Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 6.9

Time: 27.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := \frac{k}{\cos k}\\ \mathbf{if}\;k \leq -2.7246474924947304 \cdot 10^{+175} \lor \neg \left(k \leq 2.1694879221150623 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{\frac{k}{\ell} \cdot t_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t_2 \cdot \frac{k}{\ell}\right) \cdot \frac{t_1}{\ell}}\\ \end{array} \]

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))) (t_2 (/ k (cos k))))
   (if (or (<= k -2.7246474924947304e+175)
           (not (<= k 2.1694879221150623e+145)))
     (/ 2.0 (* t_2 (/ (* (/ k l) t_1) l)))
     (/ 2.0 (* (* t_2 (/ k l)) (/ t_1 l))))))

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 t_2 = k / cos(k);
	double tmp;
	if ((k <= -2.7246474924947304e+175) || !(k <= 2.1694879221150623e+145)) {
		tmp = 2.0 / (t_2 * (((k / l) * t_1) / l));
	} else {
		tmp = 2.0 / ((t_2 * (k / l)) * (t_1 / l));
	}
	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 < -2.7246474924947304e175 or 2.16948792211506227e145 < k

   1. Initial program 39.1

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

   2. Simplified 33.4

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

   3. Taylor expanded in t around 0 22.9

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

   4. Applied unpow2_binary64 22.9

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

   5. Applied associate-*l*_binary64 18.2

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

   6. Applied times-frac_binary64 14.7

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

   7. Applied unpow2_binary64 14.7

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

   8. Applied associate-/r*_binary64 9.6

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

   9. Simplified 4.9

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

   if -2.7246474924947304e175 < k < 2.16948792211506227e145

   1. Initial program 53.7

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

   2. Simplified 44.3

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

   3. Taylor expanded in t around 0 21.8

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

   4. Applied unpow2_binary64 21.8

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

   5. Applied associate-*l*_binary64 21.4

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

   6. Applied times-frac_binary64 19.5

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

   7. Applied unpow2_binary64 19.5

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

   8. Applied times-frac_binary64 9.0

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

   9. Applied associate-*r*_binary64 8.1

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.7246474924947304 \cdot 10^{+175} \lor \neg \left(k \leq 2.1694879221150623 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \frac{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022055 
(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))))