Average Error: 39.0 → 0.4
Time: 17.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \frac{\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) - \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) - \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \frac{\left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) - \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right) - \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)
double f(double x, double eps) {
        double r1913316 = x;
        double r1913317 = eps;
        double r1913318 = r1913316 + r1913317;
        double r1913319 = cos(r1913318);
        double r1913320 = cos(r1913316);
        double r1913321 = r1913319 - r1913320;
        return r1913321;
}
↓
double f(double x, double eps) {
        double r1913322 = -2.0;
        double r1913323 = eps;
        double r1913324 = 0.5;
        double r1913325 = r1913323 * r1913324;
        double r1913326 = sin(r1913325);
        double r1913327 = x;
        double r1913328 = cos(r1913327);
        double r1913329 = r1913328 * r1913326;
        double r1913330 = r1913329 * r1913329;
        double r1913331 = cos(r1913325);
        double r1913332 = sin(r1913327);
        double r1913333 = r1913331 * r1913332;
        double r1913334 = r1913333 * r1913333;
        double r1913335 = r1913330 - r1913334;
        double r1913336 = r1913329 - r1913333;
        double r1913337 = r1913335 / r1913336;
        double r1913338 = r1913326 * r1913337;
        double r1913339 = r1913322 * r1913338;
        return r1913339;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.0

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

  3. Applied diff-cos33.4

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.2

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

  6. Simplified15.2

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))