Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 17.8

Time: 2.6m

Precision: 64

• could not determine a ground truth for program body

  1. t = 2.88733846674046e-150
  2. l = -1.4028819597761525e+262
  3. k = -7.995242360819531e+16

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.66999260568644201 \cdot 10^{-136}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{2 \cdot \ell}{{t}^{\left(2 \cdot \frac{3}{2}\right)} \cdot \sin k} \cdot 1}{\tan k}}\right)}^{\left(\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\right)\\ \mathbf{elif}\;t \le 1.6050627599136617 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{1}{{t}^{\left(\frac{3}{2}\right)}}}{1} \cdot \left(\frac{\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)\right) \cdot \frac{1}{\tan k}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r286645 = 2.0;
        double r286646 = t;
        double r286647 = 3.0;
        double r286648 = pow(r286646, r286647);
        double r286649 = l;
        double r286650 = r286649 * r286649;
        double r286651 = r286648 / r286650;
        double r286652 = k;
        double r286653 = sin(r286652);
        double r286654 = r286651 * r286653;
        double r286655 = tan(r286652);
        double r286656 = r286654 * r286655;
        double r286657 = 1.0;
        double r286658 = r286652 / r286646;
        double r286659 = pow(r286658, r286645);
        double r286660 = r286657 + r286659;
        double r286661 = r286660 + r286657;
        double r286662 = r286656 * r286661;
        double r286663 = r286645 / r286662;
        return r286663;
}

↓

double f(double t, double l, double k) {
        double r286664 = t;
        double r286665 = 1.669992605686442e-136;
        bool r286666 = r286664 <= r286665;
        double r286667 = 2.0;
        double r286668 = l;
        double r286669 = r286667 * r286668;
        double r286670 = 2.0;
        double r286671 = 3.0;
        double r286672 = r286671 / r286670;
        double r286673 = r286670 * r286672;
        double r286674 = pow(r286664, r286673);
        double r286675 = k;
        double r286676 = sin(r286675);
        double r286677 = r286674 * r286676;
        double r286678 = r286669 / r286677;
        double r286679 = 1.0;
        double r286680 = r286678 * r286679;
        double r286681 = tan(r286675);
        double r286682 = r286680 / r286681;
        double r286683 = exp(r286682);
        double r286684 = 1.0;
        double r286685 = r286675 / r286664;
        double r286686 = pow(r286685, r286667);
        double r286687 = fma(r286670, r286684, r286686);
        double r286688 = r286668 / r286687;
        double r286689 = pow(r286683, r286688);
        double r286690 = log(r286689);
        double r286691 = 1.6050627599136617e-37;
        bool r286692 = r286664 <= r286691;
        double r286693 = pow(r286664, r286672);
        double r286694 = r286679 / r286693;
        double r286695 = r286694 / r286679;
        double r286696 = r286693 * r286676;
        double r286697 = r286667 / r286696;
        double r286698 = r286697 * r286668;
        double r286699 = r286698 / r286681;
        double r286700 = r286699 * r286688;
        double r286701 = r286695 * r286700;
        double r286702 = r286694 * r286698;
        double r286703 = r286679 / r286681;
        double r286704 = r286702 * r286703;
        double r286705 = r286704 * r286688;
        double r286706 = r286692 ? r286701 : r286705;
        double r286707 = r286666 ? r286690 : r286706;
        return r286707;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < 1.669992605686442e-136

   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

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 64.0

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 64.0

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 64.0

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied sqr-pow 64.0

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-*l* 64.0

       \[\leadsto \frac{\frac{2}{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied add-log-exp 64.0

       \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\right)}\]

   13. Simplified 41.0

       \[\leadsto \log \color{blue}{\left({\left(e^{\frac{\frac{2 \cdot \ell}{{t}^{\left(2 \cdot \frac{3}{2}\right)} \cdot \sin k} \cdot 1}{\tan k}}\right)}^{\left(\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\right)}\]

   if 1.669992605686442e-136 < t < 1.6050627599136617e-37

   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 42.6

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 42.6

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 42.2

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 42.5

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 40.2

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied sqr-pow 40.2

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-*l* 40.2

       \[\leadsto \frac{\frac{2}{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 40.2

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   13. Applied times-frac 40.2

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied associate-*l* 31.2

       \[\leadsto \frac{\color{blue}{\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 31.2

       \[\leadsto \frac{\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)}{\color{blue}{1 \cdot \tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   17. Applied times-frac 31.1

       \[\leadsto \color{blue}{\left(\frac{\frac{1}{{t}^{\left(\frac{3}{2}\right)}}}{1} \cdot \frac{\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell}{\tan k}\right)} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied associate-*l* 24.5

       \[\leadsto \color{blue}{\frac{\frac{1}{{t}^{\left(\frac{3}{2}\right)}}}{1} \cdot \left(\frac{\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   if 1.6050627599136617e-37 < t

   1. Initial program 22.5

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

   2. Simplified 22.1

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 22.1

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 20.9

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 17.1

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 15.7

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied sqr-pow 15.7

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied associate-*l* 13.5

       \[\leadsto \frac{\frac{2}{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 13.5

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{{t}^{\left(\frac{3}{2}\right)} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   13. Applied times-frac 13.3

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied associate-*l* 9.7

       \[\leadsto \frac{\color{blue}{\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   15. Using strategy rm

   16. Applied div-inv 9.7

       \[\leadsto \color{blue}{\left(\left(\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)\right) \cdot \frac{1}{\tan k}\right)} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.66999260568644201 \cdot 10^{-136}:\\ \;\;\;\;\log \left({\left(e^{\frac{\frac{2 \cdot \ell}{{t}^{\left(2 \cdot \frac{3}{2}\right)} \cdot \sin k} \cdot 1}{\tan k}}\right)}^{\left(\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\right)\\ \mathbf{elif}\;t \le 1.6050627599136617 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{1}{{t}^{\left(\frac{3}{2}\right)}}}{1} \cdot \left(\frac{\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{{t}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{t}^{\left(\frac{3}{2}\right)} \cdot \sin k} \cdot \ell\right)\right) \cdot \frac{1}{\tan k}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))