Error

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Target

Derivation

1. Split input into 2 regimes

2. if eps < -1.0077585081869481e-76

   1. Initial program 50.9

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

   2. Simplified7.5

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

   4. Applied add-cube-cbrt8.1

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

   5. Applied add-cube-cbrt8.0

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

   6. Applied times-frac8.0

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

   7. Applied associate-*l*7.7

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

   if -1.0077585081869481e-76 < eps

   1. Initial program 59.5

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

   2. Simplified29.9

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

   3. Taylor expanded around 0 2.5

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

4. Final simplification3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0077585081869481 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt[3]{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}{\sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*}} \cdot \left(\frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \frac{\sqrt[3]{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}{\sqrt[3]{(e^{b \cdot \varepsilon} - 1)^*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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))))

Details

Time bar (total: 28.0s)Debug log