\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
Test:
NMSE problem 3.4.2
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus eps
Time: 3.3 m
Input Error: 62.0
Output Error: 0.0
Log: ⚲
\(\begin{cases} \frac{1}{a} + \frac{1}{b} & \text{when } \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -9.296169205569568 \cdot 10^{-188} \\ \frac{1}{a} + \frac{1}{b} & \text{when } \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 7.819285243784398 \cdot 10^{-190} \\ \frac{1}{a} + \frac{1}{b} & \text{otherwise} \end{cases}\)

    if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -9.296169205569568e-188 or 7.819285243784398e-190 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    1. Started with
      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
      61.3
    2. Applied taylor to get
      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leadsto \frac{1}{a} + \frac{1}{b}\]
      0.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{a} + \frac{1}{b}} \leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
      0.0

    if -9.296169205569568e-188 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 7.819285243784398e-190

    1. Started with
      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
      62.2
    2. Applied taylor to get
      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leadsto \frac{1}{a} + \frac{1}{b}\]
      0.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{a} + \frac{1}{b}} \leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
      0.0
  1. 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)))