Input Error: 59.3b
Output Error: 0.1b
Time: 6.4s
Precision: 64b
Ground Truth: 128b
\[e^{x} - 1\]
⬇
\[{x}^2 \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x\]

Error

Bits error versus x

Derivation

  1. Initial program 59.3b

    \[e^{x} - 1\]

  2. Applied taylor 0.1b

    \[\leadsto \frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)\]

  3. Taylor expanded around 0 0.1b

    \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^2 + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]

  4. Applied simplify 0.1b

    \[\leadsto \color{blue}{{x}^2 \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x}\]
  5. Removed slow pow expressions

Runtime

Total time: 6.4s Debug log

herbie --seed '#(4129251812 871381683 3506654301 2668084457 1324419290 3276905187)'
(FPCore (x)
  :name "NMSE example 3.7"
  
  :target
  (* x (+ (+ 1 (/ x 2)) (/ (sqr x) 6)))(- (exp x) 1))