Average Error: 39.8 → 0.4
Time: 19.8s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(-2 \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) + \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(-2 \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) + \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)
double f(double x, double eps) {
        double r389366 = x;
        double r389367 = eps;
        double r389368 = r389366 + r389367;
        double r389369 = cos(r389368);
        double r389370 = cos(r389366);
        double r389371 = r389369 - r389370;
        return r389371;
}
↓
double f(double x, double eps) {
        double r389372 = -2.0;
        double r389373 = eps;
        double r389374 = 0.5;
        double r389375 = r389373 * r389374;
        double r389376 = cos(r389375);
        double r389377 = x;
        double r389378 = r389377 + r389377;
        double r389379 = r389378 * r389374;
        double r389380 = sin(r389379);
        double r389381 = r389376 * r389380;
        double r389382 = cos(r389379);
        double r389383 = sin(r389375);
        double r389384 = r389382 * r389383;
        double r389385 = r389381 + r389384;
        double r389386 = r389372 * r389385;
        double r389387 = r389386 * r389383;
        return r389387;
}

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 39.8

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

  3. Applied diff-cos34.2

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

  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 15.3

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

  6. Simplified15.3

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

  8. Applied distribute-rgt-in15.3

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

  9. Applied sin-sum0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))