\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 20.6 s
Input Error: 36.8
Output Error: 24.0
Log: ⚲
Profile: 🕒
\(\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x}\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.8
  2. Using strategy rm
    36.8
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    21.7
  4. Using strategy rm
    21.7
  5. Applied flip-- to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x}}\]
    24.0

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)))))