Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 7.8

Time: 27.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot t_1\\ t_3 := \frac{\cos k}{k}\\ \mathbf{if}\;\ell \leq -1.610639893457833 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\left(\ell \cdot \frac{1}{k}\right) \cdot t_3\right)}{t_2}\\ \mathbf{elif}\;\ell \leq -1.7779119994400633 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(t_3 \cdot \left(\ell \cdot \ell\right)\right)}{t_2}\\ \mathbf{elif}\;\ell \leq 9.083547918518802 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{k \cdot k}}{t} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{elif}\;\ell \leq 1.8106547977880653 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(t_3 \cdot \frac{\ell}{k}\right)}{t_2}\\ \end{array} \]

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 (pow (sin k) 2.0)) (t_2 (* t t_1)) (t_3 (/ (cos k) k)))
   (if (<= l -1.610639893457833e+160)
     (* 2.0 (/ (* l (* (* l (/ 1.0 k)) t_3)) t_2))
     (if (<= l -1.7779119994400633e-12)
       (* 2.0 (/ (* (/ 1.0 k) (* t_3 (* l l))) t_2))
       (if (<= l 9.083547918518802e-110)
         (* 2.0 (* (/ (* l (/ (cos k) (* k k))) t) (/ l t_1)))
         (if (<= l 1.8106547977880653e+106)
           (* 2.0 (/ (* (cos k) (* l l)) (* k (* k t_2))))
           (* 2.0 (/ (* l (* t_3 (/ l k))) 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 = pow(sin(k), 2.0);
	double t_2 = t * t_1;
	double t_3 = cos(k) / k;
	double tmp;
	if (l <= -1.610639893457833e+160) {
		tmp = 2.0 * ((l * ((l * (1.0 / k)) * t_3)) / t_2);
	} else if (l <= -1.7779119994400633e-12) {
		tmp = 2.0 * (((1.0 / k) * (t_3 * (l * l))) / t_2);
	} else if (l <= 9.083547918518802e-110) {
		tmp = 2.0 * (((l * (cos(k) / (k * k))) / t) * (l / t_1));
	} else if (l <= 1.8106547977880653e+106) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (k * (k * t_2)));
	} else {
		tmp = 2.0 * ((l * (t_3 * (l / k))) / t_2);
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if l < -1.61063989345783296e160

   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

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

   3. Taylor expanded around 0 64.0

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

   4. Simplified 64.0

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

   6. Applied associate-/r*_binary64 64.0

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

   7. Simplified 64.0

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

   9. Applied associate-*r*_binary64 45.1

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

   10. Simplified 45.1

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

   12. Applied *-un-lft-identity_binary64 45.1

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

   13. Applied times-frac_binary64 43.9

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

   14. Applied associate-*r*_binary64 15.4

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

   if -1.61063989345783296e160 < l < -1.7779119994400633e-12

   1. Initial program 48.1

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

   2. Simplified 40.5

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

   3. Taylor expanded around 0 19.2

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

   4. Simplified 19.2

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

   6. Applied associate-/r*_binary64 16.2

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

   7. Simplified 16.2

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

   9. Applied *-un-lft-identity_binary64 16.2

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

   10. Applied times-frac_binary64 15.7

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

   11. Applied associate-*l*_binary64 8.2

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

   12. Simplified 8.2

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

   if -1.7779119994400633e-12 < l < 9.08354791851880226e-110

   1. Initial program 44.9

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

   2. Simplified 36.1

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

   3. Taylor expanded around 0 16.8

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

   4. Simplified 16.8

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

   6. Applied associate-/r*_binary64 16.1

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

   7. Simplified 16.2

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

   9. Applied associate-*r*_binary64 12.5

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

   10. Simplified 12.5

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

   12. Applied times-frac_binary64 5.9

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

   if 9.08354791851880226e-110 < l < 1.8106547977880653e106

   1. Initial program 45.0

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

   2. Simplified 36.1

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

   3. Taylor expanded around 0 11.6

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

   4. Simplified 11.6

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

   6. Applied associate-*l*_binary64 6.0

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

   if 1.8106547977880653e106 < l

   1. Initial program 59.2

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

   2. Simplified 57.4

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

   3. Taylor expanded around 0 52.0

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

   4. Simplified 52.0

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

   6. Applied associate-/r*_binary64 51.2

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

   7. Simplified 51.2

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

   9. Applied associate-*r*_binary64 36.9

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

   10. Simplified 36.9

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

   12. Applied *-un-lft-identity_binary64 36.9

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

   13. Applied times-frac_binary64 35.9

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

   14. Applied associate-*r*_binary64 14.0

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

   15. Simplified 14.0

       \[\leadsto 2 \cdot \frac{\left(\color{blue}{\frac{\ell}{k}} \cdot \frac{\cos k}{k}\right) \cdot \ell}{t \cdot {\sin k}^{2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.610639893457833 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\left(\ell \cdot \frac{1}{k}\right) \cdot \frac{\cos k}{k}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq -1.7779119994400633 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\cos k}{k} \cdot \left(\ell \cdot \ell\right)\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq 9.083547918518802 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\cos k}{k \cdot k}}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 1.8106547977880653 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))))