Toniolo and Linder, Equation (7)

Average Error: 43.1 → 10.3

Time: 10.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.583977798108110900162476286547585661621 \cdot 10^{138}:\\ \;\;\;\;\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 -6.696367244110020413430084956000329612319 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -3.088634056538455682354250066621880548059 \cdot 10^{-240}:\\ \;\;\;\;\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 9.909309527651243252484098193013516257504 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\sqrt{x}} \cdot \frac{\ell}{\sqrt{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 3.291216122192683495789321781028557839903 \cdot 10^{-130}:\\ \;\;\;\;\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)}\\ \mathbf{elif}\;t \le 2.174990866497024471017537763220387342123 \cdot 10^{85}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}, 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}\]

C

double f(double x, double l, double t) {
        double r41925 = 2.0;
        double r41926 = sqrt(r41925);
        double r41927 = t;
        double r41928 = r41926 * r41927;
        double r41929 = x;
        double r41930 = 1.0;
        double r41931 = r41929 + r41930;
        double r41932 = r41929 - r41930;
        double r41933 = r41931 / r41932;
        double r41934 = l;
        double r41935 = r41934 * r41934;
        double r41936 = r41927 * r41927;
        double r41937 = r41925 * r41936;
        double r41938 = r41935 + r41937;
        double r41939 = r41933 * r41938;
        double r41940 = r41939 - r41935;
        double r41941 = sqrt(r41940);
        double r41942 = r41928 / r41941;
        return r41942;
}

↓

double f(double x, double l, double t) {
        double r41943 = t;
        double r41944 = -9.583977798108111e+138;
        bool r41945 = r41943 <= r41944;
        double r41946 = 2.0;
        double r41947 = sqrt(r41946);
        double r41948 = r41947 * r41943;
        double r41949 = 3.0;
        double r41950 = pow(r41947, r41949);
        double r41951 = x;
        double r41952 = 2.0;
        double r41953 = pow(r41951, r41952);
        double r41954 = r41950 * r41953;
        double r41955 = r41943 / r41954;
        double r41956 = r41947 * r41953;
        double r41957 = r41943 / r41956;
        double r41958 = r41947 * r41951;
        double r41959 = r41943 / r41958;
        double r41960 = r41943 * r41947;
        double r41961 = fma(r41946, r41959, r41960);
        double r41962 = fma(r41946, r41957, r41961);
        double r41963 = -r41962;
        double r41964 = fma(r41946, r41955, r41963);
        double r41965 = r41948 / r41964;
        double r41966 = -6.6963672441100204e-164;
        bool r41967 = r41943 <= r41966;
        double r41968 = pow(r41943, r41952);
        double r41969 = l;
        double r41970 = cbrt(r41951);
        double r41971 = r41970 * r41970;
        double r41972 = r41969 / r41971;
        double r41973 = r41969 / r41970;
        double r41974 = r41972 * r41973;
        double r41975 = 4.0;
        double r41976 = r41968 / r41951;
        double r41977 = r41975 * r41976;
        double r41978 = fma(r41946, r41974, r41977);
        double r41979 = fma(r41946, r41968, r41978);
        double r41980 = sqrt(r41979);
        double r41981 = r41948 / r41980;
        double r41982 = -3.0886340565384557e-240;
        bool r41983 = r41943 <= r41982;
        double r41984 = 9.909309527651243e-270;
        bool r41985 = r41943 <= r41984;
        double r41986 = sqrt(r41951);
        double r41987 = r41969 / r41986;
        double r41988 = r41987 * r41987;
        double r41989 = fma(r41946, r41988, r41977);
        double r41990 = fma(r41946, r41968, r41989);
        double r41991 = sqrt(r41990);
        double r41992 = r41948 / r41991;
        double r41993 = 3.2912161221926835e-130;
        bool r41994 = r41943 <= r41993;
        double r41995 = r41946 * r41955;
        double r41996 = r41961 - r41995;
        double r41997 = fma(r41946, r41957, r41996);
        double r41998 = r41948 / r41997;
        double r41999 = 2.1749908664970245e+85;
        bool r42000 = r41943 <= r41999;
        double r42001 = cbrt(r41969);
        double r42002 = r42001 * r42001;
        double r42003 = pow(r42002, r41952);
        double r42004 = pow(r42001, r41952);
        double r42005 = r41951 / r42004;
        double r42006 = r42003 / r42005;
        double r42007 = fma(r41946, r42006, r41977);
        double r42008 = fma(r41946, r41968, r42007);
        double r42009 = sqrt(r42008);
        double r42010 = r41948 / r42009;
        double r42011 = r42000 ? r42010 : r41998;
        double r42012 = r41994 ? r41998 : r42011;
        double r42013 = r41985 ? r41992 : r42012;
        double r42014 = r41983 ? r41965 : r42013;
        double r42015 = r41967 ? r41981 : r42014;
        double r42016 = r41945 ? r41965 : r42015;
        return r42016;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -9.583977798108111e+138 or -6.6963672441100204e-164 < t < -3.0886340565384557e-240

   1. Initial program 59.1

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

   2. Taylor expanded around -inf 9.8

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

   3. Simplified 9.8

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

   if -9.583977798108111e+138 < t < -6.6963672441100204e-164

   1. Initial program 25.3

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

   2. Taylor expanded around inf 10.3

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

   3. Simplified 10.3

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

   5. Applied add-cube-cbrt 10.4

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

   6. Applied add-sqr-sqrt 37.4

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

   7. Applied unpow-prod-down 37.4

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

   8. Applied times-frac 35.0

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

   9. Simplified 35.0

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

   10. Simplified 5.5

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

   if -3.0886340565384557e-240 < t < 9.909309527651243e-270

   1. Initial program 62.8

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

   2. Taylor expanded around inf 32.0

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

   3. Simplified 32.0

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

   5. Applied add-sqr-sqrt 32.0

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

   6. Applied add-sqr-sqrt 48.0

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

   7. Applied unpow-prod-down 48.0

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

   8. Applied times-frac 48.0

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

   9. Simplified 48.0

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

   10. Simplified 31.9

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

   if 9.909309527651243e-270 < t < 3.2912161221926835e-130 or 2.1749908664970245e+85 < t

   1. Initial program 51.7

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

   2. Taylor expanded around inf 12.2

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

   3. Simplified 12.2

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

   if 3.2912161221926835e-130 < t < 2.1749908664970245e+85

   1. Initial program 25.4

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

   2. Taylor expanded around inf 9.9

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

   3. Simplified 9.9

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

   5. Applied add-cube-cbrt 10.0

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

   6. Applied unpow-prod-down 10.0

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

   7. Applied associate-/l* 6.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \color{blue}{\frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.583977798108110900162476286547585661621 \cdot 10^{138}:\\ \;\;\;\;\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 -6.696367244110020413430084956000329612319 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -3.088634056538455682354250066621880548059 \cdot 10^{-240}:\\ \;\;\;\;\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 9.909309527651243252484098193013516257504 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\sqrt{x}} \cdot \frac{\ell}{\sqrt{x}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 3.291216122192683495789321781028557839903 \cdot 10^{-130}:\\ \;\;\;\;\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)}\\ \mathbf{elif}\;t \le 2.174990866497024471017537763220387342123 \cdot 10^{85}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}, 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 2019346 +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)))))