Average Error: 37.6 → 25.3
Time: 40.9s
Precision: 64
Internal Precision: 4416
$\frac{0.5}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r} \cdot \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)}}$
$\begin{array}{l} \mathbf{if}\;p \le -9.853934825791699 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{\sqrt{\left(1 + \frac{r}{\sqrt{r \cdot r + \left(p \cdot 4\right) \cdot p}}\right) \cdot 0.5} \cdot \left(p \cdot -2\right)}\\ \mathbf{elif}\;p \le 2.2585478044005836 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{\sqrt{1.0} \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{2.0} \cdot p + \frac{r}{\sqrt{2.0}} \cdot 0.5}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if p < -9.853934825791699e-116

1. Initial program 35.3

$\frac{0.5}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r} \cdot \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)}}$
2. Taylor expanded around -inf 19.5

$\leadsto \frac{0.5}{\color{blue}{\left(-2 \cdot p\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)}}$

## if -9.853934825791699e-116 < p < 2.2585478044005836e-87

1. Initial program 41.0

$\frac{0.5}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r} \cdot \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)}}$
2. Taylor expanded around 0 36.9

$\leadsto \frac{0.5}{\color{blue}{\sqrt{1.0} \cdot r}}$

## if 2.2585478044005836e-87 < p

1. Initial program 36.5

$\frac{0.5}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r} \cdot \sqrt{0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)}}$
2. Using strategy rm
3. Applied sqrt-unprod36.4

$\leadsto \frac{0.5}{\color{blue}{\sqrt{\left(\left(4 \cdot p\right) \cdot p + r \cdot r\right) \cdot \left(0.5 \cdot \left(1 + \frac{r}{\sqrt{\left(4 \cdot p\right) \cdot p + r \cdot r}}\right)\right)}}}$
4. Taylor expanded around inf 19.4

$\leadsto \frac{0.5}{\color{blue}{0.5 \cdot \frac{r}{\sqrt{2.0}} + p \cdot \sqrt{2.0}}}$
3. Recombined 3 regimes into one program.
4. Final simplification25.3

$\leadsto \begin{array}{l} \mathbf{if}\;p \le -9.853934825791699 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{\sqrt{\left(1 + \frac{r}{\sqrt{r \cdot r + \left(p \cdot 4\right) \cdot p}}\right) \cdot 0.5} \cdot \left(p \cdot -2\right)}\\ \mathbf{elif}\;p \le 2.2585478044005836 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5}{\sqrt{1.0} \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{2.0} \cdot p + \frac{r}{\sqrt{2.0}} \cdot 0.5}\\ \end{array}$

# Runtime

Time bar (total: 40.9s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (p r)
:name "0.5/(sqrt(4*p*p + r*r) *sqrt(0.5*(1 +r/sqrt(4*p*p + r*r))))"
(/ 0.5 (* (sqrt (+ (* (* 4 p) p) (* r r))) (sqrt (* 0.5 (+ 1 (/ r (sqrt (+ (* (* 4 p) p) (* r r))))))))))