Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 10.5

Time: 31.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1652532132985465 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \cos k\right)\\ \mathbf{elif}\;t \leq 1733.5357186408837:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot {\sin k}^{2}}{\cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\sqrt{2}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k}\right)\\ \end{array}\]

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 -2.1652532132985465e-106)
   (*
    (/ 2.0 (* (+ 2.0 (pow (/ k t) 2.0)) (* (sin k) (* t (/ (* t (sin k)) l)))))
    (* (/ l t) (cos k)))
   (if (<= t 1733.5357186408837)
     (*
      (/ l t)
      (/
       2.0
       (/
        (+
         (/ (* (* k k) (pow (sin k) 2.0)) (cos k))
         (* 2.0 (/ (* (pow (sin k) 2.0) (* t t)) (cos k))))
        l)))
     (*
      (/ (sqrt 2.0) (+ 2.0 (pow (/ k t) 2.0)))
      (* (/ l t) (/ (sqrt 2.0) (* (* t (/ (* t (sin k)) l)) (tan k))))))))

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 <= -2.1652532132985465e-106) {
		tmp = (2.0 / ((2.0 + pow((k / t), 2.0)) * (sin(k) * (t * ((t * sin(k)) / l))))) * ((l / t) * cos(k));
	} else if (t <= 1733.5357186408837) {
		tmp = (l / t) * (2.0 / (((((k * k) * pow(sin(k), 2.0)) / cos(k)) + (2.0 * ((pow(sin(k), 2.0) * (t * t)) / cos(k)))) / l));
	} else {
		tmp = (sqrt(2.0) / (2.0 + pow((k / t), 2.0))) * ((l / t) * (sqrt(2.0) / ((t * ((t * sin(k)) / l)) * tan(k))));
	}
	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 < -2.1652532132985465e-106

   1. Initial program 22.8

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

   2. Simplified 22.8

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

   4. Applied unpow3_binary64_485 22.8

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

   5. Applied times-frac_binary64_425 16.5

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

   6. Applied associate-*l*_binary64_360 14.5

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

   7. Simplified 14.5

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

   9. Applied associate-/l*_binary64_364 9.9

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

   11. Applied tan-quot_binary64_578 9.9

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

   12. Applied associate-*l/_binary64_362 9.0

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

   13. Applied frac-times_binary64_429 7.1

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

   14. Applied associate-*l/_binary64_362 6.4

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

   15. Applied associate-/r/_binary64_365 6.2

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

   16. Simplified 6.2

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

   if -2.1652532132985465e-106 < t < 1733.5357186408837

   1. Initial program 54.9

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

   2. Simplified 54.9

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

   4. Applied unpow3_binary64_485 54.9

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

   5. Applied times-frac_binary64_425 46.9

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

   6. Applied associate-*l*_binary64_360 45.5

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

   7. Simplified 45.5

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

   9. Applied associate-/l*_binary64_364 38.8

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

   11. Applied associate-*l/_binary64_362 38.8

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

   12. Applied associate-*l/_binary64_362 40.2

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

   13. Applied associate-*l/_binary64_362 37.1

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

   14. Applied associate-/r/_binary64_365 37.3

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

   15. Simplified 37.3

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

   16. Taylor expanded around inf 23.1

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

   17. Simplified 23.1

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

   if 1733.5357186408837 < t

   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. Simplified 23.2

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

   4. Applied unpow3_binary64_485 23.2

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

   5. Applied times-frac_binary64_425 16.0

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

   6. Applied associate-*l*_binary64_360 13.8

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

   7. Simplified 13.8

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

   9. Applied associate-/l*_binary64_364 7.5

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

   11. Applied associate-*l/_binary64_362 5.9

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

   12. Applied associate-*l/_binary64_362 3.7

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

   13. Applied associate-*l/_binary64_362 3.6

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

   14. Applied associate-/r/_binary64_365 3.4

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

   15. Simplified 3.3

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

   17. Applied add-sqr-sqrt_binary64_441 3.5

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

   18. Applied times-frac_binary64_425 3.3

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

   19. Applied associate-*l*_binary64_360 3.5

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

   20. Simplified 3.5

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1652532132985465 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \cos k\right)\\ \mathbf{elif}\;t \leq 1733.5357186408837:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot {\sin k}^{2}}{\cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\sqrt{2}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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))))