Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 17.6

Time: 21.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq 8.415250548228389 \cdot 10^{-300}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sin k\right)\right)}\right)\right)\\ \mathbf{elif}\;t \leq 9.836929728792166 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\left(\ell \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \sin k\right)}\right) \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{\ell}\right)\right)\\ \end{array}\]

FPCore

(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
 (if (<= t 8.415250548228389e-300)
   (*
    l
    (*
     (/ (sqrt 2.0) (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0)))
     (*
      l
      (/
       (sqrt 2.0)
       (*
        (*
         (cbrt (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0)))
         (cbrt (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0))))
        (*
         (* (tan k) (* (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))) (pow (cbrt t) 3.0)))
         (* (cbrt (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0))) (sin k))))))))
   (if (<= t 9.836929728792166e+18)
     (*
      l
      (*
       l
       (/
        2.0
        (*
         (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0))
         (*
          (sin k)
          (+
           (*
            2.0
            (*
             (pow (/ 1.0 (pow (/ 1.0 (pow t 2.0)) 1.0)) 1.0)
             (/ (sin k) (cos k))))
           (* (sin k) (/ (* k k) (cos k)))))))))
     (*
      (* (cbrt l) (cbrt l))
      (*
       (*
        l
        (/
         (sqrt 2.0)
         (*
          (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0))
          (*
           (* (tan k) (* (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))) (pow (cbrt t) 3.0)))
           (sin k)))))
       (* (/ (sqrt 2.0) (pow (* (cbrt t) (cbrt t)) (/ 3.0 2.0))) (cbrt l)))))))

C

double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) + 1.0)))));
}

↓

double code(double t, double l, double k) {
	double tmp;
	if ((t <= 8.415250548228389e-300)) {
		tmp = ((double) (l * ((double) ((((double) sqrt(2.0)) / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * ((double) (l * (((double) sqrt(2.0)) / ((double) (((double) (((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))))) * ((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))))))) * ((double) (((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow((k / t), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) (((double) cbrt(((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))))) * ((double) sin(k)))))))))))))));
	} else {
		double tmp_1;
		if ((t <= 9.836929728792166e+18)) {
			tmp_1 = ((double) (l * ((double) (l * (2.0 / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))) * ((double) (((double) sin(k)) * ((double) (((double) (2.0 * ((double) (((double) pow((1.0 / ((double) pow((1.0 / ((double) pow(t, 2.0))), 1.0))), 1.0)) * (((double) sin(k)) / ((double) cos(k))))))) + ((double) (((double) sin(k)) * (((double) (k * k)) / ((double) cos(k))))))))))))))));
		} else {
			tmp_1 = ((double) (((double) (((double) cbrt(l)) * ((double) cbrt(l)))) * ((double) (((double) (l * (((double) sqrt(2.0)) / ((double) (((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0))) * ((double) (((double) (((double) tan(k)) * ((double) (((double) (1.0 + ((double) (1.0 + ((double) pow((k / t), 2.0)))))) * ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) sin(k))))))))) * ((double) ((((double) sqrt(2.0)) / ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), (3.0 / 2.0)))) * ((double) cbrt(l))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < 8.41525054822838921e-300

   1. Initial program Error: 32.2 bits

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

   2. Simplified Error: 33.1 bits

      \[\leadsto \color{blue}{\ell \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \ell\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt Error: 33.3 bits

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

   5. Applied unpow-prod-down Error: 33.3 bits

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

   6. Applied associate-*l* Error: 30.6 bits

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

   7. Simplified Error: 27.5 bits

      \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}} \cdot \ell\right)\]
   8. Using strategy rm

   9. Applied sqr-pow Error: 27.5 bits

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

   10. Applied associate-*l* Error: 22.7 bits

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

   11. Simplified Error: 21.0 bits

       \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right)\right)}} \cdot \ell\right)\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt Error: 21.0 bits

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

   14. Applied times-frac Error: 20.9 bits

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

   15. Applied associate-*l* Error: 19.2 bits

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

   16. Simplified Error: 19.9 bits

       \[\leadsto \ell \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\right)}\right)\]
   17. Using strategy rm

   18. Applied add-cube-cbrt Error: 20.0 bits

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

   19. Applied associate-*l* Error: 20.0 bits

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

   20. Simplified Error: 19.1 bits

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

   if 8.41525054822838921e-300 < t < 9836929728792166000

   1. Initial program Error: 46.1 bits

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

   2. Simplified Error: 44.7 bits

      \[\leadsto \color{blue}{\ell \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \ell\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt Error: 44.9 bits

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

   5. Applied unpow-prod-down Error: 44.9 bits

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

   6. Applied associate-*l* Error: 39.6 bits

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

   7. Simplified Error: 39.5 bits

      \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}} \cdot \ell\right)\]
   8. Using strategy rm

   9. Applied sqr-pow Error: 39.5 bits

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

   10. Applied associate-*l* Error: 31.3 bits

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

   11. Simplified Error: 31.2 bits

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

   12. Taylor expanded around inf Error: 21.6 bits

       \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot \left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \frac{{k}^{2} \cdot \sin k}{\cos k}\right)}\right)} \cdot \ell\right)\]

   13. Simplified Error: 21.5 bits

       \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \color{blue}{\left(2 \cdot \left({\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \frac{k \cdot k}{\cos k} \cdot \sin k\right)}\right)} \cdot \ell\right)\]

   if 9836929728792166000 < t

   1. Initial program Error: 22.7 bits

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

   2. Simplified Error: 25.4 bits

      \[\leadsto \color{blue}{\ell \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \ell\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt Error: 25.5 bits

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

   5. Applied unpow-prod-down Error: 25.5 bits

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

   6. Applied associate-*l* Error: 24.4 bits

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

   7. Simplified Error: 19.4 bits

      \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}} \cdot \ell\right)\]
   8. Using strategy rm

   9. Applied sqr-pow Error: 19.4 bits

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

   10. Applied associate-*l* Error: 17.2 bits

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

   11. Simplified Error: 14.1 bits

       \[\leadsto \ell \cdot \left(\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right)\right)}} \cdot \ell\right)\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt Error: 14.1 bits

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

   14. Applied times-frac Error: 14.0 bits

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

   15. Applied associate-*l* Error: 11.7 bits

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

   16. Simplified Error: 12.9 bits

       \[\leadsto \ell \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\right)}\right)\]
   17. Using strategy rm

   18. Applied add-cube-cbrt Error: 12.9 bits

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

   19. Applied associate-*l* Error: 12.9 bits

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

   20. Simplified Error: 12.6 bits

       \[\leadsto \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\right) \cdot \left(\sqrt[3]{\ell} \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification Error: 17.6 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.415250548228389 \cdot 10^{-300}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sin k\right)\right)}\right)\right)\\ \mathbf{elif}\;t \leq 9.836929728792166 \cdot 10^{+18}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left({\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\left(\ell \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \sin k\right)}\right) \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \sqrt[3]{\ell}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))