Average Error: 43.5 → 9.8
Time: 8.7s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.81453570301648773 \cdot 10^{142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 6683.48348240185533:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -2.81453570301648773 \cdot 10^{142}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 6683.48348240185533:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r36104 = 2.0;
        double r36105 = sqrt(r36104);
        double r36106 = t;
        double r36107 = r36105 * r36106;
        double r36108 = x;
        double r36109 = 1.0;
        double r36110 = r36108 + r36109;
        double r36111 = r36108 - r36109;
        double r36112 = r36110 / r36111;
        double r36113 = l;
        double r36114 = r36113 * r36113;
        double r36115 = r36106 * r36106;
        double r36116 = r36104 * r36115;
        double r36117 = r36114 + r36116;
        double r36118 = r36112 * r36117;
        double r36119 = r36118 - r36114;
        double r36120 = sqrt(r36119);
        double r36121 = r36107 / r36120;
        return r36121;
}
↓
double f(double x, double l, double t) {
        double r36122 = t;
        double r36123 = -2.8145357030164877e+142;
        bool r36124 = r36122 <= r36123;
        double r36125 = 2.0;
        double r36126 = sqrt(r36125);
        double r36127 = r36126 * r36122;
        double r36128 = 3.0;
        double r36129 = pow(r36126, r36128);
        double r36130 = x;
        double r36131 = 2.0;
        double r36132 = pow(r36130, r36131);
        double r36133 = r36129 * r36132;
        double r36134 = r36122 / r36133;
        double r36135 = r36126 * r36132;
        double r36136 = r36122 / r36135;
        double r36137 = r36134 - r36136;
        double r36138 = r36125 * r36137;
        double r36139 = r36138 - r36127;
        double r36140 = r36126 * r36130;
        double r36141 = r36122 / r36140;
        double r36142 = r36125 * r36141;
        double r36143 = r36139 - r36142;
        double r36144 = r36127 / r36143;
        double r36145 = 6683.483482401855;
        bool r36146 = r36122 <= r36145;
        double r36147 = 4.0;
        double r36148 = pow(r36122, r36131);
        double r36149 = r36148 / r36130;
        double r36150 = r36147 * r36149;
        double r36151 = l;
        double r36152 = fabs(r36151);
        double r36153 = cbrt(r36130);
        double r36154 = r36152 / r36153;
        double r36155 = r36154 / r36153;
        double r36156 = r36155 * r36154;
        double r36157 = r36148 + r36156;
        double r36158 = r36125 * r36157;
        double r36159 = r36150 + r36158;
        double r36160 = sqrt(r36159);
        double r36161 = r36127 / r36160;
        double r36162 = r36136 + r36141;
        double r36163 = r36125 * r36162;
        double r36164 = r36125 * r36134;
        double r36165 = r36127 - r36164;
        double r36166 = r36163 + r36165;
        double r36167 = r36127 / r36166;
        double r36168 = r36146 ? r36161 : r36167;
        double r36169 = r36124 ? r36144 : r36168;
        return r36169;
}

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 < -2.8145357030164877e+142

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -2.8145357030164877e+142 < t < 6683.483482401855

    1. Initial program 39.8

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

    2. Taylor expanded around inf 18.6

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

    3. Simplified18.6

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

    5. Applied add-cube-cbrt18.7

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

    6. Applied add-sqr-sqrt18.7

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

    7. Applied times-frac18.7

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

    8. Simplified18.7

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

    9. Simplified14.6

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

    if 6683.483482401855 < t

    1. Initial program 41.5

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.81453570301648773 \cdot 10^{142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 6683.48348240185533:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))))