Average Error: 28.9 → 0.3
Time: 35.4s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -2.3403207623857413 \cdot 10^{-06}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if (- (exp (* a x)) 1) < -2.3403207623857413e-06

    1. Initial program 0.1

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm
    3. Applied add-cube-cbrt0.1

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

    if -2.3403207623857413e-06 < (- (exp (* a x)) 1)

    1. Initial program 44.1

      \[e^{a \cdot x} - 1\]
    2. Taylor expanded around 0 13.8

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)}\]
    3. Applied simplify0.4

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

Runtime

Time bar (total: 35.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (a x)
  :name "exp(a*x) - 1"
  (- (exp (* a x)) 1))