Average Error: 31.4 → 24.1
Time: 1.6m
Precision: 64
Internal Precision: 320
$\frac{t \cdot \sqrt{x - 1}}{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}$
$\begin{array}{l} \mathbf{if}\;\ell \le -1.3463465106719066 \cdot 10^{+154}:\\ \;\;\;\;\frac{t}{-\ell} \cdot \sqrt{x - 1}\\ \mathbf{if}\;\ell \le 5.603687235593547 \cdot 10^{+114}:\\ \;\;\;\;\frac{t}{\sqrt{1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x - 1}}}\\ \end{array}$

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Results

 In Out
Enter valid numbers for all inputs

Derivation

1. Split input into 3 regimes
2. if l < -1.3463465106719066e+154

1. Initial program 47.2

$\frac{t \cdot \sqrt{x - 1}}{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}$
2. Using strategy rm
3. Applied associate-/l*47.1

$\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}{\sqrt{x - 1}}}}$
4. Taylor expanded around -inf 13.3

$\leadsto \frac{t}{\frac{\color{blue}{-1 \cdot \ell}}{\sqrt{x - 1}}}$
5. Applied simplify20.3

$\leadsto \color{blue}{\frac{t}{-\ell} \cdot \sqrt{x - 1}}$

if -1.3463465106719066e+154 < l < 5.603687235593547e+114

1. Initial program 26.3

$\frac{t \cdot \sqrt{x - 1}}{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}$
2. Using strategy rm
3. Applied *-un-lft-identity26.3

$\leadsto \frac{t \cdot \sqrt{x - 1}}{\sqrt{\color{blue}{1 \cdot \left(\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t\right)}}}$
4. Applied sqrt-prod26.3

$\leadsto \frac{t \cdot \sqrt{x - 1}}{\color{blue}{\sqrt{1} \cdot \sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}}$
5. Applied times-frac26.2

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

if 5.603687235593547e+114 < l

1. Initial program 42.6

$\frac{t \cdot \sqrt{x - 1}}{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}$
2. Using strategy rm
3. Applied associate-/l*42.5

$\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\ell \cdot \ell + \left(\left(x + 1\right) \cdot t\right) \cdot t}}{\sqrt{x - 1}}}}$
4. Taylor expanded around inf 17.8

$\leadsto \frac{t}{\frac{\color{blue}{\ell}}{\sqrt{x - 1}}}$
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.6m)Debug log

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