\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 8.8 s
Input Error: 16.9
Output Error: 6.8
Log: ⚲
Profile: 🕒
\(\sin x \cdot \cos \varepsilon + \left((e^{{\left(\sqrt[3]{\log_* (1 + \cos x \cdot \sin \varepsilon)}\right)}^3} - 1)^* - \sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    16.9
  2. Using strategy rm
    16.9
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    6.4
  4. Applied associate--l+ to get
    \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
    6.4
  5. Using strategy rm
    6.4
  6. Applied expm1-log1p-u to get
    \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{(e^{\log_* (1 + \cos x \cdot \sin \varepsilon)} - 1)^*} - \sin x\right)\]
    6.5
  7. Using strategy rm
    6.5
  8. Applied add-cube-cbrt to get
    \[\sin x \cdot \cos \varepsilon + \left((e^{\color{red}{\log_* (1 + \cos x \cdot \sin \varepsilon)}} - 1)^* - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left((e^{\color{blue}{{\left(\sqrt[3]{\log_* (1 + \cos x \cdot \sin \varepsilon)}\right)}^3}} - 1)^* - \sin x\right)\]
    6.8

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)))))