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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -2.2260753947737583 \cdot 10^{-11}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(x \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)\\ \end{array}\]

Target

Original	29.3
Target	0.2
Herbie	0.6

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

Derivation

Split input into 2 regimes
if (- (exp (* a x)) 1) < -2.2260753947737583e-11
1. Initial program 0.7
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-log-exp0.7
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
if -2.2260753947737583e-11 < (- (exp (* a x)) 1)
1. Initial program 44.2
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 13.5
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)}\]
3. Taylor expanded around inf 7.8
  \[\leadsto \frac{1}{6} \cdot \color{blue}{0} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)\]
4. Applied simplify0.6
  \[\leadsto \color{blue}{x \cdot a + \left(x \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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 (- (exp (* a x)) 1) < -2.2260753947737583e-11`

`if -2.2260753947737583e-11 < (- (exp (* a x)) 1)`

Runtime