Average Error: 22.9 → 14.6
Time: 46.4s
Precision: 64
Internal Precision: 2368
\[\sqrt{16 \cdot {x}^{2} + 1} - 4 \cdot x\]
\[\begin{array}{l} \mathbf{if}\;x \le 8.840052920711461 \cdot 10^{-07}:\\ \;\;\;\;\sqrt{16 \cdot {x}^{2} + 1} - 4 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(16 - 4 \cdot 4\right) \cdot x\right) \cdot x}{\sqrt{16 \cdot {x}^{2} + 1} + 4 \cdot x}\\ \end{array}\]

Error

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 x < 8.840052920711461e-07

    1. Initial program 10.0

      \[\sqrt{16 \cdot {x}^{2} + 1} - 4 \cdot x\]

    if 8.840052920711461e-07 < x

    1. Initial program 60.1

      \[\sqrt{16 \cdot {x}^{2} + 1} - 4 \cdot x\]
    2. Using strategy rm
    3. Applied flip--59.7

      \[\leadsto \color{blue}{\frac{\sqrt{16 \cdot {x}^{2} + 1} \cdot \sqrt{16 \cdot {x}^{2} + 1} - \left(4 \cdot x\right) \cdot \left(4 \cdot x\right)}{\sqrt{16 \cdot {x}^{2} + 1} + 4 \cdot x}}\]
    4. Applied simplify29.5

      \[\leadsto \frac{\color{blue}{1 + \left(16 - 4 \cdot 4\right) \cdot \left(x \cdot x\right)}}{\sqrt{16 \cdot {x}^{2} + 1} + 4 \cdot x}\]
    5. Using strategy rm
    6. Applied associate-*r*28.0

      \[\leadsto \frac{1 + \color{blue}{\left(\left(16 - 4 \cdot 4\right) \cdot x\right) \cdot x}}{\sqrt{16 \cdot {x}^{2} + 1} + 4 \cdot x}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 46.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "sqrt(16 x^2 + 1) - 4 x"
  (- (sqrt (+ (* 16 (pow x 2)) 1)) (* 4 x)))