Average Error: 39.9 → 0.4
Time: 22.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\frac{\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) - \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right) - \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x} \cdot -2\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\frac{\left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) - \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right)}{\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right) - \cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x} \cdot -2\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)
double f(double x, double eps) {
        double r943380 = x;
        double r943381 = eps;
        double r943382 = r943380 + r943381;
        double r943383 = cos(r943382);
        double r943384 = cos(r943380);
        double r943385 = r943383 - r943384;
        return r943385;
}
↓
double f(double x, double eps) {
        double r943386 = x;
        double r943387 = cos(r943386);
        double r943388 = eps;
        double r943389 = 2.0;
        double r943390 = r943388 / r943389;
        double r943391 = sin(r943390);
        double r943392 = r943387 * r943391;
        double r943393 = r943392 * r943392;
        double r943394 = cos(r943390);
        double r943395 = sin(r943386);
        double r943396 = r943394 * r943395;
        double r943397 = r943396 * r943396;
        double r943398 = r943393 - r943397;
        double r943399 = r943392 - r943396;
        double r943400 = r943398 / r943399;
        double r943401 = -2.0;
        double r943402 = r943400 * r943401;
        double r943403 = r943402 * r943391;
        return r943403;
}

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 39.9

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

  3. Applied diff-cos34.1

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

  4. Simplified15.4

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

  5. Taylor expanded around inf 15.4

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

  6. Simplified15.4

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))