Average Error: 43.0 → 9.1
Time: 29.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.696744411039916917517605602907382612022 \cdot 10^{122}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.380741590772817149641192548059648402147 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -3.696744411039916917517605602907382612022 \cdot 10^{122}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 3.380741590772817149641192548059648402147 \cdot 10^{117}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r38037 = 2.0;
        double r38038 = sqrt(r38037);
        double r38039 = t;
        double r38040 = r38038 * r38039;
        double r38041 = x;
        double r38042 = 1.0;
        double r38043 = r38041 + r38042;
        double r38044 = r38041 - r38042;
        double r38045 = r38043 / r38044;
        double r38046 = l;
        double r38047 = r38046 * r38046;
        double r38048 = r38039 * r38039;
        double r38049 = r38037 * r38048;
        double r38050 = r38047 + r38049;
        double r38051 = r38045 * r38050;
        double r38052 = r38051 - r38047;
        double r38053 = sqrt(r38052);
        double r38054 = r38040 / r38053;
        return r38054;
}
↓
double f(double x, double l, double t) {
        double r38055 = t;
        double r38056 = -3.696744411039917e+122;
        bool r38057 = r38055 <= r38056;
        double r38058 = 2.0;
        double r38059 = sqrt(r38058);
        double r38060 = r38059 * r38055;
        double r38061 = x;
        double r38062 = r38059 * r38061;
        double r38063 = r38055 / r38062;
        double r38064 = r38058 * r38063;
        double r38065 = fma(r38055, r38059, r38064);
        double r38066 = -r38065;
        double r38067 = r38060 / r38066;
        double r38068 = 3.380741590772817e+117;
        bool r38069 = r38055 <= r38068;
        double r38070 = l;
        double r38071 = r38061 / r38070;
        double r38072 = r38070 / r38071;
        double r38073 = fma(r38055, r38055, r38072);
        double r38074 = 4.0;
        double r38075 = 2.0;
        double r38076 = pow(r38055, r38075);
        double r38077 = r38076 / r38061;
        double r38078 = r38074 * r38077;
        double r38079 = fma(r38058, r38073, r38078);
        double r38080 = sqrt(r38079);
        double r38081 = r38060 / r38080;
        double r38082 = r38060 / r38065;
        double r38083 = r38069 ? r38081 : r38082;
        double r38084 = r38057 ? r38067 : r38083;
        return r38084;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -3.696744411039917e+122

    1. Initial program 54.6

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

    2. Simplified54.6

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

    3. Taylor expanded around inf 54.9

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

    4. Simplified54.9

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

    5. Taylor expanded around -inf 2.9

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

    6. Simplified2.9

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

    if -3.696744411039917e+122 < t < 3.380741590772817e+117

    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. Simplified36.6

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

    3. Taylor expanded around inf 17.2

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

    4. Simplified17.2

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

    6. Applied unpow217.2

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

    7. Applied associate-/l*12.8

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

    if 3.380741590772817e+117 < t

    1. Initial program 53.6

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

    2. Simplified53.6

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

    3. Taylor expanded around inf 54.0

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

    4. Simplified54.0

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

    5. Taylor expanded around inf 2.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    6. Simplified2.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.696744411039916917517605602907382612022 \cdot 10^{122}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.380741590772817149641192548059648402147 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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)))))