Average Error: 39.1 → 0.4
Time: 21.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\frac{\left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) - \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}{\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x - \cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)} \cdot -2\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\frac{\left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) - \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}{\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x - \cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)} \cdot -2\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)
double f(double x, double eps) {
        double r1946300 = x;
        double r1946301 = eps;
        double r1946302 = r1946300 + r1946301;
        double r1946303 = cos(r1946302);
        double r1946304 = cos(r1946300);
        double r1946305 = r1946303 - r1946304;
        return r1946305;
}
↓
double f(double x, double eps) {
        double r1946306 = eps;
        double r1946307 = 2.0;
        double r1946308 = r1946306 / r1946307;
        double r1946309 = cos(r1946308);
        double r1946310 = x;
        double r1946311 = sin(r1946310);
        double r1946312 = r1946309 * r1946311;
        double r1946313 = r1946312 * r1946312;
        double r1946314 = cos(r1946310);
        double r1946315 = sin(r1946308);
        double r1946316 = r1946314 * r1946315;
        double r1946317 = r1946316 * r1946316;
        double r1946318 = r1946313 - r1946317;
        double r1946319 = r1946312 - r1946316;
        double r1946320 = r1946318 / r1946319;
        double r1946321 = -2.0;
        double r1946322 = r1946320 * r1946321;
        double r1946323 = r1946322 * r1946315;
        return r1946323;
}

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 39.1

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

  3. Applied diff-cos33.8

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified14.9

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))