\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.1 s
Input Error: 35.7
Output Error: 3.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon\right)}^2}{\sin x \cdot \cos \varepsilon - \cos x \cdot \sin \varepsilon} - \sin x & \text{when } \varepsilon \le -5.952085980018888 \cdot 10^{-41} \\ \varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right) & \text{when } \varepsilon \le 2.939974265627699 \cdot 10^{-99} \\ \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon\right)}^2}{\sin x \cdot \cos \varepsilon - \cos x \cdot \sin \varepsilon} - \sin x & \text{otherwise} \end{cases}\)

    if eps < -5.952085980018888e-41

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      29.5
    2. Using strategy rm
      29.5
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      3.5
    4. Using strategy rm
      3.5
    5. Applied flip-+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon\right)}^2}{\sin x \cdot \cos \varepsilon - \cos x \cdot \sin \varepsilon}} - \sin x\]
      3.6

    if -5.952085980018888e-41 < eps < 2.939974265627699e-99

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      46.4
    2. Applied taylor to get
      \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
      10.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
      10.8
    4. Applied simplify to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right)}\]
      0.1

    if 2.939974265627699e-99 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      30.0
    2. Using strategy rm
      30.0
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      7.6
    4. Using strategy rm
      7.6
    5. Applied flip-+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon\right)}^2}{\sin x \cdot \cos \varepsilon - \cos x \cdot \sin \varepsilon}} - \sin x\]
      7.7
  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)))))