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Target

Derivation

1. Split input into 2 regimes

2. if b < -5.4036214615566504e+150 or 5.841509212687738e+176 < b

   1. Initial program 50.4

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

   2. Initial simplification18.2

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

   4. Applied add-cube-cbrt18.7

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

   5. Applied associate-/r*18.7

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

   if -5.4036214615566504e+150 < b < 5.841509212687738e+176

   1. Initial program 60.5

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

   2. Initial simplification41.9

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

   3. Taylor expanded around 0 1.5

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

4. Final simplification4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4036214615566504 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}}}{\sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}}\\ \mathbf{elif}\;b \le 5.841509212687738 \cdot 10^{+176}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}} \cdot \sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}}}{\sqrt[3]{\frac{(e^{\varepsilon \cdot a} - 1)^* \cdot (e^{\varepsilon \cdot b} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018234 +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))))