Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 10.9

Time: 22.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := t \cdot \sin k\\ \mathbf{if}\;t \leq -1.0709717355558108 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot t_2\right)\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 2.6991301715389753 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \left(t_2 \cdot \frac{1}{\ell}\right)\right)\right)\right)}\\ \end{array} \]

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
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (* t (sin k))))
   (if (<= t -1.0709717355558108e-163)
     (/ 2.0 (* (* (* (/ t l) (tan k)) (* (/ t l) t_2)) t_1))
     (if (<= t 2.6991301715389753e-146)
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (* l l))
         (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
       (/ 2.0 (* t_1 (* (/ t l) (* (tan k) (* t (* t_2 (/ 1.0 l)))))))))))

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 t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = t * sin(k);
	double tmp;
	if (t <= -1.0709717355558108e-163) {
		tmp = 2.0 / ((((t / l) * tan(k)) * ((t / l) * t_2)) * t_1);
	} else if (t <= 2.6991301715389753e-146) {
		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 / (t_1 * ((t / l) * (tan(k) * (t * (t_2 * (1.0 / l))))));
	}
	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 < -1.07097173555581083e-163

   1. Initial program 26.8

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

   2. Simplified 26.8

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

   4. Applied cube-mult_binary64 26.8

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

   5. Applied times-frac_binary64 18.4

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

   6. Applied associate-*l*_binary64 16.2

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

   7. Simplified 16.2

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

   9. Applied *-un-lft-identity_binary64 16.2

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

   10. Applied times-frac_binary64 11.7

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

   11. Applied associate-*r*_binary64 10.8

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

   12. Simplified 10.8

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

   14. Applied associate-*l*_binary64 9.2

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

   15. Simplified 9.2

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

   17. Applied associate-*r*_binary64 8.3

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

   if -1.07097173555581083e-163 < t < 2.6991301715389753e-146

   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 cube-mult_binary64 64.0

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

   5. Applied times-frac_binary64 62.5

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

   6. Applied associate-*l*_binary64 62.5

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

   7. Simplified 62.5

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

   8. Taylor expanded in t around 0 40.8

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

   9. Simplified 25.0

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

   if 2.6991301715389753e-146 < t

   1. Initial program 26.1

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

   2. Simplified 26.1

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

   4. Applied cube-mult_binary64 26.1

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

   5. Applied times-frac_binary64 17.9

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

   6. Applied associate-*l*_binary64 15.3

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

   7. Simplified 15.3

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

   9. Applied *-un-lft-identity_binary64 15.3

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

   10. Applied times-frac_binary64 10.4

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

   11. Applied associate-*r*_binary64 9.5

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

   12. Simplified 9.5

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

   14. Applied associate-*l*_binary64 7.9

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

   15. Simplified 7.9

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

   17. Applied div-inv_binary64 7.9

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

   18. Applied associate-*l*_binary64 7.9

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(t \cdot \left(\frac{1}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0709717355558108 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.6991301715389753 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))