Average Error: 29.5 → 0.4

Time: 13.5s

Precision: 64

Internal Precision: 1344

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.3170629578396014:\\ \;\;\;\;\log \left(e^{\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

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In
Out

Enter valid numbers for all inputs

Target

Original	29.5
Target	0.1
Herbie	0.4

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

Derivation

Split input into 2 regimes
if (* a x) < -2.3170629578396014
1. Initial program 0
  \[e^{a \cdot x} - 1\]
2. Initial simplification0
  \[\leadsto e^{a \cdot x} - 1\]
3. Using strategy rm
4. Applied add-log-exp0.0
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
5. Using strategy rm
6. Applied add-sqr-sqrt0.0
  \[\leadsto \log \left(e^{\color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1}\right)\]
7. Applied difference-of-sqr-10.0
  \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}}\right)\]
if -2.3170629578396014 < (* a x)
1. Initial program 44.5
  \[e^{a \cdot x} - 1\]
2. Initial simplification44.5
  \[\leadsto e^{a \cdot x} - 1\]
3. Taylor expanded around 0 14.1
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
4. 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
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.3170629578396014:\\ \;\;\;\;\log \left(e^{\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	21.2	0.4	0.1	21.1	98.3%

herbie shell --seed 2018274 
(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

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Target

Derivation

`if (* a x) < -2.3170629578396014`

`if -2.3170629578396014 < (* a x)`

Runtime