Average Error: 29.6 → 0.3
Time: 11.9s
Precision: 64
Internal precision: 1408
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.863851543361235 \cdot 10^{-12}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \left(a \cdot x\right)\right)\right) \cdot \left(1 + \sqrt{e^{x \cdot a}}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Comparison0.4
Herbie0.3
\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a x) < -1.863851543361235e-12

    1. Initial program 0.7

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

    3. Applied add-sqr-sqrt 0.8

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

    4. Applied difference-of-sqr-1 0.8

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

    if -1.863851543361235e-12 < (* a x)

    1. Initial program 45.0

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

    3. Applied add-sqr-sqrt 45.0

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

    4. Applied difference-of-sqr-1 45.0

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

    5. Applied taylor 45.3

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(1 + \left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) - 1\right)\]

    6. Taylor expanded around 0 45.3

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\color{blue}{\left(1 + \left(\frac{1}{8} \cdot \left({a}^2 \cdot {x}^2\right) + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)} - 1\right)\]

    7. Applied simplify 0.0

      \[\leadsto \color{blue}{\left(\left(\left(\frac{1}{8} \cdot x\right) \cdot \left(x \cdot a\right) + \frac{1}{2} \cdot x\right) \cdot a\right) \cdot \left(1 + \sqrt{e^{x \cdot a}}\right)}\]

    8. Applied simplify 0.0

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \left(a \cdot x\right)\right)\right)} \cdot \left(1 + \sqrt{e^{x \cdot a}}\right)\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 11.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(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))