Toniolo and Linder, Equation (13)

Average Error: 34.6 → 30.7

Time: 1.3m

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;U \le -2.8641562817760440835220634548776257285 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 2.387210800340655478652547601628847274612 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3056729 = 2.0;
        double r3056730 = n;
        double r3056731 = r3056729 * r3056730;
        double r3056732 = U;
        double r3056733 = r3056731 * r3056732;
        double r3056734 = t;
        double r3056735 = l;
        double r3056736 = r3056735 * r3056735;
        double r3056737 = Om;
        double r3056738 = r3056736 / r3056737;
        double r3056739 = r3056729 * r3056738;
        double r3056740 = r3056734 - r3056739;
        double r3056741 = r3056735 / r3056737;
        double r3056742 = pow(r3056741, r3056729);
        double r3056743 = r3056730 * r3056742;
        double r3056744 = U_;
        double r3056745 = r3056732 - r3056744;
        double r3056746 = r3056743 * r3056745;
        double r3056747 = r3056740 - r3056746;
        double r3056748 = r3056733 * r3056747;
        double r3056749 = sqrt(r3056748);
        return r3056749;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3056750 = U;
        double r3056751 = -2.864156281776044e-277;
        bool r3056752 = r3056750 <= r3056751;
        double r3056753 = t;
        double r3056754 = l;
        double r3056755 = Om;
        double r3056756 = r3056754 / r3056755;
        double r3056757 = 2.0;
        double r3056758 = r3056754 * r3056757;
        double r3056759 = U_;
        double r3056760 = r3056750 - r3056759;
        double r3056761 = cbrt(r3056760);
        double r3056762 = r3056761 * r3056761;
        double r3056763 = n;
        double r3056764 = pow(r3056756, r3056757);
        double r3056765 = r3056763 * r3056764;
        double r3056766 = r3056762 * r3056765;
        double r3056767 = r3056761 * r3056766;
        double r3056768 = fma(r3056756, r3056758, r3056767);
        double r3056769 = r3056753 - r3056768;
        double r3056770 = r3056757 * r3056763;
        double r3056771 = r3056770 * r3056750;
        double r3056772 = r3056769 * r3056771;
        double r3056773 = sqrt(r3056772);
        double r3056774 = 2.3872108003406555e-169;
        bool r3056775 = r3056750 <= r3056774;
        double r3056776 = r3056760 * r3056765;
        double r3056777 = fma(r3056758, r3056756, r3056776);
        double r3056778 = r3056753 - r3056777;
        double r3056779 = r3056778 * r3056750;
        double r3056780 = r3056779 * r3056770;
        double r3056781 = sqrt(r3056780);
        double r3056782 = log1p(r3056781);
        double r3056783 = expm1(r3056782);
        double r3056784 = r3056775 ? r3056783 : r3056773;
        double r3056785 = r3056752 ? r3056773 : r3056784;
        return r3056785;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 2 regimes

2. if U < -2.864156281776044e-277 or 2.3872108003406555e-169 < U

   1. Initial program 32.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 29.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 29.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \sqrt[3]{U - U*}\right)}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\]

   5. Applied associate-*r* 29.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}}\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\]

   if -2.864156281776044e-277 < U < 2.3872108003406555e-169

   1. Initial program 42.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 40.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 40.9

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}}\]
   5. Using strategy rm

   6. Applied expm1-log1p-u 41.4

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}}\right)\right)}\]

   7. Simplified 35.6

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)\right)}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.8641562817760440835220634548776257285 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 2.387210800340655478652547601628847274612 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot 2, \sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))