Average Error: 31.1 → 0.3
Time: 17.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.03577299125405768859264910020101524423808:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{x}} - \frac{\cos x}{\sqrt{x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02817374883090965551057927029887650860474:\\
\;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\

\mathbf{elif}\;x \le 0.03577299125405768859264910020101524423808:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r23255 = 1.0;
        double r23256 = x;
        double r23257 = cos(r23256);
        double r23258 = r23255 - r23257;
        double r23259 = r23256 * r23256;
        double r23260 = r23258 / r23259;
        return r23260;
}
↓
double f(double x) {
        double r23261 = x;
        double r23262 = -0.028173748830909656;
        bool r23263 = r23261 <= r23262;
        double r23264 = 1.0;
        double r23265 = r23264 / r23261;
        double r23266 = 1.0;
        double r23267 = 3.0;
        double r23268 = pow(r23266, r23267);
        double r23269 = cos(r23261);
        double r23270 = pow(r23269, r23267);
        double r23271 = r23268 - r23270;
        double r23272 = r23266 + r23269;
        double r23273 = r23269 * r23272;
        double r23274 = fma(r23266, r23266, r23273);
        double r23275 = r23261 * r23274;
        double r23276 = r23271 / r23275;
        double r23277 = r23265 * r23276;
        double r23278 = 0.03577299125405769;
        bool r23279 = r23261 <= r23278;
        double r23280 = 0.001388888888888889;
        double r23281 = 4.0;
        double r23282 = pow(r23261, r23281);
        double r23283 = 0.5;
        double r23284 = fma(r23280, r23282, r23283);
        double r23285 = 0.041666666666666664;
        double r23286 = 2.0;
        double r23287 = pow(r23261, r23286);
        double r23288 = r23285 * r23287;
        double r23289 = r23284 - r23288;
        double r23290 = sqrt(r23261);
        double r23291 = r23265 / r23290;
        double r23292 = r23266 / r23290;
        double r23293 = r23269 / r23290;
        double r23294 = r23292 - r23293;
        double r23295 = r23291 * r23294;
        double r23296 = r23279 ? r23289 : r23295;
        double r23297 = r23263 ? r23277 : r23296;
        return r23297;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.028173748830909656

    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 flip3--0.5

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

    7. Applied associate-/l/0.5

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

    8. Simplified0.5

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

    if -0.028173748830909656 < x < 0.03577299125405769

    1. Initial program 62.3

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

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

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

    4. Applied times-frac61.3

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

    6. Applied div-sub61.4

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

    7. Taylor expanded around 0 0.0

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

    8. Simplified0.0

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

    if 0.03577299125405769 < 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 div-sub0.7

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

    8. Applied add-sqr-sqrt0.7

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

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

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

    10. Applied times-frac0.7

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

    11. Applied add-sqr-sqrt0.8

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

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

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

    13. Applied times-frac0.8

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

    14. Applied distribute-lft-out--0.7

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

    15. Applied associate-*r*0.7

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

    16. Simplified0.7

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.03577299125405768859264910020101524423808:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{x}} - \frac{\cos x}{\sqrt{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))