Average Error: 39.3 → 0.4

Time: 2.9m

Precision: 64

Internal Precision: 1408

\[e^{x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} - 1 \le -4.219088929316808 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{{\left(e^{x + x}\right)}^{3} - 1}{{\left(e^{x}\right)}^{3} + 1}}{e^{x + x} + \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus x

The X axis uses an exponential scale — Bits error versus x

Derivation

Split input into 2 regimes
if (- (exp x) 1) < -4.219088929316808e-05
1. Initial program 0.1
  \[e^{x} - 1\]
2. Using strategy rm
3. Applied flip3--0.1
  \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}\]
4. Applied simplify0.1
  \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{3} - 1}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}\]
5. Applied simplify0.1
  \[\leadsto \frac{{\left(e^{x}\right)}^{3} - 1}{\color{blue}{e^{x + x} + \left(e^{x} + 1\right)}}\]
6. Using strategy rm
7. Applied flip--0.1
  \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} \cdot {\left(e^{x}\right)}^{3} - 1 \cdot 1}{{\left(e^{x}\right)}^{3} + 1}}}{e^{x + x} + \left(e^{x} + 1\right)}\]
8. Applied simplify0.1
  \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x + x}\right)}^{3} - 1}}{{\left(e^{x}\right)}^{3} + 1}}{e^{x + x} + \left(e^{x} + 1\right)}\]
if -4.219088929316808e-05 < (- (exp x) 1)
1. Initial program 58.6
  \[e^{x} - 1\]
2. Taylor expanded around 0 0.5
  \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
3. Applied simplify0.5
  \[\leadsto \color{blue}{x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "NMSE example 3.7"
  (- (exp x) 1))