\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 8.2 s
Input Error: 16.8
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(1 \cdot \left(\cos x \cdot \sin \varepsilon - \left(\sin x - \cos \varepsilon \cdot \sin x\right)\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    16.8
  2. Using strategy rm
    16.8
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    6.3
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    6.3
  5. Using strategy rm
    6.3
  6. Applied flip-+ to get
    \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
    6.3
  7. Using strategy rm
    6.3
  8. Applied *-un-lft-identity to get
    \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{1 \cdot \left(\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}}\]
    6.3
  9. Applied *-un-lft-identity to get
    \[\frac{\color{red}{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}}{1 \cdot \left(\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)} \leadsto \frac{\color{blue}{1 \cdot \left({\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}}{1 \cdot \left(\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}\]
    6.3
  10. Applied times-frac to get
    \[\color{red}{\frac{1 \cdot \left({\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2\right)}{1 \cdot \left(\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)\right)}} \leadsto \color{blue}{\frac{1}{1} \cdot \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
    6.3
  11. Applied simplify to get
    \[\color{red}{\frac{1}{1}} \cdot \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{1} \cdot \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    6.3
  12. Applied simplify to get
    \[1 \cdot \color{red}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto 1 \cdot \color{blue}{\left(\cos x \cdot \sin \varepsilon - \left(\sin x - \cos \varepsilon \cdot \sin x\right)\right)}\]
    0.4

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE example 3.3"
  (- (sin (+ x eps)) (sin x))
  #:target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2)))))