\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 20.7 s
Input Error: 36.4
Output Error: 0.5
Log: ⚲
Profile: 🕒
\((e^{\log_* (1 + (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*)} - 1)^*\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.4
  2. Using strategy rm
    36.4
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    23.1
  4. Using strategy rm
    23.1
  5. Applied expm1-log1p-u to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{(e^{\log_* (1 + \left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\right))} - 1)^*}\]
    23.2
  6. Applied simplify to get
    \[(e^{\color{red}{\log_* (1 + \left(\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\right))}} - 1)^* \leadsto (e^{\color{blue}{\log_* (1 + (\left(\sin \varepsilon\right) * \left(\cos x\right) + \left(\cos \varepsilon \cdot \sin x - \sin x\right))_*)}} - 1)^*\]
    0.5

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)))))