Average Error: 31.1 → 0.3
Time: 19.0s
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 r23344 = 1.0;
        double r23345 = x;
        double r23346 = cos(r23345);
        double r23347 = r23344 - r23346;
        double r23348 = r23345 * r23345;
        double r23349 = r23347 / r23348;
        return r23349;
}
↓
double f(double x) {
        double r23350 = x;
        double r23351 = -0.028173748830909656;
        bool r23352 = r23350 <= r23351;
        double r23353 = 1.0;
        double r23354 = r23353 / r23350;
        double r23355 = 1.0;
        double r23356 = 3.0;
        double r23357 = pow(r23355, r23356);
        double r23358 = cos(r23350);
        double r23359 = pow(r23358, r23356);
        double r23360 = r23357 - r23359;
        double r23361 = r23355 + r23358;
        double r23362 = r23358 * r23361;
        double r23363 = fma(r23355, r23355, r23362);
        double r23364 = r23350 * r23363;
        double r23365 = r23360 / r23364;
        double r23366 = r23354 * r23365;
        double r23367 = 0.03577299125405769;
        bool r23368 = r23350 <= r23367;
        double r23369 = 0.001388888888888889;
        double r23370 = 4.0;
        double r23371 = pow(r23350, r23370);
        double r23372 = 0.5;
        double r23373 = fma(r23369, r23371, r23372);
        double r23374 = 0.041666666666666664;
        double r23375 = 2.0;
        double r23376 = pow(r23350, r23375);
        double r23377 = r23374 * r23376;
        double r23378 = r23373 - r23377;
        double r23379 = sqrt(r23350);
        double r23380 = r23354 / r23379;
        double r23381 = r23355 / r23379;
        double r23382 = r23358 / r23379;
        double r23383 = r23381 - r23382;
        double r23384 = r23380 * r23383;
        double r23385 = r23368 ? r23378 : r23384;
        double r23386 = r23352 ? r23366 : r23385;
        return r23386;
}

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)))