Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -1.6224479041462292e-81

   1. Initial program 51.9

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

   2. Simplified17.6

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

   4. Applied times-frac7.7

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

   if -1.6224479041462292e-81 < eps

   1. Initial program 59.3

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

   2. Simplified42.2

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

   3. Taylor expanded around 0 2.8

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

4. Final simplification3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.6224479041462292 \cdot 10^{-81}:\\ \;\;\;\;\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*} \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +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))))