x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\

\end{array}
double f(double x) {
        double r53995 = 1.0;
        double r53996 = x;
        double r53997 = cos(r53996);
        double r53998 = r53995 - r53997;
        double r53999 = r53996 * r53996;
        double r54000 = r53998 / r53999;
        return r54000;
}
↓
double f(double x) {
        double r54001 = x;
        double r54002 = -0.0346527842528457;
        bool r54003 = r54001 <= r54002;
        double r54004 = 0.0329978667264259;
        bool r54005 = r54001 <= r54004;
        double r54006 = !r54005;
        bool r54007 = r54003 || r54006;
        double r54008 = -1.0;
        double r54009 = r54008 / r54001;
        double r54010 = r54009 / r54001;
        double r54011 = cos(r54001);
        double r54012 = 1.0;
        double r54013 = r54012 / r54001;
        double r54014 = r54013 / r54001;
        double r54015 = r54014 * r54011;
        double r54016 = fma(r54010, r54011, r54015);
        double r54017 = 1.0;
        double r54018 = r54010 * r54011;
        double r54019 = fma(r54017, r54014, r54018);
        double r54020 = r54016 + r54019;
        double r54021 = r54001 * r54001;
        double r54022 = -0.041666666666666664;
        double r54023 = 0.001388888888888889;
        double r54024 = 4.0;
        double r54025 = pow(r54001, r54024);
        double r54026 = 0.5;
        double r54027 = fma(r54023, r54025, r54026);
        double r54028 = fma(r54021, r54022, r54027);
        double r54029 = r54007 ? r54020 : r54028;
        return r54029;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0346527842528457 or 0.0329978667264259 < x

    1. Initial program 1.0

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

    3. Applied div-sub1.1

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

    4. Simplified1.1

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

    5. Simplified0.6

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

    7. Applied *-un-lft-identity0.6

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

    8. Applied div-inv0.6

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

    9. Applied times-frac0.5

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

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

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

    11. Applied div-inv0.5

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

    12. Applied times-frac0.5

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

    13. Applied prod-diff0.5

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

    14. Simplified0.5

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

    15. Simplified0.5

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

    if -0.0346527842528457 < x < 0.0329978667264259

    1. Initial program 62.2

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03465278425284570168463105233058740850538 \lor \neg \left(x \le 0.03299786672642589929749235011513519566506\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right) + \mathsf{fma}\left(1, \frac{\frac{1}{x}}{x}, \frac{\frac{-1}{x}}{x} \cdot \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))