Average Error: 32.0 → 5.7
Time: 38.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -6.242585972483955 \cdot 10^{+198}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{t \cdot \left(t \cdot t\right)}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \ell}, 2, \frac{t}{\frac{\cos k}{\sin k \cdot \sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \frac{2}{\cos k}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array}\]
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;k \le -6.242585972483955 \cdot 10^{+198}:\\
\;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{t \cdot \left(t \cdot t\right)}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \ell}, 2, \frac{t}{\frac{\cos k}{\sin k \cdot \sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \frac{2}{\cos k}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\

\end{array}
double f(double t, double l, double k) {
        double r2149023 = 2.0;
        double r2149024 = t;
        double r2149025 = 3.0;
        double r2149026 = pow(r2149024, r2149025);
        double r2149027 = l;
        double r2149028 = r2149027 * r2149027;
        double r2149029 = r2149026 / r2149028;
        double r2149030 = k;
        double r2149031 = sin(r2149030);
        double r2149032 = r2149029 * r2149031;
        double r2149033 = tan(r2149030);
        double r2149034 = r2149032 * r2149033;
        double r2149035 = 1.0;
        double r2149036 = r2149030 / r2149024;
        double r2149037 = pow(r2149036, r2149023);
        double r2149038 = r2149035 + r2149037;
        double r2149039 = r2149038 + r2149035;
        double r2149040 = r2149034 * r2149039;
        double r2149041 = r2149023 / r2149040;
        return r2149041;
}
↓
double f(double t, double l, double k) {
        double r2149042 = k;
        double r2149043 = -6.242585972483955e+198;
        bool r2149044 = r2149042 <= r2149043;
        double r2149045 = 2.0;
        double r2149046 = t;
        double r2149047 = r2149046 * r2149046;
        double r2149048 = r2149046 * r2149047;
        double r2149049 = cos(r2149042);
        double r2149050 = r2149048 / r2149049;
        double r2149051 = sin(r2149042);
        double r2149052 = r2149051 * r2149051;
        double r2149053 = l;
        double r2149054 = r2149053 * r2149053;
        double r2149055 = r2149052 / r2149054;
        double r2149056 = r2149050 * r2149055;
        double r2149057 = r2149049 / r2149052;
        double r2149058 = r2149053 / r2149042;
        double r2149059 = r2149058 * r2149058;
        double r2149060 = r2149057 * r2149059;
        double r2149061 = r2149046 / r2149060;
        double r2149062 = fma(r2149056, r2149045, r2149061);
        double r2149063 = r2149045 / r2149062;
        double r2149064 = r2149051 / r2149049;
        double r2149065 = r2149042 / r2149058;
        double r2149066 = r2149053 / r2149046;
        double r2149067 = r2149066 / r2149051;
        double r2149068 = r2149046 / r2149067;
        double r2149069 = r2149045 / r2149049;
        double r2149070 = r2149068 * r2149069;
        double r2149071 = fma(r2149064, r2149065, r2149070);
        double r2149072 = r2149051 / r2149066;
        double r2149073 = r2149071 * r2149072;
        double r2149074 = r2149045 / r2149073;
        double r2149075 = r2149044 ? r2149063 : r2149074;
        return r2149075;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < -6.242585972483955e+198

    1. Initial program 33.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    2. Simplified21.4

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\tan k \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}}\]

    3. Taylor expanded around inf 20.6

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{\sin k \cdot {k}^{2}}{\cos k \cdot \ell}\right)} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    4. Simplified14.5

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)}{\cos k}\right)} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    5. Taylor expanded around inf 27.9

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}}\]

    6. Simplified13.8

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\sin k \cdot \sin k}{\ell \cdot \ell} \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2, \frac{t}{\frac{\cos k}{\sin k \cdot \sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right)}}\]

    if -6.242585972483955e+198 < k

    1. Initial program 31.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    2. Simplified14.0

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\tan k \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}}\]

    3. Taylor expanded around inf 13.9

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{\sin k \cdot {k}^{2}}{\cos k \cdot \ell}\right)} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    4. Simplified7.3

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)}{\cos k}\right)} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]
    5. Using strategy rm

    6. Applied associate-*l*4.7

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)}}{\cos k}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]
    7. Using strategy rm

    8. Applied div-inv4.7

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(\color{blue}{\left(t \cdot \frac{1}{\ell}\right)} \cdot \left(t \cdot \sin k\right)\right)}{\cos k}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    9. Applied associate-*l*4.7

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \color{blue}{\left(t \cdot \left(\frac{1}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)}}{\cos k}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    10. Simplified4.7

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(t \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right)}{\cos k}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]
    11. Using strategy rm

    12. Applied *-un-lft-identity4.7

      \[\leadsto \frac{2}{\color{blue}{\left(1 \cdot \mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)}{\cos k}\right)\right)} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\]

    13. Applied associate-*l*4.7

      \[\leadsto \frac{2}{\color{blue}{1 \cdot \left(\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2 \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)}{\cos k}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}}\]

    14. Simplified4.6

      \[\leadsto \frac{2}{1 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{2}{\cos k} \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -6.242585972483955 \cdot 10^{+198}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{t \cdot \left(t \cdot t\right)}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell \cdot \ell}, 2, \frac{t}{\frac{\cos k}{\sin k \cdot \sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\frac{\ell}{k}}, \frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \frac{2}{\cos k}\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))