Average Error: 33.4 → 0.2
Time: 18.4s
Precision: 64
Internal precision: 1408
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.770999575039998 \cdot 10^{-09}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^2 - {1}^2}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot {\left(a \cdot x\right)}^2 + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original33.4
Comparison8.1
Herbie0.2
\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a x) < -4.770999575039998e-09

    1. Initial program 0.3

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied flip-- 0.3

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

    if -4.770999575039998e-09 < (* a x)

    1. Initial program 48.0

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

    2. Applied taylor 22.5

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

    3. Taylor expanded around 0 22.5

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

    4. Applied taylor 12.4

      \[\leadsto \frac{1}{6} \cdot 0 + \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + a \cdot x\right)\]

    5. Taylor expanded around inf 12.4

      \[\leadsto \frac{1}{6} \cdot \color{blue}{0} + \left(\frac{1}{2} \cdot \left({a}^2 \cdot {x}^2\right) + a \cdot x\right)\]

    6. Applied simplify 0.1

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

    7. Applied simplify 0.1

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

Runtime

Time bar (total: 18.4s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (a x)
  :name "expax (section 3.5)"

  :target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))