Average Error: 22.8 → 10.9
Time: 42.4s
Precision: 64
Internal Precision: 3392
$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p + q \cdot q}}\right)}$
$\begin{array}{l} \mathbf{if}\;q \le -9.74180062808602 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{if}\;q \le 3.7899226995403642 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(q \cdot \left(\sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}} \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if q < -9.74180062808602e+175

1. Initial program 61.5

$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p + q \cdot q}}\right)}$
2. Taylor expanded around -inf 43.9

$\leadsto \sqrt{0.5 \cdot \color{blue}{0}}$
3. Applied simplify43.9

$\leadsto \color{blue}{\sqrt{0}}$

## if -9.74180062808602e+175 < q < 3.7899226995403642e+124

1. Initial program 10.4

$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p + q \cdot q}}\right)}$
2. Using strategy rm
3. Applied div-inv10.4

$\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{q \cdot \frac{1}{\sqrt{p + q \cdot q}}}\right)}$
4. Using strategy rm

$\leadsto \sqrt{0.5 \cdot \left(1 + q \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}} \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}\right)}\right)}$
6. Applied associate-*r*10.5

$\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(q \cdot \left(\sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}} \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\sqrt{p + q \cdot q}}}}\right)}$

## if 3.7899226995403642e+124 < q

1. Initial program 43.1

$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p + q \cdot q}}\right)}$
2. Taylor expanded around 0 0.3

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{q}{\color{blue}{q}}\right)}$
3. Applied simplify0.3

$\leadsto \color{blue}{\sqrt{0.5 + 0.5}}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 42.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (q p)
:name "sqrt(0.5*(1 + q/sqrt(p + q*q)))"
(sqrt (* 0.5 (+ 1 (/ q (sqrt (+ p (* q q))))))))