Average Error: 38.8 → 0.2
Time: 9.1s
Precision: 64
Internal Precision: 1344
$\left(-1\right) + \sqrt{1 - x}$
$\begin{array}{l} \mathbf{if}\;x \le -0.00013815257582370472:\\ \;\;\;\;\sqrt{1 - x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \left(\left(\frac{1}{16} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{16} \cdot \left(x \cdot x\right)\right) - \left(x \cdot \frac{1}{8} + \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{8} + \frac{1}{2}\right)\right)}{\frac{1}{16} \cdot \left(x \cdot x\right) - \left(x \cdot \frac{1}{8} + \frac{1}{2}\right)}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < -0.00013815257582370472

1. Initial program 0.1

$\left(-1\right) + \sqrt{1 - x}$
2. Initial simplification0.1

$\leadsto \sqrt{1 - x} - 1$

## if -0.00013815257582370472 < x

1. Initial program 58.8

$\left(-1\right) + \sqrt{1 - x}$
2. Initial simplification58.8

$\leadsto \sqrt{1 - x} - 1$
3. Taylor expanded around 0 0.2

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

$\leadsto \color{blue}{\left(-x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{16} + \left(\frac{1}{2} + \frac{1}{8} \cdot x\right)\right)}$
5. Using strategy rm
6. Applied flip-+0.2

$\leadsto \left(-x\right) \cdot \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right) - \left(\frac{1}{2} + \frac{1}{8} \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot x\right)}{\left(x \cdot x\right) \cdot \frac{1}{16} - \left(\frac{1}{2} + \frac{1}{8} \cdot x\right)}}$
7. Applied associate-*r/0.2

$\leadsto \color{blue}{\frac{\left(-x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{16}\right) - \left(\frac{1}{2} + \frac{1}{8} \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot x\right)\right)}{\left(x \cdot x\right) \cdot \frac{1}{16} - \left(\frac{1}{2} + \frac{1}{8} \cdot x\right)}}$
3. Recombined 2 regimes into one program.
4. Final simplification0.2

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00013815257582370472:\\ \;\;\;\;\sqrt{1 - x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \left(\left(\frac{1}{16} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{16} \cdot \left(x \cdot x\right)\right) - \left(x \cdot \frac{1}{8} + \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{8} + \frac{1}{2}\right)\right)}{\frac{1}{16} \cdot \left(x \cdot x\right) - \left(x \cdot \frac{1}{8} + \frac{1}{2}\right)}\\ \end{array}$

# Runtime

Time bar (total: 9.1s)Debug log

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