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Target

Derivation

1. Split input into 3 regimes

2. if b < 6.271620273244591e+227

   1. Initial program 59.0

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

   2. Taylor expanded around 0 3.0

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

   if 6.271620273244591e+227 < b < 7.559016357172504e+287

   1. Initial program 48.3

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

   2. Taylor expanded around 0 13.1

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt14.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{a} + \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} + \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} + \frac{1}{b}}}\]
   5. Using strategy rm

   6. Applied frac-add19.7

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

   7. Applied cbrt-div19.7

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

   8. Applied frac-add19.7

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

   9. Applied cbrt-div19.8

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

   10. Applied frac-times19.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1 \cdot b + a \cdot 1} \cdot \sqrt[3]{1 \cdot b + a \cdot 1}}{\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}}} \cdot \sqrt[3]{\frac{1}{a} + \frac{1}{b}}\]

   11. Simplified19.8

       \[\leadsto \frac{\color{blue}{\sqrt[3]{b + a} \cdot \sqrt[3]{b + a}}}{\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}} \cdot \sqrt[3]{\frac{1}{a} + \frac{1}{b}}\]

   if 7.559016357172504e+287 < b

   1. Initial program 42.0

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

   2. Taylor expanded around 0 19.2

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
   3. Using strategy rm

   4. Applied frac-add25.0

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

   5. Simplified25.0

      \[\leadsto \frac{\color{blue}{b + a}}{a \cdot b}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity25.0

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

   8. Applied times-frac19.2

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

4. Final simplification3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.271620273244591 \cdot 10^{+227}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;b \le 7.559016357172504 \cdot 10^{+287}:\\ \;\;\;\;\frac{\sqrt[3]{b + a} \cdot \sqrt[3]{b + a}}{\sqrt[3]{a \cdot b} \cdot \sqrt[3]{a \cdot b}} \cdot \sqrt[3]{\frac{1}{a} + \frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b + a}{b}\\ \end{array}\]

Runtime

Time bar (total: 37.0s)Debug logProfile

herbie shell --seed 2018225 
(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))))