\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 25.7 s
Input Error: 35.7
Output Error: 2.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)} & \text{when } \varepsilon \le -8.048717335408643 \cdot 10^{-76} \\ \varepsilon - \frac{1}{2} \cdot \left(\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right)\right) & \text{when } \varepsilon \le 1.6258231275971248 \cdot 10^{-14} \\ \frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x} & \text{otherwise} \end{cases}\)

    if eps < -8.048717335408643e-76

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      30.4
    2. Using strategy rm
      30.4
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      6.5
    4. Applied associate--l+ to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
      6.5
    5. Using strategy rm
      6.5
    6. Applied flip-+ to get
      \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
      6.7

    if -8.048717335408643e-76 < eps < 1.6258231275971248e-14

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      43.9
    2. Applied taylor to get
      \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
      8.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
      8.8
    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 1.6258231275971248e-14 < eps

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      29.6
    2. Using strategy rm
      29.6
    3. Applied sin-sum to get
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      0.9
    4. Using strategy rm
      0.9
    5. Applied flip-- to get
      \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)}^2 - {\left(\sin x\right)}^2}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) + \sin x}}\]
      1.1
  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)))))