Average Error: 29.2 → 0.4
Time: 30.3s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -2.0406204877752697 \cdot 10^{-07}:\\ \;\;\;\;\frac{\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {1}^{3}\right) \cdot \left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} - {1}^{3}\right)}{\left(\sqrt{e^{x \cdot a}} + \left(1 + e^{x \cdot a}\right)\right) \cdot \left(\left(1 + e^{x \cdot a}\right) - \sqrt{e^{x \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.2
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (exp (* a x)) 1) < -2.0406204877752697e-07

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied difference-of-sqr-10.2

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

    6. Applied flip3--0.2

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

    7. Applied flip3-+0.2

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

    8. Applied frac-times0.2

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

    9. Applied simplify0.2

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

    if -2.0406204877752697e-07 < (- (exp (* a x)) 1)

    1. Initial program 44.6

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

    2. Taylor expanded around 0 45.8

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

    3. Applied simplify0.5

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

Runtime

Time bar (total: 30.3s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))