Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 14.2

Time: 30.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -1.5482059886230863 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 4.6518583883372725 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot e^{\left(\left(\left(\log \ell + \log \ell\right) - \log k\right) + \log \cos k\right) - \left(\log k + \left(\log t + 2 \cdot \log \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

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 (<= k -1.5482059886230863e-162)
   (* 2.0 (/ (* (* l (/ l k)) (cos k)) (* k (* t (pow (sin k) 2.0)))))
   (if (<= k 4.6518583883372725e-55)
     (*
      2.0
      (exp
       (-
        (+ (- (+ (log l) (log l)) (log k)) (log (cos k)))
        (+ (log k) (+ (log t) (* 2.0 (log (sin k))))))))
     (* 2.0 (/ (* (cos k) (/ l (/ k l))) (* k (* t (pow (sin k) 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 (k <= -1.5482059886230863e-162) {
		tmp = 2.0 * (((l * (l / k)) * cos(k)) / (k * (t * pow(sin(k), 2.0))));
	} else if (k <= 4.6518583883372725e-55) {
		tmp = 2.0 * exp((((log(l) + log(l)) - log(k)) + log(cos(k))) - (log(k) + (log(t) + (2.0 * log(sin(k))))));
	} else {
		tmp = 2.0 * ((cos(k) * (l / (k / l))) / (k * (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.5482059886230863e-162

   1. Initial program 48.3

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

   2. Simplified 40.5

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

   3. Taylor expanded around 0 21.9

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

   4. Simplified 21.9

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

   6. Applied associate-*l*_binary64_360 19.9

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

   8. Applied associate-/r*_binary64_363 17.3

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

   9. Simplified 17.3

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

   11. Applied *-un-lft-identity_binary64_419 17.3

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

   12. Applied times-frac_binary64_425 12.0

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

   if -1.5482059886230863e-162 < k < 4.6518583883372725e-55

   1. Initial program 63.2

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

   2. Simplified 60.2

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

   3. Taylor expanded around 0 48.5

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

   4. Simplified 48.5

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

   6. Applied associate-*l*_binary64_360 48.5

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

   8. Applied associate-/r*_binary64_363 45.5

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

   9. Simplified 45.5

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

   11. Applied pow-to-exp_binary64_488 46.7

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

   12. Applied add-exp-log_binary64_457 56.3

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

   13. Applied prod-exp_binary64_468 56.1

       \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{k} \cdot \cos k}{k \cdot \color{blue}{e^{\log t + \log \sin k \cdot 2}}}\]

   14. Applied add-exp-log_binary64_457 56.1

       \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{k} \cdot \cos k}{\color{blue}{e^{\log k}} \cdot e^{\log t + \log \sin k \cdot 2}}\]

   15. Applied prod-exp_binary64_468 56.1

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

   16. Applied add-exp-log_binary64_457 56.1

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

   17. Applied add-exp-log_binary64_457 56.1

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

   18. Applied add-exp-log_binary64_457 60.5

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

   19. Applied add-exp-log_binary64_457 60.5

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

   20. Applied prod-exp_binary64_468 60.5

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

   21. Applied div-exp_binary64_470 59.7

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

   22. Applied prod-exp_binary64_468 59.7

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

   23. Applied div-exp_binary64_470 53.4

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

   if 4.6518583883372725e-55 < k

   1. Initial program 45.6

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

   2. Simplified 37.2

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

   3. Taylor expanded around 0 19.1

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

   4. Simplified 19.1

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

   6. Applied associate-*l*_binary64_360 16.5

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

   8. Applied associate-/r*_binary64_363 14.9

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

   9. Simplified 14.9

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

   11. Applied associate-/l*_binary64_364 9.5

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

4. Final simplification 14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.5482059886230863 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 4.6518583883372725 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot e^{\left(\left(\left(\log \ell + \log \ell\right) - \log k\right) + \log \cos k\right) - \left(\log k + \left(\log t + 2 \cdot \log \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021047 
(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))))