Average Error: 29.2 → 0.3

Time: 24.6s

Precision: 64

Internal Precision: 1408

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -4.855291007064373 \cdot 10^{-05}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(e^{e^{x \cdot a} - 1}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot a\right)\right) + x \cdot a\\ \end{array}\]

Error

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

Target

Original	29.2
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 (- (exp (* a x)) 1) < -4.855291007064373e-05
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-cbrt-cube0.1
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]
4. Applied simplify0.1
  \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{x \cdot a} - 1\right)}^{3}}}\]
5. Using strategy rm
6. Applied add-log-exp0.1
  \[\leadsto \sqrt[3]{{\color{blue}{\left(\log \left(e^{e^{x \cdot a} - 1}\right)\right)}}^{3}}\]
if -4.855291007064373e-05 < (- (exp (* a x)) 1)
1. Initial program 44.5
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-cbrt-cube44.6
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]
4. Applied simplify44.6
  \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{x \cdot a} - 1\right)}^{3}}}\]
5. Taylor expanded around 0 50.9
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(e^{\log a + \log x} \cdot \left({a}^{2} \cdot {x}^{2}\right)\right) + \left(e^{\log a + \log x} + \frac{1}{2} \cdot \left(e^{\log a + \log x} \cdot \left(a \cdot x\right)\right)\right)}\]
6. Applied simplify0.4
  \[\leadsto \color{blue}{\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot a\right)\right) + x \cdot a}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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) < -4.855291007064373e-05`

`if -4.855291007064373e-05 < (- (exp (* a x)) 1)`

Runtime