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Target

Derivation

1. Split input into 2 regimes

2. if b < 3.1206474527261573e+252 or 7.329126589161342e+307 < b

   1. Initial program 58.9

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Initial simplification28.4

      \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   3. Taylor expanded around 0 3.1

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

   if 3.1206474527261573e+252 < b < 7.329126589161342e+307

   1. Initial program 46.5

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Initial simplification15.9

      \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   3. Taylor expanded around -inf 18.7

      \[\leadsto \frac{\color{blue}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   4. Simplified15.9

      \[\leadsto \frac{\color{blue}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Recombined 2 regimes into one program.

4. Final simplification3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.1206474527261573 \cdot 10^{+252} \lor \neg \left(b \le 7.329126589161342 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))

Details

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