Average Error: 43.3 → 9.2
Time: 52.3s
Precision: 64
\[\frac{\sqrt{2.0} \cdot t}{\sqrt{\frac{x + 1.0}{x - 1.0} \cdot \left(\ell \cdot \ell + 2.0 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.1438100975772846 \cdot 10^{+85}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)} - \frac{\frac{t}{\sqrt{2.0}}}{x}\right) - \frac{\frac{t}{\sqrt{2.0}}}{x \cdot x}\right) \cdot 2.0 - \sqrt{2.0} \cdot t}\\ \mathbf{elif}\;t \le 1.311395206965946 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\ \mathbf{elif}\;t \le 3.1685097036544866 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\ \mathbf{elif}\;t \le 5.493219746550013 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\ \end{array}\]
\frac{\sqrt{2.0} \cdot t}{\sqrt{\frac{x + 1.0}{x - 1.0} \cdot \left(\ell \cdot \ell + 2.0 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.1438100975772846 \cdot 10^{+85}:\\
\;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)} - \frac{\frac{t}{\sqrt{2.0}}}{x}\right) - \frac{\frac{t}{\sqrt{2.0}}}{x \cdot x}\right) \cdot 2.0 - \sqrt{2.0} \cdot t}\\

\mathbf{elif}\;t \le 1.311395206965946 \cdot 10^{-266}:\\
\;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\

\mathbf{elif}\;t \le 3.1685097036544866 \cdot 10^{-182}:\\
\;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\

\mathbf{elif}\;t \le 5.493219746550013 \cdot 10^{+84}:\\
\;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\

\end{array}
double f(double x, double l, double t) {
        double r1681166 = 2.0;
        double r1681167 = sqrt(r1681166);
        double r1681168 = t;
        double r1681169 = r1681167 * r1681168;
        double r1681170 = x;
        double r1681171 = 1.0;
        double r1681172 = r1681170 + r1681171;
        double r1681173 = r1681170 - r1681171;
        double r1681174 = r1681172 / r1681173;
        double r1681175 = l;
        double r1681176 = r1681175 * r1681175;
        double r1681177 = r1681168 * r1681168;
        double r1681178 = r1681166 * r1681177;
        double r1681179 = r1681176 + r1681178;
        double r1681180 = r1681174 * r1681179;
        double r1681181 = r1681180 - r1681176;
        double r1681182 = sqrt(r1681181);
        double r1681183 = r1681169 / r1681182;
        return r1681183;
}
↓
double f(double x, double l, double t) {
        double r1681184 = t;
        double r1681185 = -4.1438100975772846e+85;
        bool r1681186 = r1681184 <= r1681185;
        double r1681187 = 2.0;
        double r1681188 = sqrt(r1681187);
        double r1681189 = r1681188 * r1681184;
        double r1681190 = x;
        double r1681191 = r1681190 * r1681190;
        double r1681192 = r1681187 * r1681188;
        double r1681193 = r1681191 * r1681192;
        double r1681194 = r1681184 / r1681193;
        double r1681195 = r1681184 / r1681188;
        double r1681196 = r1681195 / r1681190;
        double r1681197 = r1681194 - r1681196;
        double r1681198 = r1681195 / r1681191;
        double r1681199 = r1681197 - r1681198;
        double r1681200 = r1681199 * r1681187;
        double r1681201 = r1681200 - r1681189;
        double r1681202 = r1681189 / r1681201;
        double r1681203 = 1.311395206965946e-266;
        bool r1681204 = r1681184 <= r1681203;
        double r1681205 = l;
        double r1681206 = r1681205 / r1681190;
        double r1681207 = r1681206 * r1681205;
        double r1681208 = r1681184 * r1681184;
        double r1681209 = r1681207 + r1681208;
        double r1681210 = r1681187 * r1681209;
        double r1681211 = 4.0;
        double r1681212 = r1681190 / r1681208;
        double r1681213 = r1681211 / r1681212;
        double r1681214 = r1681210 + r1681213;
        double r1681215 = sqrt(r1681214);
        double r1681216 = r1681189 / r1681215;
        double r1681217 = 3.1685097036544866e-182;
        bool r1681218 = r1681184 <= r1681217;
        double r1681219 = r1681187 / r1681190;
        double r1681220 = r1681196 * r1681219;
        double r1681221 = r1681196 - r1681194;
        double r1681222 = r1681221 * r1681187;
        double r1681223 = r1681220 + r1681222;
        double r1681224 = r1681223 + r1681189;
        double r1681225 = r1681189 / r1681224;
        double r1681226 = 5.493219746550013e+84;
        bool r1681227 = r1681184 <= r1681226;
        double r1681228 = r1681227 ? r1681216 : r1681225;
        double r1681229 = r1681218 ? r1681225 : r1681228;
        double r1681230 = r1681204 ? r1681216 : r1681229;
        double r1681231 = r1681186 ? r1681202 : r1681230;
        return r1681231;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -4.1438100975772846e+85

