Average Error: 39.6 → 0.4
Time: 6.0s
Precision: binary64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[-2 \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + \cos x \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
-2 \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + \cos x \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\right)
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (+
   (* (* (sin x) (cos (* eps 0.5))) (sin (* eps 0.5)))
   (* (cos x) (pow (sin (* eps 0.5)) 2.0)))))
double code(double x, double eps) {
	return cos(x + eps) - cos(x);
}
↓
double code(double x, double eps) {
	return -2.0 * (((sin(x) * cos(eps * 0.5)) * sin(eps * 0.5)) + (cos(x) * pow(sin(eps * 0.5), 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 39.6

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

  3. Applied diff-cos_binary64_23034.3

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

  4. Simplified15.1

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

  5. Taylor expanded around inf 15.1

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin \left(0.5 \cdot \left(2 \cdot x + \varepsilon\right)\right)}\right)\]

  6. Simplified15.1

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin \left(x + \varepsilon \cdot 0.5\right)}\right)\]
  7. Using strategy rm

  8. Applied sin-sum_binary64_2120.4

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right)\]

  9. Applied distribute-rgt-in_binary64_290.4

    \[\leadsto -2 \cdot \color{blue}{\left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) + \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]

  10. Simplified0.4

    \[\leadsto -2 \cdot \left(\color{blue}{\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)} + \left(\cos x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

  11. Simplified0.4

    \[\leadsto -2 \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + \color{blue}{\cos x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right)\]
  12. Using strategy rm

  13. Applied pow2_binary64_1600.4

    \[\leadsto -2 \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + \cos x \cdot \color{blue}{{\sin \left(\varepsilon \cdot 0.5\right)}^{2}}\right)\]

  14. Final simplification0.4

    \[\leadsto -2 \cdot \left(\left(\sin x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right) + \cos x \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\right)\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))