Average Error: 39.5 → 0.4
Time: 4.7min
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \cos \left(\frac{\varepsilon}{2}\right) + \sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \cos \left(\frac{\varepsilon}{2}\right) + \sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)
double code(double x, double eps) {
	return ((double) (((double) cos(((double) (x + eps)))) - ((double) cos(x))));
}
↓
double code(double x, double eps) {
	return ((double) (-2.0 * ((double) (((double) (((double) (((double) sin((eps / 2.0))) * ((double) sin(x)))) * ((double) cos((eps / 2.0))))) + ((double) (((double) sin((eps / 2.0))) * ((double) (((double) cos(x)) * ((double) sin((eps / 2.0)))))))))));
}

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 39.5

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

  3. Applied diff-cos33.9

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

  4. Simplified14.8

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

  6. Applied sin-sum0.4

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

  7. Applied distribute-lft-in0.4

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

  9. Applied associate-*r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))