Average Error: 42.5 → 9.1
Time: 31.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.046010354271018 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right) - \frac{t}{x \cdot \sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 1.4541398591892197 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(t \cdot t\right) \cdot \frac{4}{x} + 2 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right) + \frac{t}{x \cdot \sqrt{2}} \cdot \frac{2}{x}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \end{array}\]
\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 -1.046010354271018 \cdot 10^{+94}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right) - \frac{t}{x \cdot \sqrt{2}} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 1.4541398591892197 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(t \cdot t\right) \cdot \frac{4}{x} + 2 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1227056 = 2.0;
        double r1227057 = sqrt(r1227056);
        double r1227058 = t;
        double r1227059 = r1227057 * r1227058;
        double r1227060 = x;
        double r1227061 = 1.0;
        double r1227062 = r1227060 + r1227061;
        double r1227063 = r1227060 - r1227061;
        double r1227064 = r1227062 / r1227063;
        double r1227065 = l;
        double r1227066 = r1227065 * r1227065;
        double r1227067 = r1227058 * r1227058;
        double r1227068 = r1227056 * r1227067;
        double r1227069 = r1227066 + r1227068;
        double r1227070 = r1227064 * r1227069;
        double r1227071 = r1227070 - r1227066;
        double r1227072 = sqrt(r1227071);
        double r1227073 = r1227059 / r1227072;
        return r1227073;
}
↓
double f(double x, double l, double t) {
        double r1227074 = t;
        double r1227075 = -1.046010354271018e+94;
        bool r1227076 = r1227074 <= r1227075;
        double r1227077 = 2.0;
        double r1227078 = sqrt(r1227077);
        double r1227079 = r1227078 * r1227074;
        double r1227080 = r1227074 / r1227078;
        double r1227081 = x;
        double r1227082 = r1227081 * r1227081;
        double r1227083 = r1227080 / r1227082;
        double r1227084 = r1227081 * r1227078;
        double r1227085 = r1227074 / r1227084;
        double r1227086 = r1227077 * r1227085;
        double r1227087 = r1227079 + r1227086;
        double r1227088 = r1227083 - r1227087;
        double r1227089 = r1227077 / r1227081;
        double r1227090 = r1227085 * r1227089;
        double r1227091 = r1227088 - r1227090;
        double r1227092 = r1227079 / r1227091;
        double r1227093 = 1.4541398591892197e+121;
        bool r1227094 = r1227074 <= r1227093;
        double r1227095 = cbrt(r1227078);
        double r1227096 = r1227095 * r1227074;
        double r1227097 = r1227095 * r1227095;
        double r1227098 = r1227096 * r1227097;
        double r1227099 = r1227074 * r1227074;
        double r1227100 = 4.0;
        double r1227101 = r1227100 / r1227081;
        double r1227102 = r1227099 * r1227101;
        double r1227103 = l;
        double r1227104 = r1227103 / r1227081;
        double r1227105 = r1227104 * r1227103;
        double r1227106 = r1227105 + r1227099;
        double r1227107 = r1227077 * r1227106;
        double r1227108 = r1227102 + r1227107;
        double r1227109 = sqrt(r1227108);
        double r1227110 = r1227098 / r1227109;
        double r1227111 = r1227087 + r1227090;
        double r1227112 = r1227111 - r1227083;
        double r1227113 = r1227079 / r1227112;
        double r1227114 = r1227094 ? r1227110 : r1227113;
        double r1227115 = r1227076 ? r1227092 : r1227114;
        return r1227115;
}

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 < -1.046010354271018e+94

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.046010354271018e+94 < t < 1.4541398591892197e+121

    1. Initial program 36.6

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

    2. Taylor expanded around inf 16.9

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

    3. Simplified16.9

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

    4. Taylor expanded around 0 16.9

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

    5. Simplified13.0

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

    7. Applied add-cube-cbrt13.0

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

    8. Applied associate-*l*13.0

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

    if 1.4541398591892197e+121 < t

    1. Initial program 53.7

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

    2. Taylor expanded around inf 3.1

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

    3. Simplified3.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.046010354271018 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right)\right) - \frac{t}{x \cdot \sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 1.4541398591892197 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(t \cdot t\right) \cdot \frac{4}{x} + 2 \cdot \left(\frac{\ell}{x} \cdot \ell + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\sqrt{2} \cdot t + 2 \cdot \frac{t}{x \cdot \sqrt{2}}\right) + \frac{t}{x \cdot \sqrt{2}} \cdot \frac{2}{x}\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))))