Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 2.9

Time: 12.4min

Precision: binary64

Cost: 20552

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -2.806566223980714 \cdot 10^{-19} \lor \neg \left(k \leq 1.0482771448169259 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array}\]

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 (or (<= k -2.806566223980714e-19) (not (<= k 1.0482771448169259e-116)))
   (* (/ l k) (/ (/ 2.0 (/ k l)) (/ (* t (pow (sin k) 2.0)) (cos k))))
   (/ 2.0 (* (/ (* k k) l) (* t (/ (* k k) 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 tmp;
	if ((k <= -2.806566223980714e-19) || !(k <= 1.0482771448169259e-116)) {
		tmp = (l / k) * ((2.0 / (k / l)) / ((t * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((k * k) / l) * (t * ((k * k) / l)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Alternatives

Alternative 1
Error	5.9
Cost	20552

\[\begin{array}{l} \mathbf{if}\;k \leq -4.252058645845645 \cdot 10^{-19} \lor \neg \left(k \leq 1.4179595266281353 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(\frac{2}{\frac{k}{\ell}} \cdot \cos k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array}\]

Alternative 2
Error	15.9
Cost	20552

\[\begin{array}{l} \mathbf{if}\;k \leq -1.0725378665625126 \cdot 10^{-35} \lor \neg \left(k \leq 9.786874362112557 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array}\]

Alternative 3
Error	24.2
Cost	960

\[\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\]

Alternative 4
Error	26.3
Cost	960

\[\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}\]

Alternative 5
Error	34.6
Cost	64

\[0\]

Alternative 6
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if k < -2.8065662239807141e-19 or 1.04827714481692591e-116 < k

   1. Initial program 45.6

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

   2. Taylor expanded around 0 20.0

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

   3. Simplified 20.0

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

   5. Applied associate-*l*_binary64_19 17.7

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

   7. Applied times-frac_binary64_84 15.6

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

   8. Applied associate-/r*_binary64_22 15.4

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

   10. Applied *-un-lft-identity_binary64_78 15.4

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

   11. Applied times-frac_binary64_84 15.4

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

   12. Applied *-un-lft-identity_binary64_78 15.4

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

   13. Applied times-frac_binary64_84 10.1

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

   14. Applied *-un-lft-identity_binary64_78 10.1

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

   15. Applied times-frac_binary64_84 9.9

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

   16. Applied times-frac_binary64_84 0.9

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

   17. Simplified 0.8

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

   18. Simplified 0.8

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

   if -2.8065662239807141e-19 < k < 1.04827714481692591e-116

   1. Initial program 62.6

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

   2. Taylor expanded around 0 47.9

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}}\]

   3. Simplified 44.7

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}}\]
   4. Using strategy rm

   5. Applied associate-/r/_binary64_24 44.7

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\color{blue}{\frac{\ell}{t} \cdot \ell}}}\]

   6. Applied sqr-pow_binary64_50 44.8

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{\frac{\ell}{t} \cdot \ell}}\]

   7. Applied times-frac_binary64_84 23.6

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{\left(\frac{4}{2}\right)}}{\frac{\ell}{t}} \cdot \frac{{k}^{\left(\frac{4}{2}\right)}}{\ell}}}\]

   8. Simplified 15.7

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot t\right)} \cdot \frac{{k}^{\left(\frac{4}{2}\right)}}{\ell}}\]

   9. Simplified 15.7

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

   10. Simplified 15.7

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.806566223980714 \cdot 10^{-19} \lor \neg \left(k \leq 1.0482771448169259 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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))))