Average Error: 29.1 → 0.5
Time: 34.1s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -2.836962126801829 \cdot 10^{-12}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(x \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot a\right)\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.1
Target0.2
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 (- (exp (* a x)) 1) < -2.836962126801829e-12

    1. Initial program 0.5

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

    3. Applied add-sqr-sqrt0.6

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

    4. Applied difference-of-sqr-10.6

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

    if -2.836962126801829e-12 < (- (exp (* a x)) 1)

    1. Initial program 44.2

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

    2. Taylor expanded around 0 13.5

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

    3. Taylor expanded around inf 7.5

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

    4. Applied simplify0.5

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

Runtime

Time bar (total: 34.1s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))