Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 9.4

Time: 53.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.958705001521343:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \cdot \left(\left(\left(\tan k \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 1.1156366225410403 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell}, \frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \cdot \left(\left(\left(\tan k \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)}{\frac{\ell}{t}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1580535 = 2.0;
        double r1580536 = t;
        double r1580537 = 3.0;
        double r1580538 = pow(r1580536, r1580537);
        double r1580539 = l;
        double r1580540 = r1580539 * r1580539;
        double r1580541 = r1580538 / r1580540;
        double r1580542 = k;
        double r1580543 = sin(r1580542);
        double r1580544 = r1580541 * r1580543;
        double r1580545 = tan(r1580542);
        double r1580546 = r1580544 * r1580545;
        double r1580547 = 1.0;
        double r1580548 = r1580542 / r1580536;
        double r1580549 = pow(r1580548, r1580535);
        double r1580550 = r1580547 + r1580549;
        double r1580551 = r1580550 + r1580547;
        double r1580552 = r1580546 * r1580551;
        double r1580553 = r1580535 / r1580552;
        return r1580553;
}

↓

double f(double t, double l, double k) {
        double r1580554 = t;
        double r1580555 = -9.958705001521343;
        bool r1580556 = r1580554 <= r1580555;
        double r1580557 = 2.0;
        double r1580558 = k;
        double r1580559 = r1580558 / r1580554;
        double r1580560 = fma(r1580559, r1580559, r1580557);
        double r1580561 = sqrt(r1580560);
        double r1580562 = tan(r1580558);
        double r1580563 = cbrt(r1580554);
        double r1580564 = r1580563 * r1580563;
        double r1580565 = l;
        double r1580566 = r1580565 / r1580554;
        double r1580567 = r1580563 / r1580566;
        double r1580568 = sin(r1580558);
        double r1580569 = r1580567 * r1580568;
        double r1580570 = r1580564 * r1580569;
        double r1580571 = r1580562 * r1580570;
        double r1580572 = cbrt(r1580561);
        double r1580573 = r1580572 * r1580572;
        double r1580574 = r1580571 * r1580573;
        double r1580575 = r1580574 * r1580572;
        double r1580576 = r1580561 * r1580575;
        double r1580577 = r1580576 / r1580566;
        double r1580578 = r1580557 / r1580577;
        double r1580579 = 1.1156366225410403e+35;
        bool r1580580 = r1580554 <= r1580579;
        double r1580581 = r1580554 * r1580554;
        double r1580582 = cos(r1580558);
        double r1580583 = r1580581 / r1580582;
        double r1580584 = r1580568 * r1580568;
        double r1580585 = r1580584 / r1580565;
        double r1580586 = r1580583 * r1580585;
        double r1580587 = r1580584 / r1580582;
        double r1580588 = r1580558 * r1580558;
        double r1580589 = r1580588 / r1580565;
        double r1580590 = r1580587 * r1580589;
        double r1580591 = fma(r1580557, r1580586, r1580590);
        double r1580592 = r1580591 / r1580566;
        double r1580593 = r1580557 / r1580592;
        double r1580594 = r1580580 ? r1580593 : r1580578;
        double r1580595 = r1580556 ? r1580578 : r1580594;
        return r1580595;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -9.958705001521343 or 1.1156366225410403e+35 < t

   1. Initial program 21.7

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

   2. Simplified 10.4

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

   4. Applied add-cube-cbrt 10.6

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

   5. Applied times-frac 10.1

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

   6. Applied associate-*l* 6.0

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

   8. Applied associate-*l/ 5.9

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

   9. Applied associate-*l/ 4.1

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

   10. Applied associate-*l/ 3.9

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

   12. Applied add-sqr-sqrt 4.0

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

   13. Applied associate-*r* 4.0

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

   15. Applied add-cube-cbrt 4.0

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

   16. Applied associate-*r* 4.0

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

   if -9.958705001521343 < t < 1.1156366225410403e+35

   1. Initial program 45.5

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

   2. Simplified 34.7

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

   4. Applied add-cube-cbrt 34.8

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

   5. Applied times-frac 34.0

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

   6. Applied associate-*l* 32.4

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

   8. Applied associate-*l/ 32.4

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

   9. Applied associate-*l/ 33.3

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

   10. Applied associate-*l/ 30.1

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

   11. Taylor expanded around -inf 20.0

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

   12. Simplified 17.3

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.958705001521343:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \cdot \left(\left(\left(\tan k \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 1.1156366225410403 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\cos k} \cdot \frac{\sin k \cdot \sin k}{\ell}, \frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \cdot \left(\left(\left(\tan k \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))