\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 22.7 s
Input Error: 36.1
Output Error: 1.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(e^{\sin x \cdot \cos \varepsilon}\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right) & \text{when } \varepsilon \le -5.086924474194238 \cdot 10^{-55} \\ \varepsilon - \left(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \frac{1}{2} & \text{when } \varepsilon \le 3.131194448353392 \cdot 10^{-13} \\ \log \left(e^{\sin x \cdot \cos \varepsilon}\right) + \left(\cos x \cdot \sin \varepsilon - \sin x\right) & \text{otherwise} \end{cases}\)

    if eps < -5.086924474194238e-55 or 3.131194448353392e-13 < 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
      \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
      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 add-log-exp to get
      \[\color{red}{\sin x \cdot \cos \varepsilon} + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \leadsto \color{blue}{\log \left(e^{\sin x \cdot \cos \varepsilon}\right)} + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\]
      3.0

    if -5.086924474194238e-55 < eps < 3.131194448353392e-13

    1. Started with
      \[\sin \left(x + \varepsilon\right) - \sin x\]
      44.4
    2. Applied taylor to get
      \[\sin \left(x + \varepsilon\right) - \sin x \leadsto \varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)\]
      9.3
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)}\]
      9.3
    4. Applied simplify to get
      \[\color{red}{\varepsilon - \left(\frac{1}{2} \cdot \left(\varepsilon \cdot {x}^2\right) + \frac{1}{2} \cdot \left({\varepsilon}^2 \cdot x\right)\right)} \leadsto \color{blue}{\varepsilon - \left(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot \frac{1}{2}}\]
      0.1
  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)))))