\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 8.6 s
Input Error: 17.0
Output Error: 2.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} & \text{when } \varepsilon \le -2.2300634f-26 \\ \varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right) & \text{when } \varepsilon \le 6.368611f-18 \\ \sin x \cdot \cos \varepsilon + \frac{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} & \text{otherwise} \end{cases}\)

    if eps < -2.2300634f-26 or 6.368611f-18 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      15.3
    2. Using strategy rm
      15.3
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      2.8
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      2.8
    5. Using strategy rm
      2.8
    6. Applied flip3-- to get
      \[\sin x \cdot \cos \varepsilon + \color{red}{\left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \color{blue}{\frac{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}}\]
      6.4
    7. Applied simplify to get
      \[\sin x \cdot \cos \varepsilon + \frac{\color{red}{{\left(\cos x \cdot \sin \varepsilon\right)}^{3} - {\left(\sin x\right)}^{3}}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)} \leadsto \sin x \cdot \cos \varepsilon + \frac{\color{blue}{{\left(\sin \varepsilon \cdot \cos x\right)}^3 - {\left(\sin x\right)}^3}}{{\left(\cos x \cdot \sin \varepsilon\right)}^2 + \left({\left(\sin x\right)}^2 + \left(\cos x \cdot \sin \varepsilon\right) \cdot \sin x\right)}\]
      2.8

    if -2.2300634f-26 < eps < 6.368611f-18

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      23.3
    2. Applied taylor to get
      \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
      5.9
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
      5.9
    4. Applied simplify to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right)}\]
      0.2
  1. 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)))))