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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.43214129231953 \cdot 10^{+250}:\\ \;\;\;\;\frac{\left(e^{\left(b + a\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Original	58.8
Target	14.5
Herbie	3.4

Derivation

Split input into 2 regimes
if a < -3.43214129231953e+250
1. Initial program 49.2
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification51.7
  \[\leadsto \frac{(\left({\left(e^{\varepsilon}\right)}^{\left(a + b\right)}\right) \cdot \varepsilon + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around inf 18.7
  \[\leadsto \frac{\color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}\]
if -3.43214129231953e+250 < a
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 simplification59.3
  \[\leadsto \frac{(\left({\left(e^{\varepsilon}\right)}^{\left(a + b\right)}\right) \cdot \varepsilon + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around 0 2.9
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Recombined 2 regimes into one program.
Final simplification3.4
\[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.43214129231953 \cdot 10^{+250}:\\ \;\;\;\;\frac{\left(e^{\left(b + a\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Runtime

herbie shell --seed 2018242 +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 a < -3.43214129231953e+250`

`if -3.43214129231953e+250 < a`

Runtime

In
Out