Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 10.9

Time: 26.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 2.3014336485196733 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{\frac{k \cdot k}{\cos k}}{\ell}}}{t_1}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5.357562004404354 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{t_1}}{k \cdot k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 3.707793217772486 \cdot 10^{+158}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k}}{t_1}\\ \end{array} \]

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 (* t (pow (sin k) 2.0))))
   (if (<= (* l l) 2.3014336485196733e-296)
     (* 2.0 (/ (/ l (/ (/ (* k k) (cos k)) l)) t_1))
     (if (<= (* l l) 5.357562004404354e+60)
       (* 2.0 (/ (* (cos k) (/ (* l l) t_1)) (* k k)))
       (if (<= (* l l) 3.707793217772486e+158)
         (* 2.0 (/ (* (cos k) (/ (* l l) k)) (* k t_1)))
         (* 2.0 (/ (/ (* (cos k) (/ l (/ k l))) k) t_1)))))))

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 = t * pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 2.3014336485196733e-296) {
		tmp = 2.0 * ((l / (((k * k) / cos(k)) / l)) / t_1);
	} else if ((l * l) <= 5.357562004404354e+60) {
		tmp = 2.0 * ((cos(k) * ((l * l) / t_1)) / (k * k));
	} else if ((l * l) <= 3.707793217772486e+158) {
		tmp = 2.0 * ((cos(k) * ((l * l) / k)) / (k * t_1));
	} else {
		tmp = 2.0 * (((cos(k) * (l / (k / l))) / k) / t_1);
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if (*.f64 l l) < 2.30143364851967326e-296

   1. Initial program 46.6

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

   2. Simplified 37.6

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

   3. Taylor expanded around 0 19.3

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

   4. Simplified 19.3

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

   6. Applied associate-/r*_binary64 19.2

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

   7. Simplified 19.2

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

   9. Applied clear-num_binary64 19.2

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

   10. Applied un-div-inv_binary64 19.2

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

   11. Applied associate-/l*_binary64 14.1

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

   if 2.30143364851967326e-296 < (*.f64 l l) < 5.3575620044043539e60

   1. Initial program 43.8

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

   2. Simplified 33.6

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

   3. Taylor expanded around 0 7.8

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

   4. Simplified 7.8

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

   6. Applied times-frac_binary64 5.8

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

   7. Applied associate-*l/_binary64 5.7

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

   if 5.3575620044043539e60 < (*.f64 l l) < 3.7077932177724858e158

   1. Initial program 42.4

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

   2. Simplified 32.9

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

   3. Taylor expanded around 0 13.7

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

   4. Simplified 13.7

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

   6. Applied associate-/r*_binary64 11.1

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

   7. Simplified 11.1

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

   9. Applied associate-*r/_binary64 11.1

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

   10. Applied associate-/r*_binary64 5.4

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

   11. Simplified 5.4

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

   13. Applied associate-/l/_binary64 1.0

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

   14. Simplified 1.0

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

   if 3.7077932177724858e158 < (*.f64 l l)

   1. Initial program 57.9

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

   2. Simplified 55.3

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

   3. Taylor expanded around 0 49.1

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

   4. Simplified 49.1

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

   6. Applied associate-/r*_binary64 47.1

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

   7. Simplified 47.1

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

   9. Applied associate-*r/_binary64 47.1

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

   10. Applied associate-/r*_binary64 40.5

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

   11. Simplified 40.5

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

   13. Applied associate-/l*_binary64 15.9

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{\frac{k}{\ell}}} \cdot \cos k}{k}}{t \cdot {\sin k}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2.3014336485196733 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{\frac{k \cdot k}{\cos k}}{\ell}}}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5.357562004404354 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}}{k \cdot k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 3.707793217772486 \cdot 10^{+158}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))))