Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 2.9

Time: 28.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.9483880244478677 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \sin k\right) \cdot \frac{k}{\frac{\cos k}{\frac{\sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}\\ \end{array} \]

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
 (if (<= (* l l) 3.9483880244478677e+307)
   (/ 2.0 (* (* (/ (* k t) l) (sin k)) (/ k (/ (cos k) (/ (sin k) l)))))
   (/
    2.0
    (*
     (* (/ k (* (cbrt l) (cbrt l))) (/ t (cbrt l)))
     (/ k (/ (cos k) (/ (pow (sin k) 2.0) 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 ((l * l) <= 3.9483880244478677e+307) {
		tmp = 2.0 / ((((k * t) / l) * sin(k)) * (k / (cos(k) / (sin(k) / l))));
	} else {
		tmp = 2.0 / (((k / (cbrt(l) * cbrt(l))) * (t / cbrt(l))) * (k / (cos(k) / (pow(sin(k), 2.0) / 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 (*.f64 l l) < 3.94838802444786774e307

   1. Initial program 44.8

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

   2. Simplified 35.9

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

   3. Taylor expanded around 0 15.7

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

   4. Simplified 15.7

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

   6. Applied associate-/l*_binary64 14.8

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

   7. Simplified 14.9

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

   9. Applied times-frac_binary64 11.4

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

   10. Applied *-un-lft-identity_binary64 11.4

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

   11. Applied times-frac_binary64 11.3

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

   12. Applied times-frac_binary64 6.8

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

   13. Simplified 6.6

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

   15. Applied *-un-lft-identity_binary64 6.6

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

   16. Applied sqr-pow_binary64 6.6

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

   17. Applied times-frac_binary64 6.3

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

   18. Applied *-un-lft-identity_binary64 6.3

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

   19. Applied times-frac_binary64 6.3

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

   20. Applied *-un-lft-identity_binary64 6.3

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

   21. Applied times-frac_binary64 6.3

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

   22. Applied associate-*r*_binary64 5.6

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

   23. Simplified 2.7

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

   if 3.94838802444786774e307 < (*.f64 l l)

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around 0 63.9

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

   4. Simplified 63.9

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

   6. Applied associate-/l*_binary64 63.8

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

   7. Simplified 63.8

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

   9. Applied times-frac_binary64 46.6

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

   10. Applied *-un-lft-identity_binary64 46.6

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

   11. Applied times-frac_binary64 46.5

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

   12. Applied times-frac_binary64 15.0

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

   13. Simplified 14.4

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

   15. Applied add-cube-cbrt_binary64 14.8

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

   16. Applied *-un-lft-identity_binary64 14.8

       \[\leadsto \frac{2}{\left(k \cdot \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right) \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}} \]

   17. Applied times-frac_binary64 14.8

       \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}\right) \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}} \]

   18. Applied associate-*r*_binary64 4.3

       \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot \frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}\right)} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}} \]

   19. Simplified 4.3

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.9483880244478677 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \sin k\right) \cdot \frac{k}{\frac{\cos k}{\frac{\sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021197 
(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))))