Average Error: 42.1 → 9.2
Time: 30.3s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.13289705956808 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 7.246088747560428 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le 8.989514490606386 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{elif}\;t \le 45772087522.28849:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]
\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 -7.13289705956808 \cdot 10^{+147}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 7.246088747560428 \cdot 10^{-307}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

\mathbf{elif}\;t \le 8.989514490606386 \cdot 10^{-225}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\

\mathbf{elif}\;t \le 45772087522.28849:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1313032 = 2.0;
        double r1313033 = sqrt(r1313032);
        double r1313034 = t;
        double r1313035 = r1313033 * r1313034;
        double r1313036 = x;
        double r1313037 = 1.0;
        double r1313038 = r1313036 + r1313037;
        double r1313039 = r1313036 - r1313037;
        double r1313040 = r1313038 / r1313039;
        double r1313041 = l;
        double r1313042 = r1313041 * r1313041;
        double r1313043 = r1313034 * r1313034;
        double r1313044 = r1313032 * r1313043;
        double r1313045 = r1313042 + r1313044;
        double r1313046 = r1313040 * r1313045;
        double r1313047 = r1313046 - r1313042;
        double r1313048 = sqrt(r1313047);
        double r1313049 = r1313035 / r1313048;
        return r1313049;
}
↓
double f(double x, double l, double t) {
        double r1313050 = t;
        double r1313051 = -7.13289705956808e+147;
        bool r1313052 = r1313050 <= r1313051;
        double r1313053 = 2.0;
        double r1313054 = sqrt(r1313053);
        double r1313055 = r1313054 * r1313050;
        double r1313056 = r1313050 / r1313054;
        double r1313057 = x;
        double r1313058 = r1313057 * r1313057;
        double r1313059 = r1313056 / r1313058;
        double r1313060 = r1313057 * r1313054;
        double r1313061 = r1313050 / r1313060;
        double r1313062 = r1313061 + r1313059;
        double r1313063 = r1313062 * r1313053;
        double r1313064 = r1313059 - r1313063;
        double r1313065 = r1313064 - r1313055;
        double r1313066 = r1313055 / r1313065;
        double r1313067 = 7.246088747560428e-307;
        bool r1313068 = r1313050 <= r1313067;
        double r1313069 = r1313053 * r1313050;
        double r1313070 = r1313069 * r1313069;
        double r1313071 = r1313070 / r1313057;
        double r1313072 = l;
        double r1313073 = r1313057 / r1313072;
        double r1313074 = r1313072 / r1313073;
        double r1313075 = r1313050 * r1313050;
        double r1313076 = r1313074 + r1313075;
        double r1313077 = r1313053 * r1313076;
        double r1313078 = r1313071 + r1313077;
        double r1313079 = sqrt(r1313078);
        double r1313080 = r1313055 / r1313079;
        double r1313081 = 8.989514490606386e-225;
        bool r1313082 = r1313050 <= r1313081;
        double r1313083 = r1313055 - r1313059;
        double r1313084 = r1313063 + r1313083;
        double r1313085 = r1313055 / r1313084;
        double r1313086 = 45772087522.28849;
        bool r1313087 = r1313050 <= r1313086;
        double r1313088 = r1313087 ? r1313080 : r1313085;
        double r1313089 = r1313082 ? r1313085 : r1313088;
        double r1313090 = r1313068 ? r1313080 : r1313089;
        double r1313091 = r1313052 ? r1313066 : r1313090;
        return r1313091;
}

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 < -7.13289705956808e+147

    1. Initial program 60.6

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

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -7.13289705956808e+147 < t < 7.246088747560428e-307 or 8.989514490606386e-225 < t < 45772087522.28849

    1. Initial program 34.8

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

    2. Taylor expanded around -inf 15.5

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

    3. Simplified15.5

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

    5. Applied associate-/l*10.9

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

    if 7.246088747560428e-307 < t < 8.989514490606386e-225 or 45772087522.28849 < t

    1. Initial program 44.4

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

    2. Taylor expanded around inf 10.0

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

    3. Simplified10.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.13289705956808 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 7.246088747560428 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le 8.989514490606386 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{elif}\;t \le 45772087522.28849:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{x} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))