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Target

Derivation

1. Split input into 2 regimes

2. if eps < -3.227794156982276e-90 or 1.2229076337163464e-62 < eps

   1. Initial program 52.0

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

   2. Initial simplification7.2

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

   4. Applied add-cube-cbrt7.8

      \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\color{blue}{\left(\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}\right) \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}}} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   5. Applied *-un-lft-identity7.8

      \[\leadsto \frac{\color{blue}{1 \cdot (e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}{\left(\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}\right) \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   6. Applied times-frac7.8

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}}\right)} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]

   7. Applied associate-*l*7.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \left(\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\right)}\]

   if -3.227794156982276e-90 < eps < 1.2229076337163464e-62

   1. Initial program 60.3

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

   2. Initial simplification33.1

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

   3. Taylor expanded around 0 1.8

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

4. Final simplification3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.227794156982276 \cdot 10^{-90} \lor \neg \left(\varepsilon \le 1.2229076337163464 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*}} \cdot \left(\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*}} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Runtime

Time bar (total: 26.9s)Debug logProfile

herbie shell --seed 2018235 +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))))