Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 10.0

Time: 23.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4431845598828684 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 5.736690439120927 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

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 -1.4431845598828684e-74)
   (/
    2.0
    (/
     (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (/ (* t (sin k)) l))))
     (/ l t)))
   (if (<= t 5.736690439120927e-49)
     (/ 2.0 (/ (/ (* (pow k 2.0) (pow (sin k) 2.0)) l) (* (/ l t) (cos k))))
     (/
      2.0
      (/
       (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (* (sin k) (/ t l)))))
       (/ 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 <= -1.4431845598828684e-74) {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * ((t * sin(k)) / l)))) / (l / t));
	} else if (t <= 5.736690439120927e-49) {
		tmp = 2.0 / (((pow(k, 2.0) * pow(sin(k), 2.0)) / l) / ((l / t) * cos(k)));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * (sin(k) * (t / l))))) / (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 < -1.4431845598828684e-74

   1. Initial program 22.2

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

   2. Simplified 22.2

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

   4. Applied unpow3_binary64_485 22.2

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

   5. Applied times-frac_binary64_425 15.8

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

   6. Applied associate-*l*_binary64_360 13.1

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

   7. Simplified 13.1

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   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 6.4

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

   12. Applied associate-*l/_binary64_362 4.9

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

   13. Applied associate-*l/_binary64_362 4.4

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

   14. Simplified 4.4

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

   if -1.4431845598828684e-74 < t < 5.7366904391209271e-49

   1. Initial program 56.5

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

   2. Simplified 56.5

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

   4. Applied unpow3_binary64_485 56.5

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

   5. Applied times-frac_binary64_425 47.7

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

   6. Applied associate-*l*_binary64_360 47.2

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

   7. Simplified 47.2

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

   9. Applied associate-/l*_binary64_364 40.4

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

   11. Applied tan-quot_binary64_578 40.4

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

   12. Applied associate-*l/_binary64_362 40.4

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

   13. Applied frac-times_binary64_429 41.7

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

   14. Applied associate-*l/_binary64_362 38.3

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

   15. Simplified 38.3

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

   16. Taylor expanded around inf 23.2

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

   if 5.7366904391209271e-49 < t

   1. Initial program 22.5

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

   2. Simplified 22.5

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

   4. Applied unpow3_binary64_485 22.5

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

   5. Applied times-frac_binary64_425 15.6

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

   6. Applied associate-*l*_binary64_360 13.6

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

   7. Simplified 13.6

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

   9. Applied associate-/l*_binary64_364 8.6

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

   11. Applied associate-*l/_binary64_362 7.5

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

   12. Applied associate-*l/_binary64_362 4.7

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

   13. Applied associate-*l/_binary64_362 4.2

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4431845598828684 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 5.736690439120927 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021013 
(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))))