Average Error: 43.2 → 9.6
Time: 9.0s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.0191951041229934 \cdot 10^{143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.7232019252513384 \cdot 10^{81}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -5.0191951041229934 \cdot 10^{143}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\

\mathbf{elif}\;t \le 5.7232019252513384 \cdot 10^{81}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r38217 = 2.0;
        double r38218 = sqrt(r38217);
        double r38219 = t;
        double r38220 = r38218 * r38219;
        double r38221 = x;
        double r38222 = 1.0;
        double r38223 = r38221 + r38222;
        double r38224 = r38221 - r38222;
        double r38225 = r38223 / r38224;
        double r38226 = l;
        double r38227 = r38226 * r38226;
        double r38228 = r38219 * r38219;
        double r38229 = r38217 * r38228;
        double r38230 = r38227 + r38229;
        double r38231 = r38225 * r38230;
        double r38232 = r38231 - r38227;
        double r38233 = sqrt(r38232);
        double r38234 = r38220 / r38233;
        return r38234;
}
↓
double f(double x, double l, double t) {
        double r38235 = t;
        double r38236 = -5.019195104122993e+143;
        bool r38237 = r38235 <= r38236;
        double r38238 = 2.0;
        double r38239 = sqrt(r38238);
        double r38240 = r38239 * r38235;
        double r38241 = 3.0;
        double r38242 = pow(r38239, r38241);
        double r38243 = x;
        double r38244 = 2.0;
        double r38245 = pow(r38243, r38244);
        double r38246 = r38242 * r38245;
        double r38247 = r38235 / r38246;
        double r38248 = r38239 * r38245;
        double r38249 = r38235 / r38248;
        double r38250 = r38239 * r38243;
        double r38251 = r38235 / r38250;
        double r38252 = r38235 * r38239;
        double r38253 = fma(r38238, r38251, r38252);
        double r38254 = fma(r38238, r38249, r38253);
        double r38255 = -r38254;
        double r38256 = fma(r38238, r38247, r38255);
        double r38257 = r38240 / r38256;
        double r38258 = 5.7232019252513384e+81;
        bool r38259 = r38235 <= r38258;
        double r38260 = cbrt(r38239);
        double r38261 = r38260 * r38260;
        double r38262 = r38260 * r38235;
        double r38263 = r38261 * r38262;
        double r38264 = pow(r38235, r38244);
        double r38265 = l;
        double r38266 = r38265 / r38243;
        double r38267 = r38265 * r38266;
        double r38268 = 4.0;
        double r38269 = r38264 / r38243;
        double r38270 = r38268 * r38269;
        double r38271 = fma(r38238, r38267, r38270);
        double r38272 = fma(r38238, r38264, r38271);
        double r38273 = sqrt(r38272);
        double r38274 = r38263 / r38273;
        double r38275 = r38238 * r38247;
        double r38276 = r38253 - r38275;
        double r38277 = fma(r38238, r38249, r38276);
        double r38278 = r38240 / r38277;
        double r38279 = r38259 ? r38274 : r38278;
        double r38280 = r38237 ? r38257 : r38279;
        return r38280;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -5.019195104122993e+143

    1. Initial program 59.4

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -5.019195104122993e+143 < t < 5.7232019252513384e+81

    1. Initial program 36.8

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 17.8

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

    3. Simplified17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\ell}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    6. Applied add-sqr-sqrt40.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{2}}{1 \cdot x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    7. Applied unpow-prod-down40.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{2} \cdot {\left(\sqrt{\ell}\right)}^{2}}}{1 \cdot x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    8. Applied times-frac38.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\frac{{\left(\sqrt{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    9. Simplified38.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\ell} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    10. Simplified13.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \color{blue}{\frac{\ell}{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt13.8

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    13. Applied associate-*l*13.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

    if 5.7232019252513384e+81 < t

    1. Initial program 48.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.0191951041229934 \cdot 10^{143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.7232019252513384 \cdot 10^{81}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))