Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 12.6

Time: 1.8min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5362665831654716 \cdot 10^{-09}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{t \cdot \left(\sin k \cdot \left(\tan k \cdot \left(t \cdot t\right)\right)\right)}\right)\\ \mathbf{elif}\;t \leq 8.924049272980926 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell \cdot 2}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \frac{\frac{\frac{\cos k}{k} \cdot \frac{\ell}{{\sin k}^{2}}}{t}}{\sqrt[3]{k}}\\ \mathbf{elif}\;t \leq 6.439006299831887 \cdot 10^{+67}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{{t}^{3}} \cdot \left(\frac{\sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)\\ \mathbf{elif}\;t \leq 1.0275830431192084 \cdot 10^{+73}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{\sqrt[3]{\frac{k}{\frac{\ell}{k}}} \cdot \left(\sqrt[3]{\frac{k}{\frac{\ell}{k}}} \cdot \sqrt[3]{\frac{k}{\frac{\ell}{k}}}\right)}}{{\sin k}^{2}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{t}^{\left(\frac{3}{2}\right)} \cdot \left(\tan k \cdot \left(\sin k \cdot {t}^{\left(\frac{3}{2}\right)}\right)\right)}\right)\\ \end{array}\]

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 -3.5362665831654716e-09)
   (*
    l
    (*
     l
     (/
      (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))
      (* t (* (sin k) (* (tan k) (* t t)))))))
   (if (<= t 8.924049272980926e-83)
     (*
      (/ (* l 2.0) (* (cbrt k) (cbrt k)))
      (/ (/ (* (/ (cos k) k) (/ l (pow (sin k) 2.0))) t) (cbrt k)))
     (if (<= t 6.439006299831887e+67)
       (*
        l
        (*
         (/
          (*
           (cbrt (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))))
           (cbrt (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))))
          (pow t 3.0))
         (*
          (/ (cbrt (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))) (sin k))
          (/ l (tan k)))))
       (if (<= t 1.0275830431192084e+73)
         (*
          l
          (*
           2.0
           (/
            (/
             (/
              (cos k)
              (*
               (cbrt (/ k (/ l k)))
               (* (cbrt (/ k (/ l k))) (cbrt (/ k (/ l k))))))
             (pow (sin k) 2.0))
            t)))
         (*
          l
          (*
           l
           (/
            (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))
            (*
             (pow t (/ 3.0 2.0))
             (* (tan k) (* (sin k) (pow t (/ 3.0 2.0)))))))))))))

C

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 tmp;
	if (t <= -3.5362665831654716e-09) {
		tmp = l * (l * ((2.0 / (2.0 + pow((k / t), 2.0))) / (t * (sin(k) * (tan(k) * (t * t))))));
	} else if (t <= 8.924049272980926e-83) {
		tmp = ((l * 2.0) / (cbrt(k) * cbrt(k))) * ((((cos(k) / k) * (l / pow(sin(k), 2.0))) / t) / cbrt(k));
	} else if (t <= 6.439006299831887e+67) {
		tmp = l * (((cbrt(2.0 / (2.0 + pow((k / t), 2.0))) * cbrt(2.0 / (2.0 + pow((k / t), 2.0)))) / pow(t, 3.0)) * ((cbrt(2.0 / (2.0 + pow((k / t), 2.0))) / sin(k)) * (l / tan(k))));
	} else if (t <= 1.0275830431192084e+73) {
		tmp = l * (2.0 * (((cos(k) / (cbrt(k / (l / k)) * (cbrt(k / (l / k)) * cbrt(k / (l / k))))) / pow(sin(k), 2.0)) / t));
	} else {
		tmp = l * (l * ((2.0 / (2.0 + pow((k / t), 2.0))) / (pow(t, (3.0 / 2.0)) * (tan(k) * (sin(k) * pow(t, (3.0 / 2.0)))))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if t < -3.5362665831654716e-9

