Average Error: 39.1 → 0.2
Time: 22.7s
Precision: 64
Internal Precision: 1344
\[{\left(1 + x\right)}^{\left(\frac{1}{2}\right)} - 1\]
\[\begin{array}{l} \mathbf{if}\;x \le 0.00021811841864165541:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{2}\right)} - 1\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < 0.00021811841864165541

    1. Initial program 58.8

      \[{\left(1 + x\right)}^{\left(\frac{1}{2}\right)} - 1\]
    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot {x}^{3} + \frac{1}{2} \cdot x\right) - \frac{1}{8} \cdot {x}^{2}}\]
    3. Simplified0.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{16} \cdot x - \frac{1}{8}\right) \cdot \left(x \cdot x\right)}\]

    if 0.00021811841864165541 < x

    1. Initial program 0.1

      \[{\left(1 + x\right)}^{\left(\frac{1}{2}\right)} - 1\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00021811841864165541:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{2}\right)} - 1\\ \end{array}\]

Runtime

Time bar (total: 22.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "(1+x)^(1/2) - 1"
  (- (pow (+ 1 x) (/ 1 2)) 1))