cos2 (problem 3.4.1)

Average Error: 31.5 → 0.3

Time: 12.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03465278425284570168463105233058740850538:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03597822376690750639793847653891134541482:\\ \;\;\;\;\left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right) + {x}^{4} \cdot \frac{1}{720}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r18993 = 1.0;
        double r18994 = x;
        double r18995 = cos(r18994);
        double r18996 = r18993 - r18995;
        double r18997 = r18994 * r18994;
        double r18998 = r18996 / r18997;
        return r18998;
}

↓

double f(double x) {
        double r18999 = x;
        double r19000 = -0.0346527842528457;
        bool r19001 = r18999 <= r19000;
        double r19002 = 1.0;
        double r19003 = r19002 / r18999;
        double r19004 = r19003 / r18999;
        double r19005 = 1.0;
        double r19006 = cos(r18999);
        double r19007 = r19005 - r19006;
        double r19008 = r19004 * r19007;
        double r19009 = 0.035978223766907506;
        bool r19010 = r18999 <= r19009;
        double r19011 = 0.5;
        double r19012 = 0.041666666666666664;
        double r19013 = 2.0;
        double r19014 = pow(r18999, r19013);
        double r19015 = r19012 * r19014;
        double r19016 = r19011 - r19015;
        double r19017 = 4.0;
        double r19018 = pow(r18999, r19017);
        double r19019 = 0.001388888888888889;
        double r19020 = r19018 * r19019;
        double r19021 = r19016 + r19020;
        double r19022 = r19005 / r18999;
        double r19023 = r19006 / r18999;
        double r19024 = r19022 - r19023;
        double r19025 = r19024 / r18999;
        double r19026 = r19010 ? r19021 : r19025;
        double r19027 = r19001 ? r19008 : r19026;
        return r19027;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.0346527842528457

   1. Initial program 0.9

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

   3. Applied div-inv 1.0

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

   4. Simplified 0.5

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

   if -0.0346527842528457 < x < 0.035978223766907506

   1. Initial program 62.2

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

   3. Applied div-inv 62.2

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

   4. Simplified 62.2

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

   6. Applied pow1 62.2

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

   7. Applied pow1 62.2

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

   8. Applied pow-prod-down 62.2

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

   9. Simplified 61.3

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

   11. Applied div-sub 61.3

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

   12. Taylor expanded around 0 0.0

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

   13. Simplified 0.0

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

   if 0.035978223766907506 < x

   1. Initial program 1.0

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

   3. Applied div-inv 1.0

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

   4. Simplified 0.5

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

   6. Applied pow1 0.5

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

   7. Applied pow1 0.5

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

   8. Applied pow-prod-down 0.5

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

   9. Simplified 0.4

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

   11. Applied div-sub 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03465278425284570168463105233058740850538:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.03597822376690750639793847653891134541482:\\ \;\;\;\;\left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right) + {x}^{4} \cdot \frac{1}{720}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

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