Average Error: 32.4 → 12.7
Time: 30.2s
Precision: 64
Internal precision: 128
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.5153768898317694 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{x \cdot \left(a + a\right)}}{e^{a \cdot x} + 1} - \frac{1}{1 + e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({a}^3 \cdot {x}^3\right) \cdot \frac{1}{6} + x \cdot a\right) + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original32.4
Comparison5.9
Herbie12.7
\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a x) < -1.5153768898317694e-23

    1. Initial program 2.3

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

    3. Applied flip-- 2.3

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

    5. Applied add-exp-log 2.3

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

    6. Applied simplify 2.3

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

    8. Applied div-sub 2.3

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

    9. Applied simplify 2.3

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

    if -1.5153768898317694e-23 < (* a x)

    1. Initial program 46.3

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

    2. Applied taylor 17.5

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

    3. Taylor expanded around 0 17.5

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

    4. Applied simplify 17.5

      \[\leadsto \color{blue}{\left(\left({a}^3 \cdot {x}^3\right) \cdot \frac{1}{6} + x \cdot a\right) + \frac{1}{2} \cdot {\left(x \cdot a\right)}^2}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 30.2s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2828272200 303340187 932035863 914110578 2524963528 4077377947)'
(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))