2cos (problem 3.3.5)

Average Error: 39.1 → 0.5

Time: 22.1s

Precision: 64

Math

\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[-2 \cdot \left(\frac{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}{\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

C

double f(double x, double eps) {
        double r34390 = x;
        double r34391 = eps;
        double r34392 = r34390 + r34391;
        double r34393 = cos(r34392);
        double r34394 = cos(r34390);
        double r34395 = r34393 - r34394;
        return r34395;
}

↓

double f(double x, double eps) {
        double r34396 = -2.0;
        double r34397 = 0.5;
        double r34398 = 2.0;
        double r34399 = x;
        double r34400 = r34398 * r34399;
        double r34401 = r34397 * r34400;
        double r34402 = sin(r34401);
        double r34403 = eps;
        double r34404 = r34397 * r34403;
        double r34405 = cos(r34404);
        double r34406 = r34402 * r34405;
        double r34407 = r34406 * r34406;
        double r34408 = r34400 * r34397;
        double r34409 = cos(r34408);
        double r34410 = r34403 * r34397;
        double r34411 = sin(r34410);
        double r34412 = r34409 * r34411;
        double r34413 = r34412 * r34412;
        double r34414 = r34407 - r34413;
        double r34415 = r34406 - r34412;
        double r34416 = r34414 / r34415;
        double r34417 = sin(r34404);
        double r34418 = r34416 * r34417;
        double r34419 = r34396 * r34418;
        return r34419;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Initial program 39.1

   \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm

3. Applied diff-cos 33.5

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

4. Simplified 14.9

   \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

5. Taylor expanded around inf 14.9

   \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
6. Using strategy rm

7. Applied distribute-lft-in 14.9

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

8. Applied sin-sum 0.4

   \[\leadsto -2 \cdot \left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) + \cos \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

9. Simplified 0.4

   \[\leadsto -2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) + \color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
10. Using strategy rm

11. Applied flip-+ 0.5

    \[\leadsto -2 \cdot \left(\color{blue}{\frac{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}{\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

12. Final simplification 0.5

    \[\leadsto -2 \cdot \left(\frac{\left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right)\right) - \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)}{\sin \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))