\[e^{a \cdot x} - 1\]
Test:
NMSE section 3.5
Bits:
128 bits
Bits error versus a
Bits error versus x
Time: 10.5 s
Input Error: 13.0
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right)}^3 & \text{when } a \cdot x \le -0.0009205923f0 \\ \left(\sqrt{e^{x \cdot a}} + 1\right) \cdot \left(x \cdot \left(a \cdot \frac{1}{2} + \left(\frac{1}{8} \cdot a\right) \cdot \left(x \cdot a\right)\right)\right) & \text{otherwise} \end{cases}\)

    if (* a x) < -0.0009205923f0

    1. Started with
      \[e^{a \cdot x} - 1\]
      0.2
    2. Using strategy rm
      0.2
    3. Applied add-cube-cbrt to get
      \[\color{red}{e^{a \cdot x} - 1} \leadsto \color{blue}{{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}^3}\]
      0.3
    4. Using strategy rm
      0.3
    5. Applied add-log-exp to get
      \[{\left(\sqrt[3]{\color{red}{e^{a \cdot x} - 1}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}}\right)}^3\]
      0.3

    if -0.0009205923f0 < (* a x)

    1. Started with
      \[e^{a \cdot x} - 1\]
      20.1
    2. Using strategy rm
      20.1
    3. Applied add-sqr-sqrt to get
      \[\color{red}{e^{a \cdot x}} - 1 \leadsto \color{blue}{{\left(\sqrt{e^{a \cdot x}}\right)}^2} - 1\]
      20.2
    4. Applied difference-of-sqr-1 to get
      \[\color{red}{{\left(\sqrt{e^{a \cdot x}}\right)}^2 - 1} \leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]
      20.2
    5. Applied taylor to get
      \[\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right) \leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) - 1\right)\]
      21.1
    6. Taylor expanded around 0 to get
      \[\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\color{red}{\left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)} - 1\right) \leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\color{blue}{\left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)} - 1\right)\]
      21.1
    7. Applied simplify to get
      \[\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \left(1 + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) - 1\right) \leadsto \left(1 + \sqrt{e^{x \cdot a}}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(a \cdot \left(\frac{1}{8} \cdot a\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot a + \left(1 - 1\right)\right)\right)\]
      3.8
    8. Applied final simplification
    9. Applied simplify to get
      \[\color{red}{\left(1 + \sqrt{e^{x \cdot a}}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(a \cdot \left(\frac{1}{8} \cdot a\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot a + \left(1 - 1\right)\right)\right)} \leadsto \color{blue}{\left(\sqrt{e^{x \cdot a}} + 1\right) \cdot \left(x \cdot \left(a \cdot \frac{1}{2} + \left(\frac{1}{8} \cdot a\right) \cdot \left(x \cdot a\right)\right)\right)}\]
      0.2
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (x default))
  #:name "NMSE section 3.5"
  (- (exp (* a x)) 1)
  #:target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1)))