Error

Target

Derivation

1. Split input into 2 regimes

2. if a < 1.5572463218337037e+75

   1. Initial program 59.6

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

   2. Simplified36.0

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

   3. Taylor expanded around 0 2.4

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

   if 1.5572463218337037e+75 < a

   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. Simplified24.3

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

   4. Applied *-commutative24.3

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

   5. Applied associate-/l*16.9

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

   6. Applied associate-/l/16.9

      \[\leadsto \color{blue}{\frac{\varepsilon}{(e^{\varepsilon \cdot b} - 1)^* \cdot \frac{(e^{\varepsilon \cdot a} - 1)^*}{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.5572463218337037 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*} \cdot (e^{\varepsilon \cdot b} - 1)^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +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))))