Average Error: 32.6 → 13.0
Time: 16.8s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.291768009392636 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)}^{3} \cdot \tan k\right) \cdot t_2}\\ \mathbf{elif}\;t \leq 1.4741695502041248 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot {\left(\frac{\frac{t \cdot t_1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]
\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}
t_1 := \sqrt[3]{\sin k}\\
t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -2.291768009392636 \cdot 10^{-57}:\\
\;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)}^{3} \cdot \tan k\right) \cdot t_2}\\

\mathbf{elif}\;t \leq 1.4741695502041248 \cdot 10^{-108}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot {\left(\frac{\frac{t \cdot t_1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\


\end{array}

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k))) (t_2 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -2.291768009392636e-57)
     (/
      2.0
      (*
       (* (pow (* (/ t (pow (cbrt l) 2.0)) (log1p (expm1 t_1))) 3.0) (tan k))
       t_2))
     (if (<= t 1.4741695502041248e-108)
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (* l l))
         (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
       (/
        2.0
        (* t_2 (* (tan k) (pow (/ (/ (* t t_1) (cbrt l)) (cbrt l)) 3.0))))))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}
↓
double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -2.291768009392636e-57) {
		tmp = 2.0 / ((pow(((t / pow(cbrt(l), 2.0)) * log1p(expm1(t_1))), 3.0) * tan(k)) * t_2);
	} else if (t <= 1.4741695502041248e-108) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * pow((((t * t_1) / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -2.29176800939263592e-57

    1. Initial program 23.2

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

    2. Simplified23.2

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

    3. Applied egg-rr8.2

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

    4. Applied egg-rr8.2

      \[\leadsto \frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

    if -2.29176800939263592e-57 < t < 1.4741695502041248e-108

    1. Initial program 59.2

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

    2. Simplified59.2

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

    3. Taylor expanded in t around 0 38.0

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

    4. Simplified25.2

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

    if 1.4741695502041248e-108 < t

    1. Initial program 24.3

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

    2. Simplified24.3

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

    3. Applied egg-rr9.7

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

    4. Applied egg-rr9.7

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.291768009392636 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.4741695502041248 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot {\left(\frac{\frac{t \cdot \sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))