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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -2.633327163628067 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \left(x \cdot a\right) + x \cdot a\\ \end{array}\]

Target

Original	29.1
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 (- (exp (* a x)) 1) < -2.633327163628067e-08
1. Initial program 0.2
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.2
  \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]
4. Applied difference-of-sqr-10.2
  \[\leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]
if -2.633327163628067e-08 < (- (exp (* a x)) 1)
1. Initial program 44.3
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 45.5
  \[\leadsto \color{blue}{\left(a \cdot x + \left(1 + \frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)\right)} - 1\]
3. Applied simplify0.7
  \[\leadsto \color{blue}{\left(\left(x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \left(x \cdot a\right) + x \cdot a}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018207 
(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 (- (exp (* a x)) 1) < -2.633327163628067e-08`

`if -2.633327163628067e-08 < (- (exp (* a x)) 1)`

Runtime

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