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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.240954335863827:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Target

Original	29.4
Target	0.2
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

Split input into 2 regimes
if (* a x) < -3.240954335863827
1. Initial program 0
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around inf 0
  \[\leadsto \color{blue}{e^{a \cdot x}} - 1\]
if -3.240954335863827 < (* a x)
1. Initial program 43.9
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 14.1
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.7
  \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x}\]
Recombined 2 regimes into one program.
Final simplification0.5
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.240954335863827:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Runtime

herbie shell --seed 2018278 
(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

Try it out

Target

Derivation

`if (* a x) < -3.240954335863827`

`if -3.240954335863827 < (* a x)`

Runtime

In
Out