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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.339969652804326 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*}}}{\sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Original	58.6
Target	14.2
Herbie	3.3

Derivation

Split input into 2 regimes
if eps < -1.339969652804326e-76
1. Initial program 52.5
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification7.3
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Using strategy rm
4. Applied add-cube-cbrt7.9
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{\color{blue}{\left(\sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}\right) \cdot \sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}}}\]
5. Applied associate-/r*7.9
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \color{blue}{\frac{\frac{\varepsilon}{\sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}}}{\sqrt[3]{(e^{\varepsilon \cdot a} - 1)^*}}}\]
if -1.339969652804326e-76 < eps
1. Initial program 59.3
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Initial simplification30.2
  \[\leadsto \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\]
3. Taylor expanded around 0 2.7
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Recombined 2 regimes into one program.
Final simplification3.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.339969652804326 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*}}}{\sqrt[3]{(e^{a \cdot \varepsilon} - 1)^*}} \cdot \frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	3.4	3.3	0.0	3.4	3.6%

herbie shell --seed 2018354 +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 < -1.339969652804326e-76`

`if -1.339969652804326e-76 < eps`

Runtime

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