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

⬇

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.230530163534557 \cdot 10^{-06}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a\\ \end{array}\]

Target

Original	33.4
Comparison	8.1
Herbie	0.1

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

Derivation

Split input into 2 regimes.
if (* a x) < -4.230530163534557e-06
1. Initial program 0.1
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-log-exp 0.1
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
if -4.230530163534557e-06 < (* a x)
1. Initial program 47.5
  \[e^{a \cdot x} - 1\]
2. Applied taylor 43.6
  \[\leadsto \left(a \cdot x + \left(1 + \frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)\right) - 1\]
3. Taylor expanded around 0 43.6
  
  \[\leadsto \color{blue}{\left(a \cdot x + \left(1 + \frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)\right)} - 1\]
4. Applied simplify 0.1
  \[\leadsto \color{blue}{\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1067901057 3396600083 3715501224 3126139233 3908045574 1593683916)'
(FPCore (a x)
  :name "expax (section 3.5)"

  :target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))

Error

Target

Derivation

`if (* a x) < -4.230530163534557e-06`

`if -4.230530163534557e-06 < (* a x)`

Runtime