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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot \varepsilon \le 4543654.312817573:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{a} + \frac{1}{b}\right)}^{3}}\\ \end{array}\]

Original	62.0
Target	14.7
Herbie	4.4

Derivation

Split input into 2 regimes
if (* a eps) < 4543654.312817573
1. Initial program 61.9
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
if 4543654.312817573 < (* a eps)
1. Initial program 62.4
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
3. Using strategy rm
4. Applied add-cbrt-cube35.9
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{b} + \frac{1}{a}\right) \cdot \left(\frac{1}{b} + \frac{1}{a}\right)\right) \cdot \left(\frac{1}{b} + \frac{1}{a}\right)}}\]
5. Applied simplify35.9
  \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{a} + \frac{1}{b}\right)}^{3}}}\]
Recombined 2 regimes into one program.
Removed slow pow expressions.

Runtime

herbie shell --seed '#(151349756 408087815 228312487 2538703040 1980610373 1250971417)' 
(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

Target

Derivation

`if (* a eps) < 4543654.312817573`

`if 4543654.312817573 < (* a eps)`

Runtime