Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 3.6

Time: 25.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{k}{\ell \cdot \cos k}\\ \mathbf{if}\;t \leq -4.1579964146031276 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -3.36110731526155 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{t_2}{\cos k}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot t_2\right)\right)}\\ \end{array}\\ \end{array} \]

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 (/ k (* l (cos k)))))
   (if (<= t -4.1579964146031276e+123)
     (/ 2.0 (* (* (* (sin k) (* t (sin k))) (/ k l)) t_1))
     (let* ((t_2 (pow (sin k) 2.0)))
       (if (<= t -3.36110731526155e-187)
         (* (/ (/ l t) k) (/ 2.0 (/ k (/ l (/ t_2 (cos k))))))
         (/ 2.0 (* t_1 (* t (* (/ k l) t_2)))))))))

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 = k / (l * cos(k));
	double tmp;
	if (t <= -4.1579964146031276e+123) {
		tmp = 2.0 / (((sin(k) * (t * sin(k))) * (k / l)) * t_1);
	} else {
		double t_2 = pow(sin(k), 2.0);
		double tmp_1;
		if (t <= -3.36110731526155e-187) {
			tmp_1 = ((l / t) / k) * (2.0 / (k / (l / (t_2 / cos(k)))));
		} else {
			tmp_1 = 2.0 / (t_1 * (t * ((k / l) * t_2)));
		}
		tmp = tmp_1;
	}
	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 < -4.15799641460312759e123

   1. Initial program 54.3

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

   2. Simplified 39.1

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

   3. Taylor expanded around 0 20.9

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

   4. Simplified 20.9

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

   6. Applied associate-/l*_binary64 18.6

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

   7. Simplified 18.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{\frac{t \cdot {\sin k}^{2}}{\cos k}}}}} \]
   8. Using strategy rm

   9. Applied div-inv_binary64 18.7

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

   10. Applied times-frac_binary64 12.4

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{t \cdot {\sin k}^{2}} \cdot \frac{\ell}{\frac{1}{\cos k}}}}} \]

   11. Applied times-frac_binary64 5.0

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{t \cdot {\sin k}^{2}}} \cdot \frac{k}{\frac{\ell}{\frac{1}{\cos k}}}}} \]

   12. Simplified 5.0

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

   13. Simplified 5.0

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

   15. Applied sqr-pow_binary64 5.0

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

   16. Applied associate-*r*_binary64 0.8

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

   17. Simplified 0.8

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

   if -4.15799641460312759e123 < t < -3.3611073152615502e-187

   1. Initial program 39.3

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

   2. Simplified 33.7

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

   3. Taylor expanded around 0 22.9

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

   4. Simplified 22.9

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

   6. Applied associate-/l*_binary64 21.6

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

   7. Simplified 21.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{\frac{t \cdot {\sin k}^{2}}{\cos k}}}}} \]
   8. Using strategy rm

   9. Applied *-un-lft-identity_binary64 21.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\frac{t \cdot {\sin k}^{2}}{\color{blue}{1 \cdot \cos k}}}}} \]

   10. Applied times-frac_binary64 21.6

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\color{blue}{\frac{t}{1} \cdot \frac{{\sin k}^{2}}{\cos k}}}}} \]

   11. Applied times-frac_binary64 17.2

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{t}{1}} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}} \]

   12. Applied times-frac_binary64 5.3

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{1}}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}} \]

   13. Applied *-un-lft-identity_binary64 5.3

       \[\leadsto \frac{\color{blue}{1 \cdot 2}}{\frac{k}{\frac{\ell}{\frac{t}{1}}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}} \]

   14. Applied times-frac_binary64 5.0

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{\frac{t}{1}}}} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}} \]

   15. Simplified 5.0

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

   if -3.3611073152615502e-187 < t

   1. Initial program 50.2

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

   2. Simplified 43.3

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

   3. Taylor expanded around 0 24.3

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

   4. Simplified 24.3

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

   6. Applied associate-/l*_binary64 24.1

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

   7. Simplified 24.1

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{\frac{t \cdot {\sin k}^{2}}{\cos k}}}}} \]
   8. Using strategy rm

   9. Applied div-inv_binary64 24.1

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

   10. Applied times-frac_binary64 19.3

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{t \cdot {\sin k}^{2}} \cdot \frac{\ell}{\frac{1}{\cos k}}}}} \]

   11. Applied times-frac_binary64 9.4

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{t \cdot {\sin k}^{2}}} \cdot \frac{k}{\frac{\ell}{\frac{1}{\cos k}}}}} \]

   12. Simplified 5.0

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

   13. Simplified 5.0

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

   15. Applied associate-*l*_binary64 3.8

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1579964146031276 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq -3.36110731526155 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021196 
(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))))