\[\frac{1 - \cos x}{\sin x}\]
Test:
NMSE example 3.4
Bits:
128 bits
Bits error versus x
Time: 1.4 m
Input Error: 30.2
Output Error: 0.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left({1}^2 + \left({\left(\cos x\right)}^2 + 1 \cdot \cos x\right)\right)} & \text{when } x \le -1.128700823931291 \cdot 10^{-06} \\ (e^{\log_* (1 + \frac{\sin x}{\cos x + 1})} - 1)^* & \text{when } x \le 1.675811971431754 \cdot 10^{-80} \\ \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x} & \text{otherwise} \end{cases}\)

    if x < -1.128700823931291e-06

    1. Started with
      \[\frac{1 - \cos x}{\sin x}\]
      1.2
    2. Using strategy rm
      1.2
    3. Applied flip3-- to get
      \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{{1}^2 + \left({\left(\cos x\right)}^2 + 1 \cdot \cos x\right)}}}{\sin x}\]
      1.3
    4. Applied associate-/l/ to get
      \[\color{red}{\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{{1}^2 + \left({\left(\cos x\right)}^2 + 1 \cdot \cos x\right)}}{\sin x}} \leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left({1}^2 + \left({\left(\cos x\right)}^2 + 1 \cdot \cos x\right)\right)}}\]
      1.3

    if -1.128700823931291e-06 < x < 1.675811971431754e-80

    1. Started with
      \[\frac{1 - \cos x}{\sin x}\]
      60.2
    2. Using strategy rm
      60.2
    3. Applied flip-- to get
      \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]
      60.2
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{\sin x} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
      33.6
    5. Using strategy rm
      33.6
    6. Applied expm1-log1p-u to get
      \[\color{red}{\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}} \leadsto \color{blue}{(e^{\log_* (1 + \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x})} - 1)^*}\]
      33.6
    7. Applied simplify to get
      \[(e^{\color{red}{\log_* (1 + \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x})}} - 1)^* \leadsto (e^{\color{blue}{\log_* (1 + \frac{\sin x}{\cos x + 1})}} - 1)^*\]
      0.0

    if 1.675811971431754e-80 < x

    1. Started with
      \[\frac{1 - \cos x}{\sin x}\]
      12.2
    2. Using strategy rm
      12.2
    3. Applied flip-- to get
      \[\frac{\color{red}{1 - \cos x}}{\sin x} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]
      12.6
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{1}^2 - {\left(\cos x\right)}^2}}{1 + \cos x}}{\sin x} \leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
      0.8
  1. Removed slow pow expressions

Original test:


(lambda ((x default))
  #:name "NMSE example 3.4"
  (/ (- 1 (cos x)) (sin x))
  #:target
  (tan (/ x 2)))