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

# Try it out

Results

 In Out
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)))