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Target

Derivation

1. Split input into 2 regimes

2. if (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)) < -0.005329997397605313 or 7.680189817449445e-12 < (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))

   1. Initial program 9.7

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

   3. Applied clear-num9.8

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

   if -0.005329997397605313 < (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)) < 7.680189817449445e-12

   1. Initial program 62.1

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

   2. Taylor expanded around 0 0.1

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

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right) \le -0.005329997397605313 \lor \neg \left(\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right) \le 7.680189817449445 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{\frac{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}{\left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right) \cdot \varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018274 
(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))))