Toniolo and Linder, Equation (10-)

Average Error: 41.5 → 4.5

Time: 18.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 9.06101675763 \cdot 10^{-313}:\\ \;\;\;\;\left(t_1 \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t_2}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2.9764153496865264 \cdot 10^{+165}:\\ \;\;\;\;t_1 \cdot \frac{2}{\frac{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{2}{\frac{t_2 \cdot \frac{k}{\ell}}{\ell}}\\ \end{array} \]

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 (/ (cos k) k)) (t_2 (* t (pow (sin k) 2.0))))
   (if (<= (* l l) 9.06101675763e-313)
     (* (* t_1 (/ l k)) (/ 2.0 (/ t_2 l)))
     (if (<= (* l l) 2.9764153496865264e+165)
       (* t_1 (/ 2.0 (/ (* k (* (sin k) (* t (sin k)))) (* l l))))
       (* t_1 (/ 2.0 (/ (* t_2 (/ k l)) 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 = cos(k) / k;
	double t_2 = t * pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 9.06101675763e-313) {
		tmp = (t_1 * (l / k)) * (2.0 / (t_2 / l));
	} else if ((l * l) <= 2.9764153496865264e+165) {
		tmp = t_1 * (2.0 / ((k * (sin(k) * (t * sin(k)))) / (l * l)));
	} else {
		tmp = t_1 * (2.0 / ((t_2 * (k / l)) / l));
	}
	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) < 9.06101675763e-313

   1. Initial program 49.4

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

   2. Simplified 42.8

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

   3. Taylor expanded in t around 0 28.6

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

   4. Applied unpow2_binary64 28.6

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

   5. Applied associate-*l*_binary64 28.6

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

   6. Applied times-frac_binary64 28.4

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

   7. Applied *-un-lft-identity_binary64 28.4

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

   8. Applied times-frac_binary64 28.4

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

   9. Simplified 28.4

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

   10. Simplified 28.4

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

   11. Applied times-frac_binary64 8.7

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

   12. Applied *-un-lft-identity_binary64 8.7

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

   13. Applied times-frac_binary64 8.5

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

   14. Applied associate-*r*_binary64 7.6

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

   15. Simplified 7.6

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

   if 9.06101675763e-313 < (*.f64 l l) < 2.97641534968652642e165

   1. Initial program 37.2

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

   2. Simplified 30.2

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

   3. Taylor expanded in t around 0 6.5

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

   4. Applied unpow2_binary64 6.5

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

   5. Applied associate-*l*_binary64 4.3

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

   6. Applied times-frac_binary64 2.1

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

   7. Applied *-un-lft-identity_binary64 2.1

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

   8. Applied times-frac_binary64 2.1

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

   9. Simplified 2.1

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

   10. Simplified 2.1

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

   11. Applied unpow2_binary64 2.1

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

   12. Applied associate-*r*_binary64 1.8

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

   if 2.97641534968652642e165 < (*.f64 l l)

   1. Initial program 41.0

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

   2. Simplified 40.1

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

   3. Taylor expanded in t around 0 21.1

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

   4. Applied unpow2_binary64 21.1

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

   5. Applied associate-*l*_binary64 18.7

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

   6. Applied times-frac_binary64 17.3

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

   7. Applied *-un-lft-identity_binary64 17.3

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

   8. Applied times-frac_binary64 17.3

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

   9. Simplified 17.3

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

   10. Simplified 17.3

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

   11. Applied associate-/r*_binary64 10.6

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

   12. Simplified 5.2

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

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 9.06101675763 \cdot 10^{-313}:\\ \;\;\;\;\left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2.9764153496865264 \cdot 10^{+165}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{2}{\frac{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot \frac{k}{\ell}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))))