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Target

Derivation

1. Split input into 2 regimes

2. if a < 1.911891629676131e+196 or 4.488836011909856e+279 < a

   1. Initial program 59.3

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

   2. Initial simplification28.0

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

   3. Taylor expanded around 0 2.7

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

   if 1.911891629676131e+196 < a < 4.488836011909856e+279

   1. Initial program 48.8

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

   2. Initial simplification16.1

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

   4. Applied associate-*r/16.1

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

4. Final simplification3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.911891629676131 \cdot 10^{+196} \lor \neg \left(a \le 4.488836011909856 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}\]

Runtime

Time bar (total: 24.0s)Debug logProfile

herbie shell --seed 2018339 +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))))