Average Error: 29.1 → 0.3

Time: 12.1s

Precision: 64

Internal Precision: 128

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0014230971540968266:\\ \;\;\;\;\log \left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	29.1
Target	0.2
Herbie	0.3

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

Derivation

Split input into 2 regimes
if (* a x) < -0.0014230971540968266
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{e^{a \cdot x} - 1}\]
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-cube-cbrt0.0
  \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}}\right)\]
7. Applied exp-prod0.0
  \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right)}^{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}\right)}\]
8. Applied log-pow0.0
  \[\leadsto \color{blue}{\sqrt[3]{e^{a \cdot x} - 1} \cdot \log \left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right)}\]
if -0.0014230971540968266 < (* a x)
1. Initial program 43.8
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 14.3
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.5
  \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x}\]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0014230971540968266:\\ \;\;\;\;\log \left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	21.0	0.3	0.1	20.9	98.7%

herbie shell --seed 2018353 
(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) < -0.0014230971540968266`

`if -0.0014230971540968266 < (* a x)`

Runtime