Toniolo and Linder, Equation (10+)

Average Error: 32.9 → 9.1

Time: 14.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -7.807885226879273 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\sqrt[3]{2}}{\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 4.210309937890153 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

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 -7.807885226879273e+49)
   (*
    (/ (* (cbrt 2.0) (cbrt 2.0)) (+ 2.0 (pow (/ k t) 2.0)))
    (* (/ l t) (/ (cbrt 2.0) (* (tan k) (* t (* (/ t l) (sin k)))))))
   (if (<= t 4.210309937890153e-37)
     (*
      (/ l t)
      (/
       2.0
       (*
        (/ (pow (sin k) 2.0) (cos k))
        (+ (* 2.0 (* t (/ t l))) (/ (* k k) l)))))
     (/
      2.0
      (/
       (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (* (/ t l) (sin k)))))
       (/ l t))))))

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 <= -7.807885226879273e+49) {
		tmp = ((cbrt(2.0) * cbrt(2.0)) / (2.0 + pow((k / t), 2.0))) * ((l / t) * (cbrt(2.0) / (tan(k) * (t * ((t / l) * sin(k))))));
	} else if (t <= 4.210309937890153e-37) {
		tmp = (l / t) * (2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((2.0 * (t * (t / l))) + ((k * k) / l))));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * ((t / l) * sin(k))))) / (l / t));
	}
	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 < -7.80788522687927267e49

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   4. Applied unpow3_binary64_485 23.0

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

   5. Applied times-frac_binary64_425 15.1

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

   6. Applied associate-*l*_binary64_360 13.9

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

   8. Applied associate-/l*_binary64_364 6.8

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

   10. Applied associate-*l/_binary64_362 5.1

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

   11. Applied associate-*l/_binary64_362 1.9

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

   12. Applied associate-*l/_binary64_362 2.0

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

   13. Applied associate-/r/_binary64_365 1.9

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

   14. Simplified 1.9

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

   16. Applied add-cube-cbrt_binary64_454 2.3

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

   17. Applied times-frac_binary64_425 2.4

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

   18. Applied associate-*l*_binary64_360 2.3

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

   19. Simplified 2.3

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

   if -7.80788522687927267e49 < t < 4.210309937890153e-37

   1. Initial program 48.2

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

   2. Simplified 48.2

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

   4. Applied unpow3_binary64_485 48.2

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

   5. Applied times-frac_binary64_425 40.7

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

   6. Applied associate-*l*_binary64_360 39.1

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

   8. Applied associate-/l*_binary64_364 33.7

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

   10. Applied associate-*l/_binary64_362 33.7

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

   11. Applied associate-*l/_binary64_362 34.8

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

   12. Applied associate-*l/_binary64_362 31.7

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

   13. Applied associate-/r/_binary64_365 31.7

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

   14. Simplified 31.7

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

   15. Taylor expanded around inf 20.9

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

   16. Simplified 18.0

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

   if 4.210309937890153e-37 < t

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   4. Applied unpow3_binary64_485 23.0

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

   5. Applied times-frac_binary64_425 16.4

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

   6. Applied associate-*l*_binary64_360 14.1

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

   8. Applied associate-/l*_binary64_364 8.7

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

   10. Applied associate-*l/_binary64_362 7.1

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

   11. Applied associate-*l/_binary64_362 4.7

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

   12. Applied associate-*l/_binary64_362 4.1

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

   13. Simplified 4.1

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.807885226879273 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\sqrt[3]{2}}{\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 4.210309937890153 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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))))