Average Error: 23.6 → 16.5
Time: 33.1s
Precision: 64
Internal Precision: 2624
$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p \cdot p + q \cdot q}}\right)}$
$\begin{array}{l} \mathbf{if}\;q \le -9.72744973566335 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{if}\;q \le 6.294595312968477 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{q}{\sqrt{p \cdot p + q \cdot q}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if q < -9.72744973566335e+175

1. Initial program 55.5

$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p \cdot p + q \cdot q}}\right)}$
2. Taylor expanded around -inf 35.2

$\leadsto \sqrt{0.5 \cdot \color{blue}{0}}$
3. Applied simplify35.2

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

## if -9.72744973566335e+175 < q < 6.294595312968477e+146

1. Initial program 15.4

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

$\leadsto \sqrt{0.5 \cdot \color{blue}{\log \left(e^{1 + \frac{q}{\sqrt{p \cdot p + q \cdot q}}}\right)}}$

## if 6.294595312968477e+146 < q

1. Initial program 43.0

$\sqrt{0.5 \cdot \left(1 + \frac{q}{\sqrt{p \cdot p + q \cdot q}}\right)}$
2. Taylor expanded around 0 7.3

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{q}{\color{blue}{q}}\right)}$
3. Applied simplify7.3

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

# Runtime

Time bar (total: 33.1s)Debug log

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