Error

Try it out

Target

Derivation

1. Split input into 2 regimes

2. if b < -4.418679367205824e+243 or 3.380573644616822e+183 < b < 4.27530359699961e+272

   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. Simplified15.8

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

   4. Applied div-inv16.8

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

   if -4.418679367205824e+243 < b < 3.380573644616822e+183 or 4.27530359699961e+272 < b

   1. Initial program 59.7

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

   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 2.3

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

4. Final simplification3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.418679367205824 \cdot 10^{+243} \lor \neg \left(b \le 3.380573644616822 \cdot 10^{+183}\right) \land b \le 4.27530359699961 \cdot 10^{+272}:\\ \;\;\;\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \left(\frac{1}{(e^{a \cdot \varepsilon} - 1)^*} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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: 32.4s)Debug log