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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 7.572989311479111 \cdot 10^{+219}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;(e^{\varepsilon \cdot \left(b + a\right)} - 1)^* \cdot \left(\frac{1}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}\right)\\ \end{array}\]

Original	58.6
Target	14.6
Herbie	3.3

Derivation

Split input into 2 regimes
if b < 7.572989311479111e+219
1. Initial program 59.2
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification27.9
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around 0 2.8
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
if 7.572989311479111e+219 < b
1. Initial program 47.4
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification13.8
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Using strategy rm
4. Applied div-inv13.8
  \[\leadsto \color{blue}{\left((e^{\left(a + b\right) \cdot \varepsilon} - 1)^* \cdot \frac{1}{(e^{\varepsilon \cdot b} - 1)^*}\right)} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
5. Applied associate-*l*13.8
  \[\leadsto \color{blue}{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^* \cdot \left(\frac{1}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\right)}\]
Recombined 2 regimes into one program.
Final simplification3.3
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.572989311479111 \cdot 10^{+219}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;(e^{\varepsilon \cdot \left(b + a\right)} - 1)^* \cdot \left(\frac{1}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}\right)\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	3.3	3.3	0.0	3.3	0.4%

herbie shell --seed 2018295 +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))))

Error

Try it out

Target

Derivation

`if b < 7.572989311479111e+219`

`if 7.572989311479111e+219 < b`

Runtime

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