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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le 1.1751478576497096 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}\\ \end{array}\]

Original	58.4
Target	14.0
Herbie	3.4

Derivation

Split input into 2 regimes
if eps < 1.1751478576497096e-77
1. Initial program 59.1
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification29.9
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around 0 2.9
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
if 1.1751478576497096e-77 < eps
1. Initial program 52.9
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification7.6
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around inf 29.2
  \[\leadsto \frac{\color{blue}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
4. Simplified7.6
  \[\leadsto \frac{\color{blue}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
Recombined 2 regimes into one program.
Final simplification3.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le 1.1751478576497096 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{a \cdot \varepsilon} - 1)^*}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	3.6	3.4	0.0	3.5	2.9%

herbie shell --seed 2018290 +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 eps < 1.1751478576497096e-77`

`if 1.1751478576497096e-77 < eps`

Runtime

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