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Target

Derivation

1. Split input into 2 regimes

2. if b < -1.9610380610541195e+146

   1. Initial program 50.6

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

   2. Initial simplification18.0

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

   4. Applied associate-*l/18.0

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

   if -1.9610380610541195e+146 < b

   1. Initial program 59.6

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

   2. Initial simplification28.5

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

   3. Taylor expanded around 0 2.4

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

4. Final simplification4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9610380610541195 \cdot 10^{+146}:\\ \;\;\;\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^* \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot b} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Runtime

Time bar (total: 33.4s)Debug logProfile

herbie shell --seed 2018296 +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))))