Toniolo and Linder, Equation (7)

Average Error: 43.8 → 9.8

Time: 17.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r38612 = 2.0;
        double r38613 = sqrt(r38612);
        double r38614 = t;
        double r38615 = r38613 * r38614;
        double r38616 = x;
        double r38617 = 1.0;
        double r38618 = r38616 + r38617;
        double r38619 = r38616 - r38617;
        double r38620 = r38618 / r38619;
        double r38621 = l;
        double r38622 = r38621 * r38621;
        double r38623 = r38614 * r38614;
        double r38624 = r38612 * r38623;
        double r38625 = r38622 + r38624;
        double r38626 = r38620 * r38625;
        double r38627 = r38626 - r38622;
        double r38628 = sqrt(r38627);
        double r38629 = r38615 / r38628;
        return r38629;
}

↓

double f(double x, double l, double t) {
        double r38630 = t;
        double r38631 = -8.171560713962325e+60;
        bool r38632 = r38630 <= r38631;
        double r38633 = 2.0;
        double r38634 = sqrt(r38633);
        double r38635 = r38634 * r38630;
        double r38636 = 3.0;
        double r38637 = pow(r38634, r38636);
        double r38638 = x;
        double r38639 = 2.0;
        double r38640 = pow(r38638, r38639);
        double r38641 = r38637 * r38640;
        double r38642 = r38630 / r38641;
        double r38643 = r38634 * r38640;
        double r38644 = r38630 / r38643;
        double r38645 = r38642 - r38644;
        double r38646 = r38633 * r38645;
        double r38647 = r38634 * r38638;
        double r38648 = r38630 / r38647;
        double r38649 = r38630 * r38634;
        double r38650 = fma(r38633, r38648, r38649);
        double r38651 = r38646 - r38650;
        double r38652 = r38635 / r38651;
        double r38653 = -1.1776830540197237e-181;
        bool r38654 = r38630 <= r38653;
        double r38655 = r38630 * r38630;
        double r38656 = l;
        double r38657 = r38638 / r38656;
        double r38658 = r38656 / r38657;
        double r38659 = 4.0;
        double r38660 = pow(r38630, r38639);
        double r38661 = r38660 / r38638;
        double r38662 = r38659 * r38661;
        double r38663 = fma(r38633, r38658, r38662);
        double r38664 = fma(r38633, r38655, r38663);
        double r38665 = sqrt(r38664);
        double r38666 = r38649 / r38665;
        double r38667 = -5.07338813022938e-287;
        bool r38668 = r38630 <= r38667;
        double r38669 = 6.769189022626266e+24;
        bool r38670 = r38630 <= r38669;
        double r38671 = sqrt(r38634);
        double r38672 = r38671 * r38630;
        double r38673 = r38671 * r38672;
        double r38674 = r38673 / r38665;
        double r38675 = r38633 * r38648;
        double r38676 = fma(r38630, r38634, r38675);
        double r38677 = r38635 / r38676;
        double r38678 = r38670 ? r38674 : r38677;
        double r38679 = r38668 ? r38652 : r38678;
        double r38680 = r38654 ? r38666 : r38679;
        double r38681 = r38632 ? r38652 : r38680;
        return r38681;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -8.171560713962325e+60 or -1.1776830540197237e-181 < t < -5.07338813022938e-287

   1. Initial program 51.2

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around -inf 11.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)\right)}}\]

   3. Simplified 11.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}}\]

   if -8.171560713962325e+60 < t < -1.1776830540197237e-181

   1. Initial program 31.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 11.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 11.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{{\ell}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\]
   4. Using strategy rm

   5. Applied unpow2 11.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   6. Applied associate-/l* 7.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{x}{\ell}}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 7.0

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   9. Applied associate-*l* 7.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   10. Taylor expanded around 0 7.0

       \[\leadsto \frac{\color{blue}{t \cdot \sqrt{2}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   if -5.07338813022938e-287 < t < 6.769189022626266e+24

   1. Initial program 45.2

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 19.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 19.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{{\ell}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\]
   4. Using strategy rm

   5. Applied unpow2 19.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   6. Applied associate-/l* 15.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{x}{\ell}}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 15.8

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   9. Applied sqrt-prod 15.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   10. Applied associate-*l* 15.8

       \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\]

   if 6.769189022626266e+24 < t

   1. Initial program 41.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 39.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 39.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{{\ell}^{2}}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}}\]

   4. Taylor expanded around inf 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

   5. Simplified 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))