Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if a < -1.1993054013943756e+171 or 6.289267259125741e+197 < a < 7.702453821883354e+225

   1. Initial program 51.0

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

   2. Simplified17.5

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

   4. Applied add-cube-cbrt17.7

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

   5. Applied associate-/r*17.7

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

   if -1.1993054013943756e+171 < a < 6.289267259125741e+197 or 7.702453821883354e+225 < a

   1. Initial program 59.8

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

   2. Simplified28.7

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

   3. Taylor expanded around 0 2.2

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

4. Final simplification3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.1993054013943756 \cdot 10^{+171} \lor \neg \left(a \le 6.289267259125741 \cdot 10^{+197}\right) \land a \le 7.702453821883354 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}}}{\sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}} \cdot \frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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))))