   1. Initial program 22.4

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

   2. Simplified 22.8

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

   4. Applied cube-mult_binary64_790 22.8

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

   5. Applied associate-*l*_binary64_701 21.6

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

   6. Simplified 15.8

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

   if -3.5362665831654716e-9 < t < 8.9240492729809263e-83

   1. Initial program 53.9

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

   2. Simplified 53.5

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

   3. Taylor expanded around inf 23.7

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

   4. Simplified 22.7

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

   6. Applied *-un-lft-identity_binary64_760 22.7

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

   7. Applied *-un-lft-identity_binary64_760 22.7

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

   8. Applied unpow-prod-down_binary64_839 22.7

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

   9. Applied *-un-lft-identity_binary64_760 22.7

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

   10. Applied times-frac_binary64_766 16.8

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

   11. Applied *-un-lft-identity_binary64_760 16.8

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

   12. Applied times-frac_binary64_766 16.0

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

   13. Applied times-frac_binary64_766 16.0

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

   14. Applied times-frac_binary64_766 7.8

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

   15. Simplified 7.8

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

   17. Applied associate-*l/_binary64_703 7.8

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

   18. Applied associate-*r/_binary64_702 7.8

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

   19. Applied associate-*r/_binary64_702 11.3

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

   20. Simplified 11.3

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

   22. Applied add-cube-cbrt_binary64_795 11.7

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

   23. Applied times-frac_binary64_766 6.1

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \frac{\frac{\frac{\cos k}{k} \cdot \frac{\ell}{{\sin k}^{2}}}{t}}{\sqrt[3]{k}}}\]

   if 8.9240492729809263e-83 < t < 6.4390062998318865e67

   1. Initial program 22.3

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

   2. Simplified 19.9

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

   4. Applied add-cube-cbrt_binary64_795 20.0

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

   5. Applied times-frac_binary64_766 20.1

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

   6. Applied associate-*l*_binary64_701 18.1

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

   7. Simplified 13.6

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

   if 6.4390062998318865e67 < t < 1.02758304311920835e73

   1. Initial program 17.1

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

   2. Simplified 16.3

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

   3. Taylor expanded around inf 35.5

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

   4. Simplified 30.8

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

   6. Applied add-cube-cbrt_binary64_795 30.9

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

   7. Simplified 30.9

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

   8. Simplified 24.6

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

   if 1.02758304311920835e73 < t

   1. Initial program 24.2

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

   2. Simplified 26.3

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

   4. Applied sqr-pow_binary64_732 26.3

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

   5. Applied associate-*l*_binary64_701 24.1

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

   6. Simplified 15.6

      \[\leadsto \ell \cdot \left(\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{t}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {t}^{\left(\frac{3}{2}\right)}\right)\right)}} \cdot \ell\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5362665831654716 \cdot 10^{-09}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{t \cdot \left(\sin k \cdot \left(\tan k \cdot \left(t \cdot t\right)\right)\right)}\right)\\ \mathbf{elif}\;t \leq 8.924049272980926 \cdot 10^{-83}:\\ \;\;\;\;\frac{\ell \cdot 2}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \frac{\frac{\frac{\cos k}{k} \cdot \frac{\ell}{{\sin k}^{2}}}{t}}{\sqrt[3]{k}}\\ \mathbf{elif}\;t \leq 6.439006299831887 \cdot 10^{+67}:\\ \;\;\;\;\ell \cdot \left(\frac{\sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{{t}^{3}} \cdot \left(\frac{\sqrt[3]{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)\\ \mathbf{elif}\;t \leq 1.0275830431192084 \cdot 10^{+73}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{\sqrt[3]{\frac{k}{\frac{\ell}{k}}} \cdot \left(\sqrt[3]{\frac{k}{\frac{\ell}{k}}} \cdot \sqrt[3]{\frac{k}{\frac{\ell}{k}}}\right)}}{{\sin k}^{2}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{t}^{\left(\frac{3}{2}\right)} \cdot \left(\tan k \cdot \left(\sin k \cdot {t}^{\left(\frac{3}{2}\right)}\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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))))