\[(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a\]
Test:
(- (expm1 (- (tan (* a a)) a)) a)
Bits:
128 bits
Bits error versus a
Time: 7.4 s
Input Error: 5.4
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} (e^{\log_* (1 + \left((e^{\tan \left({a}^2\right) - a} - 1)^* - a\right))} - 1)^* & \text{when } a \le 3.63006f0 \\ (e^{\frac{\sin \left(\frac{1}{{a}^2}\right)}{\cos \left(\frac{1}{{a}^2}\right)} - a} - 1)^* - a & \text{otherwise} \end{cases}\)

    if a < 3.63006f0

    1. Started with
      \[(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a\]
      0.2
    2. Applied simplify to get
      \[\color{red}{(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a} \leadsto \color{blue}{(e^{\tan \left({a}^2\right) - a} - 1)^* - a}\]
      0.2
    3. Using strategy rm
      0.2
    4. Applied expm1-log1p-u to get
      \[\color{red}{(e^{\tan \left({a}^2\right) - a} - 1)^* - a} \leadsto \color{blue}{(e^{\log_* (1 + \left((e^{\tan \left({a}^2\right) - a} - 1)^* - a\right))} - 1)^*}\]
      0.2

    if 3.63006f0 < a

    1. Started with
      \[(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a\]
      16.4
    2. Applied simplify to get
      \[\color{red}{(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a} \leadsto \color{blue}{(e^{\tan \left({a}^2\right) - a} - 1)^* - a}\]
      16.4
    3. Applied taylor to get
      \[(e^{\tan \left({a}^2\right) - a} - 1)^* - a \leadsto (e^{\frac{\sin \left(\frac{1}{{a}^2}\right)}{\cos \left(\frac{1}{{a}^2}\right)} - a} - 1)^* - a\]
      0.0
    4. Taylor expanded around inf to get
      \[(e^{\color{red}{\frac{\sin \left(\frac{1}{{a}^2}\right)}{\cos \left(\frac{1}{{a}^2}\right)}} - a} - 1)^* - a \leadsto (e^{\color{blue}{\frac{\sin \left(\frac{1}{{a}^2}\right)}{\cos \left(\frac{1}{{a}^2}\right)}} - a} - 1)^* - a\]
      0.0
  1. Removed slow pow expressions

Original test:


(lambda ((a default))
  #:name "(- (expm1 (- (tan (* a a)) a)) a)"
  (- (expm1 (- (tan (* a a)) a)) a))