Average Error: 42.6 → 9.8
Time: 2.3m
Precision: 64
Internal Precision: 1344
$\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) - \ell \cdot \ell}}$
$\begin{array}{l} \mathbf{if}\;t \le -9.078422274793759 \cdot 10^{+92}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{if}\;t \le 7.410915467559403 \cdot 10^{-282} \lor \neg \left(t \le 1.0569840575233042 \cdot 10^{-159} \lor \neg \left(t \le 2.060264808456829 \cdot 10^{+67}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\ell \cdot \frac{2}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(2 - 1\right) \cdot \frac{\frac{t}{x \cdot x}}{\sqrt{2}}}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if t < -9.078422274793759e+92

1. Initial program 50.3

$\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) - \ell \cdot \ell}}$
2. Taylor expanded around -inf 3.4

$\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right)}}$
3. Applied simplify3.4

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

## if -9.078422274793759e+92 < t < 7.410915467559403e-282 or 1.0569840575233042e-159 < t < 2.060264808456829e+67

1. Initial program 34.5

$\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) - \ell \cdot \ell}}$
2. Taylor expanded around inf 15.6

$\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}\right)}}}$
3. Applied simplify15.6

$\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\sqrt{\left(\ell \cdot \ell\right) \cdot \frac{2}{x} + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}}$
4. Using strategy rm
5. Applied associate-*l*11.1

$\leadsto \frac{t \cdot \sqrt{2}}{\sqrt{\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)} + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}$

## if 7.410915467559403e-282 < t < 1.0569840575233042e-159 or 2.060264808456829e+67 < t

1. Initial program 50.5

$\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + \left(2 \cdot t\right) \cdot t\right) - \ell \cdot \ell}}$
2. Taylor expanded around inf 12.4

$\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right) - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}}}$
3. Applied simplify12.4

$\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) + \frac{\frac{t}{x \cdot x}}{\sqrt{2}} \cdot \left(2 - 1\right)}}$
3. Recombined 3 regimes into one program.
4. Applied simplify9.8

$\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -9.078422274793759 \cdot 10^{+92}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(1 - 2\right) \cdot \frac{t}{\left(x \cdot x\right) \cdot \sqrt{2}} - \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{if}\;t \le 7.410915467559403 \cdot 10^{-282} \lor \neg \left(t \le 1.0569840575233042 \cdot 10^{-159} \lor \neg \left(t \le 2.060264808456829 \cdot 10^{+67}\right)\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\ell \cdot \frac{2}{x}\right) + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(2 - 1\right) \cdot \frac{\frac{t}{x \cdot x}}{\sqrt{2}}}\\ \end{array}}$

# Runtime

Time bar (total: 2.3m)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (t x l)
:name "sqrt(2)*t/sqrt((x+1)/(x-1)*(l*l + 2*t*t)-l*l)"
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* (* 2 t) t))) (* l l)))))