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Target

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ 1/2 (* eps (* 1/6 a)))) (* a eps)) (- (exp (* b eps)) 1))) < -1.535950601578352e-203 or 4.548241810026706e-261 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ 1/2 (* eps (* 1/6 a)))) (* a eps)) (- (exp (* b eps)) 1)))

   1. Initial program 61.3

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

   2. Taylor expanded around 0 0.8

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

   if -1.535950601578352e-203 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* (* (* a eps) (* a eps)) (+ 1/2 (* eps (* 1/6 a)))) (* a eps)) (- (exp (* b eps)) 1))) < 4.548241810026706e-261

   1. Initial program 3.5

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

   3. Applied add-sqr-sqrt3.6

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

   4. Applied difference-of-sqr-13.6

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

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\varepsilon \cdot a + \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot \frac{1}{6}\right) \cdot \varepsilon\right)\right)} \le -1.535950601578352 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(\varepsilon \cdot a + \left(\left(\varepsilon \cdot a\right) \cdot \left(\varepsilon \cdot a\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot \frac{1}{6}\right) \cdot \varepsilon\right)\right)} \le 4.548241810026706 \cdot 10^{-261}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(\left(\sqrt{e^{\varepsilon \cdot a}} + 1\right) \cdot \left(\sqrt{e^{\varepsilon \cdot a}} - 1\right)\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Runtime

Time bar (total: 46.8s)Debug logProfile

herbie shell --seed 2018217 
(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))))