    1. Initial program 47.1

      \[\frac{\sqrt{2.0} \cdot t}{\sqrt{\frac{x + 1.0}{x - 1.0} \cdot \left(\ell \cdot \ell + 2.0 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 2.5

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\color{blue}{2.0 \cdot \frac{t}{{\left(\sqrt{2.0}\right)}^{3} \cdot {x}^{2}} - \left(2.0 \cdot \frac{t}{\sqrt{2.0} \cdot x} + \left(\sqrt{2.0} \cdot t + 2.0 \cdot \frac{t}{\sqrt{2.0} \cdot {x}^{2}}\right)\right)}}\]

    3. Simplified2.5

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\color{blue}{2.0 \cdot \left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(\sqrt{2.0} \cdot 2.0\right)} - \frac{\frac{t}{\sqrt{2.0}}}{x}\right) - \frac{\frac{t}{\sqrt{2.0}}}{x \cdot x}\right) - t \cdot \sqrt{2.0}}}\]

    if -4.1438100975772846e+85 < t < 1.311395206965946e-266 or 3.1685097036544866e-182 < t < 5.493219746550013e+84

    1. Initial program 37.6

      \[\frac{\sqrt{2.0} \cdot t}{\sqrt{\frac{x + 1.0}{x - 1.0} \cdot \left(\ell \cdot \ell + 2.0 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 16.1

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\sqrt{\color{blue}{2.0 \cdot {t}^{2} + \left(2.0 \cdot \frac{{\ell}^{2}}{x} + 4.0 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified16.1

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\sqrt{\color{blue}{2.0 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity16.1

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}}\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\]

    6. Applied times-frac11.7

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(t \cdot t + \color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{x}}\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\]

    7. Simplified11.7

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(t \cdot t + \color{blue}{\ell} \cdot \frac{\ell}{x}\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\]

    if 1.311395206965946e-266 < t < 3.1685097036544866e-182 or 5.493219746550013e+84 < t

    1. Initial program 51.4

      \[\frac{\sqrt{2.0} \cdot t}{\sqrt{\frac{x + 1.0}{x - 1.0} \cdot \left(\ell \cdot \ell + 2.0 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 9.9

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\color{blue}{\left(2.0 \cdot \frac{t}{\sqrt{2.0} \cdot x} + \left(\sqrt{2.0} \cdot t + 2.0 \cdot \frac{t}{\sqrt{2.0} \cdot {x}^{2}}\right)\right) - 2.0 \cdot \frac{t}{{\left(\sqrt{2.0}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified9.9

      \[\leadsto \frac{\sqrt{2.0} \cdot t}{\color{blue}{t \cdot \sqrt{2.0} + \left(\frac{2.0}{x} \cdot \frac{\frac{t}{\sqrt{2.0}}}{x} + 2.0 \cdot \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(\sqrt{2.0} \cdot 2.0\right)}\right)\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.1438100975772846 \cdot 10^{+85}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\left(\frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)} - \frac{\frac{t}{\sqrt{2.0}}}{x}\right) - \frac{\frac{t}{\sqrt{2.0}}}{x \cdot x}\right) \cdot 2.0 - \sqrt{2.0} \cdot t}\\ \mathbf{elif}\;t \le 1.311395206965946 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\ \mathbf{elif}\;t \le 3.1685097036544866 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\ \mathbf{elif}\;t \le 5.493219746550013 \cdot 10^{+84}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\sqrt{2.0 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right) + \frac{4.0}{\frac{x}{t \cdot t}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2.0} \cdot t}{\left(\frac{\frac{t}{\sqrt{2.0}}}{x} \cdot \frac{2.0}{x} + \left(\frac{\frac{t}{\sqrt{2.0}}}{x} - \frac{t}{\left(x \cdot x\right) \cdot \left(2.0 \cdot \sqrt{2.0}\right)}\right) \cdot 2.0\right) + \sqrt{2.0} \cdot t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))