Average Error: 27.2 → 24.2
Time: 29.5s
Precision: 64
Internal Precision: 2368
$\frac{2.0}{NdotD + \sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}}$
$\begin{array}{l} \mathbf{if}\;NdotD \le 8.124670755501723 \cdot 10^{+148}:\\ \;\;\;\;\frac{2.0}{\sqrt{\sqrt{\left(alpha_y \cdot YdotD\right) \cdot \left(alpha_y \cdot YdotD\right) + \left(\left(alpha_x \cdot XdotD\right) \cdot \left(alpha_x \cdot XdotD\right) + NdotD \cdot NdotD\right)}} \cdot \sqrt{\sqrt{\left(alpha_y \cdot YdotD\right) \cdot \left(alpha_y \cdot YdotD\right) + \left(\left(alpha_x \cdot XdotD\right) \cdot \left(alpha_x \cdot XdotD\right) + NdotD \cdot NdotD\right)}} + NdotD}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0}{NdotD + NdotD}\\ \end{array}$

# Try it out

Your Program's Arguments

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if NdotD < 8.124670755501723e+148

1. Initial program 24.4

$\frac{2.0}{NdotD + \sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}}$
2. Using strategy rm
3. Applied add-sqr-sqrt24.4

$\leadsto \frac{2.0}{NdotD + \sqrt{\color{blue}{\sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)} \cdot \sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}}}}$
4. Applied sqrt-prod24.6

$\leadsto \frac{2.0}{NdotD + \color{blue}{\sqrt{\sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}} \cdot \sqrt{\sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}}}}$

## if 8.124670755501723e+148 < NdotD

1. Initial program 44.0

$\frac{2.0}{NdotD + \sqrt{\left(NdotD \cdot NdotD + \left(XdotD \cdot alpha_x\right) \cdot \left(XdotD \cdot alpha_x\right)\right) + \left(YdotD \cdot alpha_y\right) \cdot \left(YdotD \cdot alpha_y\right)}}$
2. Taylor expanded around inf 22.1

$\leadsto \frac{2.0}{NdotD + \color{blue}{NdotD}}$
3. Recombined 2 regimes into one program.
4. Final simplification24.2

$\leadsto \begin{array}{l} \mathbf{if}\;NdotD \le 8.124670755501723 \cdot 10^{+148}:\\ \;\;\;\;\frac{2.0}{\sqrt{\sqrt{\left(alpha_y \cdot YdotD\right) \cdot \left(alpha_y \cdot YdotD\right) + \left(\left(alpha_x \cdot XdotD\right) \cdot \left(alpha_x \cdot XdotD\right) + NdotD \cdot NdotD\right)}} \cdot \sqrt{\sqrt{\left(alpha_y \cdot YdotD\right) \cdot \left(alpha_y \cdot YdotD\right) + \left(\left(alpha_x \cdot XdotD\right) \cdot \left(alpha_x \cdot XdotD\right) + NdotD \cdot NdotD\right)}} + NdotD}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0}{NdotD + NdotD}\\ \end{array}$

# Runtime

Time bar (total: 29.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (NdotD XdotD alpha_x YdotD alpha_y)
:name "2.0 / (NdotD + sqrt((NdotD*NdotD) + (XdotD*alpha_x)*(XdotD*alpha_x) + (YdotD*alpha_y)*(YdotD*alpha_y)))"
(/ 2.0 (+ NdotD (sqrt (+ (+ (* NdotD NdotD) (* (* XdotD alpha_x) (* XdotD alpha_x))) (* (* YdotD alpha_y) (* YdotD alpha_y)))))))