Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 6.8

Time: 1.0min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.404420770027635 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 9.42696047899692 \cdot 10^{+123}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array}\]

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
 (if (<= (* l l) 5.404420770027635e-199)
   (* 2.0 (/ (* (* (/ l k) (/ l k)) (cos k)) (* (sin k) (* t (sin k)))))
   (if (<= (* l l) 9.42696047899692e+123)
     (* 2.0 (/ (* (cos k) (* l (/ l k))) (* k (* t (pow (sin k) 2.0)))))
     (* 2.0 (* (/ (* (/ l k) (/ l k)) t) (/ (cos k) (pow (sin k) 2.0)))))))

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 ((l * l) <= 5.404420770027635e-199) {
		tmp = 2.0 * ((((l / k) * (l / k)) * cos(k)) / (sin(k) * (t * sin(k))));
	} else if ((l * l) <= 9.42696047899692e+123) {
		tmp = 2.0 * ((cos(k) * (l * (l / k))) / (k * (t * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) / t) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.4044207700276351e-199

   1. Initial program 46.1

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

   2. Simplified 36.5

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

   3. Taylor expanded around 0 17.7

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

   4. Simplified 17.7

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

   6. Applied associate-/r*_binary64_363 17.2

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

   7. Simplified 17.2

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

   9. Applied times-frac_binary64_425 12.5

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

   11. Applied unpow2_binary64_484 12.5

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

   12. Applied associate-*r*_binary64_359 9.4

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

   if 5.4044207700276351e-199 < (*.f64 l l) < 9.42696047899691992e123

   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 34.7

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

   3. Taylor expanded around 0 8.8

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

   4. Simplified 8.8

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

   6. Applied associate-/r*_binary64_363 7.9

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

   7. Simplified 7.9

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

   9. Applied times-frac_binary64_425 6.9

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

   11. Applied associate-*l/_binary64_362 6.8

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

   12. Applied associate-*l/_binary64_362 6.8

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

   13. Applied associate-/l/_binary64_366 1.8

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

   if 9.42696047899691992e123 < (*.f64 l l)

   1. Initial program 56.2

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

   2. Simplified 52.9

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

   3. Taylor expanded around 0 45.2

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

   4. Simplified 45.2

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

   6. Applied associate-/r*_binary64_363 43.2

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

   7. Simplified 43.2

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

   9. Applied times-frac_binary64_425 7.6

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

   11. Applied times-frac_binary64_425 7.6

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.404420770027635 \cdot 10^{-199}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 9.42696047899692 \cdot 10^{+123}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))))