\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 39.7 s
Input Error: 36.6
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(1 \cdot (\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.6
  2. Using strategy rm
    36.6
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    21.9
  4. Using strategy rm
    21.9
  5. Applied *-un-lft-identity to get
    \[\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \color{red}{\sin x} \leadsto \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \color{blue}{1 \cdot \sin x}\]
    21.9
  6. Applied *-un-lft-identity to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - 1 \cdot \sin x \leadsto \color{blue}{1 \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - 1 \cdot \sin x\]
    21.9
  7. Applied distribute-lft-out-- to get
    \[\color{red}{1 \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - 1 \cdot \sin x} \leadsto \color{blue}{1 \cdot \left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\right)}\]
    21.9
  8. Applied simplify to get
    \[1 \cdot \color{red}{\left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\right)} \leadsto 1 \cdot \color{blue}{(\left(\cos x\right) * \left(\sin \varepsilon\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*}\]
    0.4
  9. Removed slow pow expressions

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)))))