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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.19087205089669393:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Original	29.6
Target	0.2
Herbie	0.4

Derivation

Split input into 2 regimes
if (* a x) < -0.19087205089669393
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-log-exp0.0
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
if -0.19087205089669393 < (* 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. Simplified0.6
  \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x}\]
Recombined 2 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.19087205089669393:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Runtime

herbie shell --seed 2018277 
(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) < -0.19087205089669393`

`if -0.19087205089669393 < (* a x)`

Runtime

In
Out