Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 16.1

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -2.42796361625678879 \cdot 10^{162} \lor \neg \left(\ell \le 1.253217422869759 \cdot 10^{154}\right):\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{\sin k}}{\sin k}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r85539 = 2.0;
        double r85540 = t;
        double r85541 = 3.0;
        double r85542 = pow(r85540, r85541);
        double r85543 = l;
        double r85544 = r85543 * r85543;
        double r85545 = r85542 / r85544;
        double r85546 = k;
        double r85547 = sin(r85546);
        double r85548 = r85545 * r85547;
        double r85549 = tan(r85546);
        double r85550 = r85548 * r85549;
        double r85551 = 1.0;
        double r85552 = r85546 / r85540;
        double r85553 = pow(r85552, r85539);
        double r85554 = r85551 + r85553;
        double r85555 = r85554 - r85551;
        double r85556 = r85550 * r85555;
        double r85557 = r85539 / r85556;
        return r85557;
}

↓

double f(double t, double l, double k) {
        double r85558 = l;
        double r85559 = -2.4279636162567888e+162;
        bool r85560 = r85558 <= r85559;
        double r85561 = 1.2532174228697586e+154;
        bool r85562 = r85558 <= r85561;
        double r85563 = !r85562;
        bool r85564 = r85560 || r85563;
        double r85565 = 2.0;
        double r85566 = sqrt(r85565);
        double r85567 = t;
        double r85568 = cbrt(r85567);
        double r85569 = r85568 * r85568;
        double r85570 = 3.0;
        double r85571 = pow(r85569, r85570);
        double r85572 = r85571 / r85558;
        double r85573 = r85566 / r85572;
        double r85574 = k;
        double r85575 = r85574 / r85567;
        double r85576 = pow(r85575, r85565);
        double r85577 = sin(r85574);
        double r85578 = tan(r85574);
        double r85579 = r85577 * r85578;
        double r85580 = r85576 * r85579;
        double r85581 = pow(r85568, r85570);
        double r85582 = r85581 / r85558;
        double r85583 = r85566 / r85582;
        double r85584 = r85580 / r85583;
        double r85585 = r85573 / r85584;
        double r85586 = 1.0;
        double r85587 = 1.0;
        double r85588 = pow(r85567, r85587);
        double r85589 = 2.0;
        double r85590 = r85565 / r85589;
        double r85591 = pow(r85574, r85590);
        double r85592 = r85588 * r85591;
        double r85593 = r85586 / r85592;
        double r85594 = pow(r85593, r85587);
        double r85595 = r85586 / r85591;
        double r85596 = pow(r85595, r85587);
        double r85597 = cos(r85574);
        double r85598 = pow(r85558, r85589);
        double r85599 = r85597 * r85598;
        double r85600 = r85599 / r85577;
        double r85601 = r85600 / r85577;
        double r85602 = r85596 * r85601;
        double r85603 = r85594 * r85602;
        double r85604 = r85565 * r85603;
        double r85605 = r85564 ? r85585 : r85604;
        return r85605;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if l < -2.4279636162567888e+162 or 1.2532174228697586e+154 < l

   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

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

   4. Applied add-cube-cbrt 64.0

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

   5. Applied unpow-prod-down 64.0

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

   6. Applied times-frac 49.9

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

   7. Applied add-sqr-sqrt 49.9

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

   8. Applied times-frac 49.9

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

   9. Applied associate-/l* 46.7

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

   if -2.4279636162567888e+162 < l < 1.2532174228697586e+154

   1. Initial program 45.1

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

   2. Simplified 36.8

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

   3. Taylor expanded around inf 15.0

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

   5. Applied sqr-pow 15.0

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

   6. Applied associate-*r* 12.5

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

   8. Applied *-un-lft-identity 12.5

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

   9. Applied times-frac 12.3

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

   10. Applied unpow-prod-down 12.3

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

   11. Applied associate-*l* 10.9

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

   13. Applied unpow2 10.9

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

   14. Applied associate-/r* 10.7

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.42796361625678879 \cdot 10^{162} \lor \neg \left(\ell \le 1.253217422869759 \cdot 10^{154}\right):\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{\sin k}}{\sin k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))