Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 12.0

Time: 25.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2.972876280547332 \cdot 10^{-201}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\frac{1}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\sqrt[3]{\frac{\ell}{t}} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 9.316965900380896 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

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 -2.972876280547332e-201)
   (/
    2.0
    (/
     (*
      (*
       (sin k)
       (* (/ 1.0 (* (cbrt (/ l t)) (cbrt (/ l t)))) (* t (/ (* t (sin k)) l))))
      (+ 2.0 (pow (/ k t) 2.0)))
     (* (cbrt (/ l t)) (cos k))))
   (if (<= t 9.316965900380896e-154)
     (/
      2.0
      (*
       (/ (pow (sin k) 2.0) (* l l))
       (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
     (/
      2.0
      (/
       (* (+ 2.0 (pow (/ k t) 2.0)) (* (* t (/ (* t (sin k)) l)) (tan 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 <= -2.972876280547332e-201) {
		tmp = 2.0 / (((sin(k) * ((1.0 / (cbrt(l / t) * cbrt(l / t))) * (t * ((t * sin(k)) / l)))) * (2.0 + pow((k / t), 2.0))) / (cbrt(l / t) * cos(k)));
	} else if (t <= 9.316965900380896e-154) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * ((t * ((t * sin(k)) / l)) * tan(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 < -2.97287628054733196e-201

   1. Initial program 28.9

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

   2. Simplified 28.9

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

   4. Applied unpow3_binary64_485 28.9

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

   5. Applied times-frac_binary64_425 20.1

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

   6. Applied associate-*l*_binary64_360 18.0

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

   7. Simplified 18.0

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

   9. Applied associate-/l*_binary64_364 13.1

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

   11. Applied add-cube-cbrt_binary64_454 13.3

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

   12. Applied *-un-lft-identity_binary64_419 13.3

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

   13. Applied times-frac_binary64_425 13.3

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

   14. Applied associate-*l*_binary64_360 12.2

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

   15. Simplified 12.5

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

   17. Applied tan-quot_binary64_578 12.5

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

   18. Applied associate-*r/_binary64_361 12.3

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

   19. Applied associate-*r/_binary64_361 12.3

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

   20. Applied frac-times_binary64_429 11.2

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

   21. Applied associate-*l/_binary64_362 10.9

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

   if -2.97287628054733196e-201 < t < 9.3169659003808956e-154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied unpow3_binary64_485 64.0

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

   5. Applied times-frac_binary64_425 62.6

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

   6. Applied associate-*l*_binary64_360 62.6

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

   7. Simplified 62.6

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

   8. Taylor expanded around 0 42.3

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

   9. Simplified 27.6

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

   if 9.3169659003808956e-154 < t

   1. Initial program 25.6

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

   2. Simplified 25.6

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

   4. Applied unpow3_binary64_485 25.6

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

   5. Applied times-frac_binary64_425 17.8

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

   6. Applied associate-*l*_binary64_360 15.5

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

   7. Simplified 15.5

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

   9. Applied associate-/l*_binary64_364 11.3

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

   11. Applied associate-*l/_binary64_362 10.3

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

   12. Applied associate-*l/_binary64_362 9.2

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

   13. Applied associate-*l/_binary64_362 8.0

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

   14. Simplified 7.9

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.972876280547332 \cdot 10^{-201}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\frac{1}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\sqrt[3]{\frac{\ell}{t}} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 9.316965900380896 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))))