Toniolo and Linder, Equation (10-)

Average Error: 48.7 → 8.6

Time: 35.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{{\sin k}^{2}}\\ \mathbf{if}\;k \leq -1.0800721622191825 \cdot 10^{+154} \lor \neg \left(k \leq 1.4364428154027302 \cdot 10^{+90}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{k} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{t}\right)\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\ell \cdot \frac{\cos k}{{k}^{2}}}{t}\right)\\ \end{array} \]

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 (/ l (pow (sin k) 2.0))))
   (if (or (<= k -1.0800721622191825e+154) (not (<= k 1.4364428154027302e+90)))
     (* 2.0 (* (* (/ 1.0 k) (* (/ (cos k) k) (/ l t))) t_1))
     (* 2.0 (* t_1 (/ (* l (/ (cos k) (pow k 2.0))) t))))))

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 = l / pow(sin(k), 2.0);
	double tmp;
	if ((k <= -1.0800721622191825e+154) || !(k <= 1.4364428154027302e+90)) {
		tmp = 2.0 * (((1.0 / k) * ((cos(k) / k) * (l / t))) * t_1);
	} else {
		tmp = 2.0 * (t_1 * ((l * (cos(k) / pow(k, 2.0))) / t));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -1.0800721622191825e154 or 1.43644281540273021e90 < k

   1. Initial program 41.3

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

   2. Simplified 35.0

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

   3. Taylor expanded in t around 0 22.9

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

   4. Applied times-frac_binary64 23.0

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

   5. Applied unpow2_binary64 23.0

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

   6. Applied times-frac_binary64 20.9

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

   7. Applied associate-*r*_binary64 19.9

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

   8. Applied unpow2_binary64 19.9

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

   9. Applied *-un-lft-identity_binary64 19.9

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

   10. Applied times-frac_binary64 19.4

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

   11. Applied associate-*l*_binary64 11.7

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

   if -1.0800721622191825e154 < k < 1.43644281540273021e90

   1. Initial program 55.7

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

   2. Simplified 46.6

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

   3. Taylor expanded in t around 0 22.5

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

   4. Applied times-frac_binary64 20.5

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

   5. Applied unpow2_binary64 20.5

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

   6. Applied times-frac_binary64 11.9

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

   7. Applied associate-*r*_binary64 8.5

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

   8. Applied associate-*r/_binary64 5.7

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.0800721622191825 \cdot 10^{+154} \lor \neg \left(k \leq 1.4364428154027302 \cdot 10^{+90}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{k} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{t}\right)\right) \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\ell \cdot \frac{\cos k}{{k}^{2}}}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))