Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 15.7

Time: 15.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -5.682272213353497 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 8.735486909965795 \cdot 10^{-203}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\right)}\\ \end{array}\]

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 (<= t -5.682272213353497e-132)
   (/
    2.0
    (*
     (* (* (/ t l) (* (* t (/ t l)) (sin k))) (tan k))
     (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t 8.735486909965795e-203)
     (/
      2.0
      (*
       (/ (pow (sin k) 2.0) (cos k))
       (+ (/ (* t (* k k)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l l))))))
     (/
      2.0
      (*
       (cbrt
        (*
         (+ 2.0 (pow (/ k t) 2.0))
         (*
          (tan k)
          (*
           (/ t l)
           (* (/ t (* (cbrt l) (cbrt l))) (* (sin k) (/ t (cbrt l))))))))
       (*
        (cbrt
         (*
          (+ 2.0 (pow (/ k t) 2.0))
          (*
           (tan k)
           (*
            (/ t l)
            (* (/ t (* (cbrt l) (cbrt l))) (* (sin k) (/ t (cbrt l))))))))
        (cbrt
         (*
          (+ 2.0 (pow (/ k t) 2.0))
          (*
           (tan k)
           (*
            (/ t l)
            (*
             (/ t (* (cbrt l) (cbrt l)))
             (* (sin k) (/ t (cbrt 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 (t <= -5.682272213353497e-132) {
		tmp = 2.0 / ((((t / l) * ((t * (t / l)) * sin(k))) * tan(k)) * (2.0 + pow((k / t), 2.0)));
	} else if (t <= 8.735486909965795e-203) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((t * (k * k)) / (l * l)) + (2.0 * (pow(t, 3.0) / (l * l)))));
	} else {
		tmp = 2.0 / (cbrt((2.0 + pow((k / t), 2.0)) * (tan(k) * ((t / l) * ((t / (cbrt(l) * cbrt(l))) * (sin(k) * (t / cbrt(l))))))) * (cbrt((2.0 + pow((k / t), 2.0)) * (tan(k) * ((t / l) * ((t / (cbrt(l) * cbrt(l))) * (sin(k) * (t / cbrt(l))))))) * cbrt((2.0 + pow((k / t), 2.0)) * (tan(k) * ((t / l) * ((t / (cbrt(l) * cbrt(l))) * (sin(k) * (t / cbrt(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 t < -5.682272213353497e-132

   1. Initial program 25.0

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

   2. Simplified 25.0

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

   4. Applied cube-mult_binary64_441 25.0

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

   5. Applied times-frac_binary64_420 17.7

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

   6. Applied associate-*l*_binary64_357 15.5

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

   8. Applied *-un-lft-identity_binary64_414 15.5

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

   9. Applied times-frac_binary64_420 10.2

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

   10. Simplified 10.2

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

   if -5.682272213353497e-132 < t < 8.7354869099657953e-203

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around inf 41.2

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

   4. Simplified 40.0

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

   if 8.7354869099657953e-203 < t

   1. Initial program 28.8

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

   2. Simplified 28.8

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

   4. Applied cube-mult_binary64_441 28.8

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

   5. Applied times-frac_binary64_420 21.2

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

   6. Applied associate-*l*_binary64_357 18.8

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

   8. Applied add-cube-cbrt_binary64_446 19.0

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

   9. Applied times-frac_binary64_420 13.7

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

   10. Applied associate-*l*_binary64_357 12.5

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

   11. Simplified 12.5

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)}\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt_binary64_446 12.6

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin k \cdot \frac{t}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\sin k \cdot \frac{t}{\sqrt[3]{\ell}}}\right) \cdot \sqrt[3]{\sin k \cdot \frac{t}{\sqrt[3]{\ell}}}\right)}\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt_binary64_446 12.7

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

   16. Simplified 12.6

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

   17. Simplified 12.6

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

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.682272213353497 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 8.735486909965795 \cdot 10^{-203}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)} \cdot \left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\sin k \cdot \frac{t}{\sqrt[3]{\ell}}\right)\right)\right)\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))))