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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -173.7775306990116:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) + a \cdot x\\ \end{array}\]

Original	29.6
Target	0.2
Herbie	0.6

Derivation

Split input into 2 regimes
if (* a x) < -173.7775306990116
1. Initial program 0
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-log-exp0
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
if -173.7775306990116 < (* a x)
1. Initial program 43.9
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 14.2
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.9
  \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + a \cdot \left(x \cdot \frac{1}{6}\right)\right) + a \cdot x}\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -173.7775306990116:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) + a \cdot x\\ \end{array}\]

Runtime

herbie shell --seed 2018221 
(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) < -173.7775306990116`

`if -173.7775306990116 < (* a x)`

Runtime

In
Out