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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.518689892188866 \cdot 10^{+163}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*}\right) \cdot \frac{1}{(e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Original	58.6
Target	14.7
Herbie	3.7

Derivation

Split input into 2 regimes
if b < -4.518689892188866e+163
1. Initial program 47.9
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification16.4
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Using strategy rm
4. Applied div-inv17.1
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{(e^{\varepsilon \cdot a} - 1)^*}\right)}\]
5. Applied associate-*r*17.1
  \[\leadsto \color{blue}{\left(\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \varepsilon\right) \cdot \frac{1}{(e^{\varepsilon \cdot a} - 1)^*}}\]
if -4.518689892188866e+163 < 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. Initial simplification28.4
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around 0 2.3
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Recombined 2 regimes into one program.
Final simplification3.7
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.518689892188866 \cdot 10^{+163}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*}\right) \cdot \frac{1}{(e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	3.4	3.7	0.0	3.3	-8.9%

herbie shell --seed 2018353 +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 < -4.518689892188866e+163`

`if -4.518689892188866e+163 < b`

Runtime

In
Out