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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01114501368054272:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]

Original	29.8
Target	0.1
Herbie	0.3

Derivation

Split input into 2 regimes
if (* a x) < -0.01114501368054272
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{e^{a \cdot x} - 1}\]
if -0.01114501368054272 < (* a x)
1. Initial program 45.0
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 13.8
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.4
  \[\leadsto \color{blue}{\left(a \cdot x + x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01114501368054272:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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))

Error

Target

Derivation

`if (* a x) < -0.01114501368054272`

`if -0.01114501368054272 < (* a x)`

Reproduce