Average Error: 29.3 → 0.0
Time: 3.7s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[(e^{a \cdot x} - 1)^*\]

Error

Bits error versus a

Bits error versus x

Target

Original29.3
Target0.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Initial program 29.3

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

  2. Simplified0.0

    \[\leadsto \color{blue}{(e^{a \cdot x} - 1)^*}\]

  3. Final simplification0.0

    \[\leadsto (e^{a \cdot x} - 1)^*\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(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))