Average Error: 43.1 → 10.0
Time: 29.8s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.303460822673397315215412157251637183073 \cdot 10^{149}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -1.230065150000774851732639496826971723637 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le -4.303938162079978611630039682650481103959 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.190616919639676067581902350380230729983 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]
\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 -2.303460822673397315215412157251637183073 \cdot 10^{149}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le -1.230065150000774851732639496826971723637 \cdot 10^{-149}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

\mathbf{elif}\;t \le -4.303938162079978611630039682650481103959 \cdot 10^{-243}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 3.190616919639676067581902350380230729983 \cdot 10^{-45}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r44027 = 2.0;
        double r44028 = sqrt(r44027);
        double r44029 = t;
        double r44030 = r44028 * r44029;
        double r44031 = x;
        double r44032 = 1.0;
        double r44033 = r44031 + r44032;
        double r44034 = r44031 - r44032;
        double r44035 = r44033 / r44034;
        double r44036 = l;
        double r44037 = r44036 * r44036;
        double r44038 = r44029 * r44029;
        double r44039 = r44027 * r44038;
        double r44040 = r44037 + r44039;
        double r44041 = r44035 * r44040;
        double r44042 = r44041 - r44037;
        double r44043 = sqrt(r44042);
        double r44044 = r44030 / r44043;
        return r44044;
}
↓
double f(double x, double l, double t) {
        double r44045 = t;
        double r44046 = -2.3034608226733973e+149;
        bool r44047 = r44045 <= r44046;
        double r44048 = 2.0;
        double r44049 = sqrt(r44048);
        double r44050 = r44049 * r44045;
        double r44051 = x;
        double r44052 = 2.0;
        double r44053 = pow(r44051, r44052);
        double r44054 = r44045 / r44053;
        double r44055 = r44049 * r44048;
        double r44056 = r44048 / r44055;
        double r44057 = r44048 / r44049;
        double r44058 = r44056 - r44057;
        double r44059 = r44054 * r44058;
        double r44060 = r44059 - r44050;
        double r44061 = r44049 * r44051;
        double r44062 = r44045 / r44061;
        double r44063 = r44048 * r44062;
        double r44064 = r44060 - r44063;
        double r44065 = r44050 / r44064;
        double r44066 = -1.2300651500007749e-149;
        bool r44067 = r44045 <= r44066;
        double r44068 = 4.0;
        double r44069 = pow(r44045, r44052);
        double r44070 = r44069 / r44051;
        double r44071 = r44068 * r44070;
        double r44072 = r44045 * r44045;
        double r44073 = l;
        double r44074 = fabs(r44073);
        double r44075 = r44074 / r44051;
        double r44076 = r44074 * r44075;
        double r44077 = r44072 + r44076;
        double r44078 = r44048 * r44077;
        double r44079 = r44071 + r44078;
        double r44080 = sqrt(r44079);
        double r44081 = r44050 / r44080;
        double r44082 = -4.3039381620799786e-243;
        bool r44083 = r44045 <= r44082;
        double r44084 = 3.190616919639676e-45;
        bool r44085 = r44045 <= r44084;
        double r44086 = r44045 * r44049;
        double r44087 = r44063 + r44086;
        double r44088 = r44087 - r44059;
        double r44089 = r44050 / r44088;
        double r44090 = r44085 ? r44081 : r44089;
        double r44091 = r44083 ? r44065 : r44090;
        double r44092 = r44067 ? r44081 : r44091;
        double r44093 = r44047 ? r44065 : r44092;
        return r44093;
}

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 < -2.3034608226733973e+149 or -1.2300651500007749e-149 < t < -4.3039381620799786e-243

    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 10.9

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

    3. Simplified10.9

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

    if -2.3034608226733973e+149 < t < -1.2300651500007749e-149 or -4.3039381620799786e-243 < t < 3.190616919639676e-45

    1. Initial program 37.2

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

    2. Taylor expanded around inf 17.3

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

    3. Simplified17.3

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

    5. Applied *-un-lft-identity17.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

    6. Applied add-sqr-sqrt17.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{\ell}^{2}}}}{1 \cdot x}\right)}}\]

    7. Applied times-frac17.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \color{blue}{\frac{\sqrt{{\ell}^{2}}}{1} \cdot \frac{\sqrt{{\ell}^{2}}}{x}}\right)}}\]

    8. Simplified17.3

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

    9. Simplified12.7

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

    if 3.190616919639676e-45 < t

    1. Initial program 39.5

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

    2. Taylor expanded around inf 5.6

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

    3. Simplified5.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.303460822673397315215412157251637183073 \cdot 10^{149}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -1.230065150000774851732639496826971723637 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le -4.303938162079978611630039682650481103959 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.190616919639676067581902350380230729983 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))))