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Target

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.02142687041441532

   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-cube-cbrt0.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt0.0

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

   if -0.02142687041441532 < (* a x)

   1. Initial program 43.6

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

   2. Initial simplification43.6

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

   3. Taylor expanded around 0 14.3

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

   4. Simplified0.4

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.02142687041441532:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 15.2s)Debug logProfile

herbie shell --seed 2018354 
(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))