cos2 (problem 3.4.1)

Average Error: 31.4 → 0.3

Time: 31.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02740531929745840256096300890931161120534:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \mathbf{elif}\;x \le 0.03110675776409119533405522872726578498259:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} - \frac{1}{24}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \cos x} \cdot \frac{\frac{\sqrt{1 - \cos x}}{x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r877244 = 1.0;
        double r877245 = x;
        double r877246 = cos(r877245);
        double r877247 = r877244 - r877246;
        double r877248 = r877245 * r877245;
        double r877249 = r877247 / r877248;
        return r877249;
}

↓

double f(double x) {
        double r877250 = x;
        double r877251 = -0.027405319297458403;
        bool r877252 = r877250 <= r877251;
        double r877253 = 1.0;
        double r877254 = cos(r877250);
        double r877255 = r877253 - r877254;
        double r877256 = exp(r877255);
        double r877257 = log(r877256);
        double r877258 = sqrt(r877257);
        double r877259 = r877258 / r877250;
        double r877260 = r877259 / r877250;
        double r877261 = sqrt(r877255);
        double r877262 = r877260 * r877261;
        double r877263 = 0.031106757764091195;
        bool r877264 = r877250 <= r877263;
        double r877265 = r877250 * r877250;
        double r877266 = 0.001388888888888889;
        double r877267 = r877265 * r877266;
        double r877268 = 0.041666666666666664;
        double r877269 = r877267 - r877268;
        double r877270 = r877265 * r877269;
        double r877271 = 0.5;
        double r877272 = r877270 + r877271;
        double r877273 = r877261 / r877250;
        double r877274 = r877273 / r877250;
        double r877275 = r877261 * r877274;
        double r877276 = r877264 ? r877272 : r877275;
        double r877277 = r877252 ? r877262 : r877276;
        return r877277;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.027405319297458403

   1. Initial program 1.2

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

   3. Applied associate-/r* 0.5

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

   5. Applied *-un-lft-identity 0.5

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

   6. Applied *-un-lft-identity 0.5

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

   7. Applied add-sqr-sqrt 0.6

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

   8. Applied times-frac 0.6

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

   9. Applied times-frac 0.6

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

   10. Simplified 0.6

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

   12. Applied add-log-exp 0.6

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

   13. Applied add-log-exp 0.6

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

   14. Applied diff-log 0.6

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

   15. Simplified 0.6

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

   if -0.027405319297458403 < x < 0.031106757764091195

   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. Simplified 0.0

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

   if 0.031106757764091195 < x

   1. Initial program 1.0

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

   3. Applied associate-/r* 0.5

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

   5. Applied *-un-lft-identity 0.5

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

   6. Applied *-un-lft-identity 0.5

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

   7. Applied add-sqr-sqrt 0.6

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

   8. Applied times-frac 0.6

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

   9. Applied times-frac 0.6

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

   10. Simplified 0.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02740531929745840256096300890931161120534:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \mathbf{elif}\;x \le 0.03110675776409119533405522872726578498259:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} - \frac{1}{24}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \cos x} \cdot \frac{\frac{\sqrt{1 - \cos x}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))