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Target

Derivation

1. Split input into 2 regimes

2. if a < -2.628102819609262e+126 or 5.294256156442987e-29 < a

   1. Initial program 20.7

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

   3. Applied flip--20.7

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

   5. Applied flip--20.8

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

   6. Applied associate-/l/20.8

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

   7. Simplified20.7

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

   9. Applied add-cube-cbrt20.7

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

   10. Applied associate-/l*20.7

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

   if -2.628102819609262e+126 < a < 5.294256156442987e-29

   1. Initial program 33.9

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

   2. Taylor expanded around 0 18.8

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

   3. Simplified10.9

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

4. Final simplification14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.628102819609262 \cdot 10^{+126} \lor \neg \left(a \le 5.294256156442987 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1} \cdot \sqrt[3]{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}}{\frac{\left(e^{x \cdot a} \cdot e^{x \cdot a} + 1\right) \cdot \left(e^{x \cdot a} + 1\right)}{\sqrt[3]{e^{\left(x + x\right) \cdot \left(a + a\right)} + -1}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 22.7s)Debug logProfile

herbie shell --seed 2018295 
(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))