Toniolo and Linder, Equation (7)

Average Error: 43.4 → 9.7

Time: 11.9s

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 -7.7906296943154337 \cdot 10^{114}:\\ \;\;\;\;\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 -2.3984297611885329 \cdot 10^{-164}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}} \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\right) \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\\ \mathbf{elif}\;t \le -7.1589711936441436 \cdot 10^{-250}:\\ \;\;\;\;\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 2.8121907514337819 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{\left|\sqrt[3]{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}\right|}{t}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\\ \mathbf{elif}\;t \le 4.05887515224947638 \cdot 10^{-230}:\\ \;\;\;\;\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 3.08908601063149641 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}{\sqrt{2} \cdot t}}\\ \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 r43945 = 2.0;
        double r43946 = sqrt(r43945);
        double r43947 = t;
        double r43948 = r43946 * r43947;
        double r43949 = x;
        double r43950 = 1.0;
        double r43951 = r43949 + r43950;
        double r43952 = r43949 - r43950;
        double r43953 = r43951 / r43952;
        double r43954 = l;
        double r43955 = r43954 * r43954;
        double r43956 = r43947 * r43947;
        double r43957 = r43945 * r43956;
        double r43958 = r43955 + r43957;
        double r43959 = r43953 * r43958;
        double r43960 = r43959 - r43955;
        double r43961 = sqrt(r43960);
        double r43962 = r43948 / r43961;
        return r43962;
}

↓

double f(double x, double l, double t) {
        double r43963 = t;
        double r43964 = -7.790629694315434e+114;
        bool r43965 = r43963 <= r43964;
        double r43966 = 2.0;
        double r43967 = sqrt(r43966);
        double r43968 = r43967 * r43963;
        double r43969 = 3.0;
        double r43970 = pow(r43967, r43969);
        double r43971 = x;
        double r43972 = 2.0;
        double r43973 = pow(r43971, r43972);
        double r43974 = r43970 * r43973;
        double r43975 = r43963 / r43974;
        double r43976 = r43967 * r43973;
        double r43977 = r43963 / r43976;
        double r43978 = r43967 * r43971;
        double r43979 = r43963 / r43978;
        double r43980 = r43963 * r43967;
        double r43981 = fma(r43966, r43979, r43980);
        double r43982 = fma(r43966, r43977, r43981);
        double r43983 = -r43982;
        double r43984 = fma(r43966, r43975, r43983);
        double r43985 = r43968 / r43984;
        double r43986 = -2.398429761188533e-164;
        bool r43987 = r43963 <= r43986;
        double r43988 = pow(r43963, r43972);
        double r43989 = l;
        double r43990 = r43971 / r43989;
        double r43991 = r43989 / r43990;
        double r43992 = 4.0;
        double r43993 = r43988 / r43971;
        double r43994 = r43992 * r43993;
        double r43995 = fma(r43966, r43991, r43994);
        double r43996 = fma(r43966, r43988, r43995);
        double r43997 = sqrt(r43996);
        double r43998 = r43968 / r43997;
        double r43999 = cbrt(r43998);
        double r44000 = r43999 * r43999;
        double r44001 = r44000 * r43999;
        double r44002 = -7.158971193644144e-250;
        bool r44003 = r43963 <= r44002;
        double r44004 = 2.812190751433782e-269;
        bool r44005 = r43963 <= r44004;
        double r44006 = cbrt(r43996);
        double r44007 = fabs(r44006);
        double r44008 = r44007 / r43963;
        double r44009 = r43967 / r44008;
        double r44010 = sqrt(r44006);
        double r44011 = r44009 / r44010;
        double r44012 = 4.0588751522494764e-230;
        bool r44013 = r43963 <= r44012;
        double r44014 = r43966 * r43975;
        double r44015 = r43981 - r44014;
        double r44016 = fma(r43966, r43977, r44015);
        double r44017 = r43968 / r44016;
        double r44018 = 3.0890860106314964e+41;
        bool r44019 = r43963 <= r44018;
        double r44020 = 1.0;
        double r44021 = r43997 / r43968;
        double r44022 = r44020 / r44021;
        double r44023 = r44019 ? r44022 : r44017;
        double r44024 = r44013 ? r44017 : r44023;
        double r44025 = r44005 ? r44011 : r44024;
        double r44026 = r44003 ? r43985 : r44025;
        double r44027 = r43987 ? r44001 : r44026;
        double r44028 = r43965 ? r43985 : r44027;
        return r44028;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -7.790629694315434e+114 or -2.398429761188533e-164 < t < -7.158971193644144e-250

   1. Initial program 55.7

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

   2. Taylor expanded around -inf 10.0

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

   3. Simplified 10.0

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

   if -7.790629694315434e+114 < t < -2.398429761188533e-164

   1. Initial program 25.9

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

   2. Taylor expanded around inf 10.6

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

   3. Simplified 10.6

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

   5. Applied unpow2 10.6

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

   6. Applied associate-/l* 5.7

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

   8. Applied add-cube-cbrt 5.6

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

   if -7.158971193644144e-250 < t < 2.812190751433782e-269

   1. Initial program 62.5

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

   2. Taylor expanded around inf 29.9

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

   3. Simplified 29.9

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

   5. Applied unpow2 29.9

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

   6. Applied associate-/l* 29.4

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

   8. Applied add-cube-cbrt 29.6

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

   9. Applied sqrt-prod 29.6

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

   10. Applied associate-/r* 29.6

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

   11. Simplified 29.7

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

   if 2.812190751433782e-269 < t < 4.0588751522494764e-230 or 3.0890860106314964e+41 < t

   1. Initial program 45.9

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

   2. Taylor expanded around inf 7.1

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

   3. Simplified 7.1

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

   if 4.0588751522494764e-230 < t < 3.0890860106314964e+41

   1. Initial program 39.3

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

   2. Taylor expanded around inf 17.0

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

   3. Simplified 17.0

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

   5. Applied unpow2 17.0

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

   6. Applied associate-/l* 11.5

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

   8. Applied clear-num 11.7

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}{\sqrt{2} \cdot t}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.7906296943154337 \cdot 10^{114}:\\ \;\;\;\;\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 -2.3984297611885329 \cdot 10^{-164}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}} \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\right) \cdot \sqrt[3]{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\\ \mathbf{elif}\;t \le -7.1589711936441436 \cdot 10^{-250}:\\ \;\;\;\;\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 2.8121907514337819 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{\left|\sqrt[3]{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}\right|}{t}}}{\sqrt{\sqrt[3]{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\\ \mathbf{elif}\;t \le 4.05887515224947638 \cdot 10^{-230}:\\ \;\;\;\;\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 3.08908601063149641 \cdot 10^{41}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}{\sqrt{2} \cdot t}}\\ \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 2020083 +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)))))