\[\sin \left(x + \varepsilon\right) - \sin x\]
Test:
NMSE example 3.3
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.3 s
Input Error: 36.8
Output Error: 20.3
Log: ⚲
Profile: 🕒
\(\sqrt[3]{{\left(\sin \varepsilon\right)}^3 \cdot {\left(\cos x\right)}^3} + \left(\cos \varepsilon \cdot \sin x - \sin x\right)\)
  1. Started with
    \[\sin \left(x + \varepsilon\right) - \sin x\]
    36.8
  2. Using strategy rm
    36.8
  3. Applied sin-sum to get
    \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
    21.7
  4. Using strategy rm
    21.7
  5. Applied add-cbrt-cube to get
    \[\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \color{red}{\sin \varepsilon}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}}\right) - \sin x\]
    25.7
  6. Applied add-cbrt-cube to get
    \[\left(\sin x \cdot \cos \varepsilon + \color{red}{\cos x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}\right) - \sin x\]
    25.7
  7. Applied cbrt-unprod to get
    \[\left(\sin x \cdot \cos \varepsilon + \color{red}{\sqrt[3]{{\left(\cos x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}}\right) - \sin x\]
    25.7
  8. Applied taylor to get
    \[\left(\sin x \cdot \cos \varepsilon + \sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right) - \sin x\]
    25.7
  9. Taylor expanded around 0 to get
    \[\left(\sin x \cdot \cos \varepsilon + \color{red}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}}\right) - \sin x \leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}}\right) - \sin x\]
    25.7
  10. Applied simplify to get
    \[\left(\sin x \cdot \cos \varepsilon + \sqrt[3]{{\left(\cos x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}\right) - \sin x \leadsto \sqrt[3]{{\left(\sin \varepsilon\right)}^3 \cdot {\left(\cos x\right)}^3} + \left(\cos \varepsilon \cdot \sin x - \sin x\right)\]
    20.3
  11. Applied final simplification

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)))))