Error

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Target

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0016894500806073423

   1. Initial program 0.0

      \[e^{a \cdot x} - 1\]

   2. Initial simplification0.0

      \[\leadsto e^{a \cdot x} - 1\]
   3. Using strategy rm

   4. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt0.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}}\]

   if -0.0016894500806073423 < (* a x)

   1. Initial program 44.3

      \[e^{a \cdot x} - 1\]

   2. Initial simplification44.3

      \[\leadsto e^{a \cdot x} - 1\]

   3. Taylor expanded around 0 13.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

   4. Simplified0.3

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0016894500806073423:\\ \;\;\;\;\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))

Details

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