Average Error: 22.9 → 12.5
Time: 52.7s
Precision: 64
Internal Precision: 576
$\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}}$
$\begin{array}{l} \mathbf{if}\;\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}} \le 4.84850573732997 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{x}{{y}^{3}} + \frac{x}{y}\right) - \frac{1}{2} \cdot \frac{1}{x \cdot y}\\ \mathbf{if}\;\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}} \le 5.569068589266447 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{x}{{y}^{3}} + \frac{x}{y}\right) - \frac{1}{2} \cdot \frac{1}{x \cdot y}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if (sqrt (/ (- 1 (* x x)) (- 1 (* y y)))) < 4.84850573732997e-310 or 5.569068589266447e+73 < (sqrt (/ (- 1 (* x x)) (- 1 (* y y))))

1. Initial program 58.7

$\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}}$
2. Taylor expanded around inf 32.2

$\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{y}^{3}} + \frac{x}{y}\right) - \frac{1}{2} \cdot \frac{1}{x \cdot y}}$

## if 4.84850573732997e-310 < (sqrt (/ (- 1 (* x x)) (- 1 (* y y)))) < 5.569068589266447e+73

1. Initial program 0.0

$\sqrt{\frac{1 - x \cdot x}{1 - y \cdot y}}$
3. Recombined 2 regimes into one program.

# Runtime

Time bar (total: 52.7s)Debug log

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