Average Error: 29.4 → 0.4
Time: 18.6s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -1.7449511921346047 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
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) < -1.7449511921346047e-08

    1. Initial program 0.2

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

    2. Initial simplification0.2

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

    4. Applied add-log-exp0.3

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

    6. Applied add-cube-cbrt0.3

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

    8. Applied add-sqr-sqrt0.3

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

    9. Applied log-prod0.3

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

    if -1.7449511921346047e-08 < (- (exp (* a x)) 1)

    1. Initial program 44.1

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

    2. Initial simplification44.1

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

    3. Taylor expanded around 0 14.2

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

    4. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -1.7449511921346047 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Runtime

Time bar (total: 18.6s)Debug logProfile

herbie shell --seed 2018225 
(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))