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Target

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0007656553037481052

   1. Initial program 0.0

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

   3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
   4. Using strategy rm

   5. Applied flip--0.0

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

   7. Applied flip--0.0

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

   if -0.0007656553037481052 < (* a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 13.7

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

   3. Simplified0.5

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0007656553037481052:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(e^{a \cdot x} \cdot e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x}\right)\right) \cdot \left(\left(e^{a \cdot x} \cdot e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x}\right)\right) - 1}{1 + \left(e^{a \cdot x} \cdot e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x}\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + 1}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + a \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

Time bar (total: 11.4s)Debug log