Average Error: 29.3 → 0.5
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.903874562583787 \cdot 10^{-06}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.3
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -4.903874562583787e-06

    1. Initial program 0.1

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

    if -4.903874562583787e-06 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 45.4

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

    3. Applied simplify0.6

      \[\leadsto \color{blue}{\left(a \cdot x + 0\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}}\]

    4. Applied simplify0.6

      \[\leadsto \color{blue}{a \cdot x} + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.9m)Debug log

herbie shell --seed '#(2094665535 3654809497 1731717781 3082199462 566033875 575777438)' 
(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))