\[(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a\]
Test:
(- (expm1 (- (tan (* a a)) a)) a)
Bits:
128 bits
Bits error versus a
Time: 6.9 s
Input Error: 10.8
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} (e^{(\left({a}^{6}\right) * \frac{1}{3} + \left((\left({a}^{10}\right) * \frac{2}{15} + \left({a}^2 - a\right))_*\right))_*} - 1)^* - a & \text{when } a \le 17.12856537901564 \\ (e^{\frac{\sin \left(\frac{1}{{a}^2}\right)}{\cos \left(\frac{1}{{a}^2}\right)} - a} - 1)^* - a & \text{otherwise} \end{cases}\)

    if a < 17.12856537901564

    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. Applied taylor to get
      \[(e^{\tan \left({a}^2\right) - a} - 1)^* - a \leadsto (e^{\left({a}^2 + \left(\frac{1}{3} \cdot {a}^{6} + \frac{2}{15} \cdot {a}^{10}\right)\right) - a} - 1)^* - a\]
      0.0
    4. Taylor expanded around 0 to get
      \[(e^{\color{red}{\left({a}^2 + \left(\frac{1}{3} \cdot {a}^{6} + \frac{2}{15} \cdot {a}^{10}\right)\right)} - a} - 1)^* - a \leadsto (e^{\color{blue}{\left({a}^2 + \left(\frac{1}{3} \cdot {a}^{6} + \frac{2}{15} \cdot {a}^{10}\right)\right)} - a} - 1)^* - a\]
      0.0
    5. Applied simplify to get
      \[\color{red}{(e^{\left({a}^2 + \left(\frac{1}{3} \cdot {a}^{6} + \frac{2}{15} \cdot {a}^{10}\right)\right) - a} - 1)^* - a} \leadsto \color{blue}{(e^{(\left({a}^{6}\right) * \frac{1}{3} + \left((\left({a}^{10}\right) * \frac{2}{15} + \left({a}^2 - a\right))_*\right))_*} - 1)^* - a}\]
      0.0

    if 17.12856537901564 < a

    1. Started with
      \[(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a\]
      31.6
    2. Applied simplify to get
      \[\color{red}{(e^{\tan \left(a \cdot a\right) - a} - 1)^* - a} \leadsto \color{blue}{(e^{\tan \left({a}^2\right) - a} - 1)^* - a}\]
      31.6
    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
    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
  1. Removed slow pow expressions

Original test:


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