Toniolo and Linder, Equation (7)

Average Error: 42.9 → 10.2

Time: 17.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2404699594045954599540100621504872448:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.492099203484521869436081722227008598355 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\left|\ell\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -7.231141217156239276359457821248495233406 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.53909661729405029639481098041358476011 \cdot 10^{-299} \lor \neg \left(t \le 1.464783658109499293496367809888962941264 \cdot 10^{-233} \lor \neg \left(t \le 7.020103372014630694165473293132646183048 \cdot 10^{147}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r26632 = 2.0;
        double r26633 = sqrt(r26632);
        double r26634 = t;
        double r26635 = r26633 * r26634;
        double r26636 = x;
        double r26637 = 1.0;
        double r26638 = r26636 + r26637;
        double r26639 = r26636 - r26637;
        double r26640 = r26638 / r26639;
        double r26641 = l;
        double r26642 = r26641 * r26641;
        double r26643 = r26634 * r26634;
        double r26644 = r26632 * r26643;
        double r26645 = r26642 + r26644;
        double r26646 = r26640 * r26645;
        double r26647 = r26646 - r26642;
        double r26648 = sqrt(r26647);
        double r26649 = r26635 / r26648;
        return r26649;
}

↓

double f(double x, double l, double t) {
        double r26650 = t;
        double r26651 = -2.4046995940459546e+36;
        bool r26652 = r26650 <= r26651;
        double r26653 = 2.0;
        double r26654 = sqrt(r26653);
        double r26655 = r26654 * r26650;
        double r26656 = 3.0;
        double r26657 = pow(r26654, r26656);
        double r26658 = x;
        double r26659 = 2.0;
        double r26660 = pow(r26658, r26659);
        double r26661 = r26657 * r26660;
        double r26662 = r26650 / r26661;
        double r26663 = r26654 * r26658;
        double r26664 = r26650 / r26663;
        double r26665 = r26662 - r26664;
        double r26666 = r26653 * r26665;
        double r26667 = r26650 * r26654;
        double r26668 = r26666 - r26667;
        double r26669 = r26655 / r26668;
        double r26670 = -6.492099203484522e-157;
        bool r26671 = r26650 <= r26670;
        double r26672 = l;
        double r26673 = fabs(r26672);
        double r26674 = cbrt(r26658);
        double r26675 = r26674 * r26674;
        double r26676 = r26673 / r26675;
        double r26677 = r26673 / r26674;
        double r26678 = r26676 * r26677;
        double r26679 = fma(r26650, r26650, r26678);
        double r26680 = 4.0;
        double r26681 = pow(r26650, r26659);
        double r26682 = r26681 / r26658;
        double r26683 = r26680 * r26682;
        double r26684 = fma(r26653, r26679, r26683);
        double r26685 = sqrt(r26684);
        double r26686 = r26655 / r26685;
        double r26687 = -7.231141217156239e-225;
        bool r26688 = r26650 <= r26687;
        double r26689 = 1.5390966172940503e-299;
        bool r26690 = r26650 <= r26689;
        double r26691 = 1.4647836581094993e-233;
        bool r26692 = r26650 <= r26691;
        double r26693 = 7.02010337201463e+147;
        bool r26694 = r26650 <= r26693;
        double r26695 = !r26694;
        bool r26696 = r26692 || r26695;
        double r26697 = !r26696;
        bool r26698 = r26690 || r26697;
        double r26699 = r26673 / r26658;
        double r26700 = r26673 * r26699;
        double r26701 = fma(r26650, r26650, r26700);
        double r26702 = fma(r26653, r26701, r26683);
        double r26703 = sqrt(r26702);
        double r26704 = r26655 / r26703;
        double r26705 = r26653 * r26664;
        double r26706 = fma(r26650, r26654, r26705);
        double r26707 = r26655 / r26706;
        double r26708 = r26698 ? r26704 : r26707;
        double r26709 = r26688 ? r26669 : r26708;
        double r26710 = r26671 ? r26686 : r26709;
        double r26711 = r26652 ? r26669 : r26710;
        return r26711;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -2.4046995940459546e+36 or -6.492099203484522e-157 < t < -7.231141217156239e-225

   1. Initial program 46.4

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

   2. Simplified 46.4

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 40.8

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

   4. Simplified 40.8

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

   6. Applied *-un-lft-identity 40.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   7. Applied add-sqr-sqrt 40.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{\ell}^{2}}}}{1 \cdot x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   8. Applied times-frac 40.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \color{blue}{\frac{\sqrt{{\ell}^{2}}}{1} \cdot \frac{\sqrt{{\ell}^{2}}}{x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   9. Simplified 40.8

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

   10. Simplified 38.1

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

   11. Taylor expanded around -inf 9.0

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

   12. Simplified 9.0

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

   if -2.4046995940459546e+36 < t < -6.492099203484522e-157

   1. Initial program 29.4

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

   2. Simplified 29.4

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 9.5

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

   4. Simplified 9.5

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

   6. Applied add-cube-cbrt 9.5

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

   7. Applied add-sqr-sqrt 9.5

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

   8. Applied times-frac 9.6

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

   9. Simplified 9.6

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

   10. Simplified 5.0

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

   if -7.231141217156239e-225 < t < 1.5390966172940503e-299 or 1.4647836581094993e-233 < t < 7.02010337201463e+147

   1. Initial program 35.8

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

   2. Simplified 35.8

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 17.6

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

   4. Simplified 17.6

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

   6. Applied *-un-lft-identity 17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   7. Applied add-sqr-sqrt 17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{\ell}^{2}}}}{1 \cdot x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   8. Applied times-frac 17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \color{blue}{\frac{\sqrt{{\ell}^{2}}}{1} \cdot \frac{\sqrt{{\ell}^{2}}}{x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\]

   9. Simplified 17.6

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

   10. Simplified 13.4

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

   if 1.5390966172940503e-299 < t < 1.4647836581094993e-233 or 7.02010337201463e+147 < t

   1. Initial program 60.9

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

   2. Simplified 60.9

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 54.5

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

   4. Simplified 54.5

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

   5. Taylor expanded around inf 10.5

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

   6. Simplified 10.5

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2404699594045954599540100621504872448:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.492099203484521869436081722227008598355 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\left|\ell\right|}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -7.231141217156239276359457821248495233406 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.53909661729405029639481098041358476011 \cdot 10^{-299} \lor \neg \left(t \le 1.464783658109499293496367809888962941264 \cdot 10^{-233} \lor \neg \left(t \le 7.020103372014630694165473293132646183048 \cdot 10^{147}\right)\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))))