Average Error: 22.9 → 0.0
Time: 34.1s
Precision: 64
Internal Precision: 2368
$\frac{-1}{\left(-t\right) + \sqrt{1 + t \cdot t}}$
$\begin{array}{l} \mathbf{if}\;t \le -63.65339348579835:\\ \;\;\;\;\frac{-1}{\frac{\frac{\frac{1}{8}}{t}}{t \cdot t} - \left(\left(t + t\right) + \frac{\frac{1}{2}}{t}\right)}\\ \mathbf{if}\;t \le 255.19911368715447:\\ \;\;\;\;\frac{-1}{-1} \cdot \left(\left(-t\right) - \sqrt{1 + t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{8}}{t}}{t \cdot t} - \left(2 \cdot t + \frac{\frac{1}{2}}{t}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if t < -63.65339348579835

1. Initial program 30.1

$\frac{-1}{\left(-t\right) + \sqrt{1 + t \cdot t}}$
2. Taylor expanded around -inf 0.1

$\leadsto \frac{-1}{\left(-t\right) + \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{t}^{3}} - \left(t + \frac{1}{2} \cdot \frac{1}{t}\right)\right)}}$
3. Applied simplify0.1

$\leadsto \color{blue}{\frac{-1}{\frac{\frac{\frac{1}{8}}{t}}{t \cdot t} - \left(\left(t + t\right) + \frac{\frac{1}{2}}{t}\right)}}$

## if -63.65339348579835 < t < 255.19911368715447

1. Initial program 0.1

$\frac{-1}{\left(-t\right) + \sqrt{1 + t \cdot t}}$
2. Using strategy rm
3. Applied flip-+0.1

$\leadsto \frac{-1}{\color{blue}{\frac{\left(-t\right) \cdot \left(-t\right) - \sqrt{1 + t \cdot t} \cdot \sqrt{1 + t \cdot t}}{\left(-t\right) - \sqrt{1 + t \cdot t}}}}$
4. Applied associate-/r/0.1

$\leadsto \color{blue}{\frac{-1}{\left(-t\right) \cdot \left(-t\right) - \sqrt{1 + t \cdot t} \cdot \sqrt{1 + t \cdot t}} \cdot \left(\left(-t\right) - \sqrt{1 + t \cdot t}\right)}$
5. Applied simplify0.0

$\leadsto \color{blue}{\frac{-1}{-1}} \cdot \left(\left(-t\right) - \sqrt{1 + t \cdot t}\right)$

## if 255.19911368715447 < t

1. Initial program 61.6

$\frac{-1}{\left(-t\right) + \sqrt{1 + t \cdot t}}$
2. Taylor expanded around inf 0.0

$\leadsto \color{blue}{\frac{1}{8} \cdot \frac{1}{{t}^{3}} - \left(2 \cdot t + \frac{1}{2} \cdot \frac{1}{t}\right)}$
3. Applied simplify0.0

$\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{t}}{t \cdot t} - \left(2 \cdot t + \frac{\frac{1}{2}}{t}\right)}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 34.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (t)
:name "-1/(-t + sqrt(1 + t*t))"
(/ (- 1) (+ (- t) (sqrt (+ 1 (* t t))))))