Average Error: 39.9 → 0.4
Time: 22.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \cos \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x + \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \cos \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x + \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)
double f(double x, double eps) {
        double r833943 = x;
        double r833944 = eps;
        double r833945 = r833943 + r833944;
        double r833946 = cos(r833945);
        double r833947 = cos(r833943);
        double r833948 = r833946 - r833947;
        return r833948;
}
↓
double f(double x, double eps) {
        double r833949 = eps;
        double r833950 = 2.0;
        double r833951 = r833949 / r833950;
        double r833952 = sin(r833951);
        double r833953 = -2.0;
        double r833954 = r833952 * r833953;
        double r833955 = cos(r833951);
        double r833956 = r833954 * r833955;
        double r833957 = x;
        double r833958 = sin(r833957);
        double r833959 = r833956 * r833958;
        double r833960 = cos(r833957);
        double r833961 = r833960 * r833952;
        double r833962 = r833961 * r833954;
        double r833963 = r833959 + r833962;
        return r833963;
}

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

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

  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

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

  8. Applied sin-sum0.4

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

  9. Applied distribute-lft-in0.4

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

  11. Applied associate-*r*0.4

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

  12. Final simplification0.4

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

Reproduce

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