Average Error: 29.4 → 0.6
Time: 28.4s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.690133559154876 \cdot 10^{-12}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.690133559154876e-12

    1. Initial program 0.6

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

    3. Applied add-cube-cbrt0.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.6

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.6

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

    8. Applied difference-of-sqr-10.6

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

    if -3.690133559154876e-12 < (* a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 46.0

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

    3. Applied simplify0.6

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

Runtime

Time bar (total: 28.4s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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))