Average Error: 32.2 → 0.3
Time: 9.8s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\ \;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\
\;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\

\mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\

\end{array}
double f(double x) {
        double r33573 = 1.0;
        double r33574 = x;
        double r33575 = cos(r33574);
        double r33576 = r33573 - r33575;
        double r33577 = r33574 * r33574;
        double r33578 = r33576 / r33577;
        return r33578;
}
↓
double f(double x) {
        double r33579 = x;
        double r33580 = -0.02623489072028306;
        bool r33581 = r33579 <= r33580;
        double r33582 = 1.0;
        double r33583 = r33582 / r33579;
        double r33584 = 1.0;
        double r33585 = cos(r33579);
        double r33586 = r33584 - r33585;
        double r33587 = sqrt(r33586);
        double r33588 = r33587 / r33579;
        double r33589 = r33587 * r33588;
        double r33590 = r33583 * r33589;
        double r33591 = 0.034744172255630555;
        bool r33592 = r33579 <= r33591;
        double r33593 = r33579 * r33579;
        double r33594 = -0.041666666666666664;
        double r33595 = 0.001388888888888889;
        double r33596 = 4.0;
        double r33597 = pow(r33579, r33596);
        double r33598 = 0.5;
        double r33599 = fma(r33595, r33597, r33598);
        double r33600 = fma(r33593, r33594, r33599);
        double r33601 = exp(r33586);
        double r33602 = log(r33601);
        double r33603 = r33602 / r33579;
        double r33604 = r33583 * r33603;
        double r33605 = r33592 ? r33600 : r33604;
        double r33606 = r33581 ? r33590 : r33605;
        return r33606;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02623489072028306

    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-frac0.6

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

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

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

    7. Applied add-sqr-sqrt0.7

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

    8. Applied times-frac0.7

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

    9. Simplified0.7

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

    if -0.02623489072028306 < x < 0.034744172255630555

    1. Initial program 62.4

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

    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

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

    if 0.034744172255630555 < x

    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 add-log-exp0.6

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

    7. Applied add-log-exp0.6

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

    8. Applied diff-log0.7

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

    9. Simplified0.6

      \[\leadsto \frac{1}{x} \cdot \frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02623489072028305918449220257571141701192:\\ \;\;\;\;\frac{1}{x} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{\sqrt{1 - \cos x}}{x}\right)\\ \mathbf{elif}\;x \le 0.03474417225563055516834864988595654722303:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))