\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Test:
NMSE problem 3.4.2
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus eps
Time: 1.4 m
Input Error: 58.6
Output Error: 28.4
Log: ⚲
Profile: 🕒
\(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \left(\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right) \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}\right)\)
  1. Started with
    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    58.6
  2. Applied simplify to get
    \[\color{red}{\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \leadsto \color{blue}{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}}\]
    28.1
  3. Using strategy rm
    28.1
  4. Applied add-cube-cbrt to get
    \[\color{red}{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right)}^3} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}\]
    28.3
  5. Using strategy rm
    28.3
  6. Applied cube-mult to get
    \[\color{red}{{\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right)}^3} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*} \leadsto \color{blue}{\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right)\right)} \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}\]
    28.3
  7. Applied associate-*l* to get
    \[\color{red}{\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right)\right) \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}} \leadsto \color{blue}{\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \left(\left(\sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{a \cdot \varepsilon} - 1)^*}}\right) \cdot \frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}\right)}\]
    28.4
  8. Removed slow pow expressions

Original test:


(lambda ((a default) (b default) (eps default))
  #:name "NMSE problem 3.4.2"
  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))
  #:target
  (/ (+ a b) (* a b)))