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Target

Derivation

1. Split input into 2 regimes

2. if a < -6.810032246131824e+212 or 7.690864542863468e+190 < a < 1.4768088829653062e+258

   1. Initial program 47.6

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   2. Using strategy rm

   3. Applied associate-/r*47.6

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

   4. Simplified24.9

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

   if -6.810032246131824e+212 < a < 7.690864542863468e+190 or 1.4768088829653062e+258 < a

   1. Initial program 59.5

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

   2. Taylor expanded around 0 2.5

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.810032246131824 \cdot 10^{+212} \lor \neg \left(a \le 7.690864542863468 \cdot 10^{+190}\right) \land a \le 1.4768088829653062 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{e^{\varepsilon \cdot a} - 1}}{(e^{\varepsilon \cdot b} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Runtime

Time bar (total: 39.6s)Debug logProfile

herbie shell --seed 2018219 +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))))