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 r2317498 = 2.0;
        double r2317499 = n;
        double r2317500 = r2317498 * r2317499;
        double r2317501 = U;
        double r2317502 = r2317500 * r2317501;
        double r2317503 = t;
        double r2317504 = l;
        double r2317505 = r2317504 * r2317504;
        double r2317506 = Om;
        double r2317507 = r2317505 / r2317506;
        double r2317508 = r2317498 * r2317507;
        double r2317509 = r2317503 - r2317508;
        double r2317510 = r2317504 / r2317506;
        double r2317511 = pow(r2317510, r2317498);
        double r2317512 = r2317499 * r2317511;
        double r2317513 = U_;
        double r2317514 = r2317501 - r2317513;
        double r2317515 = r2317512 * r2317514;
        double r2317516 = r2317509 - r2317515;
        double r2317517 = r2317502 * r2317516;
        double r2317518 = sqrt(r2317517);
        return r2317518;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2317519 = U;
        double r2317520 = -2.864156281776044e-277;
        bool r2317521 = r2317519 <= r2317520;
        double r2317522 = t;
        double r2317523 = l;
        double r2317524 = Om;
        double r2317525 = r2317523 / r2317524;
        double r2317526 = 2.0;
        double r2317527 = r2317523 * r2317526;
        double r2317528 = U_;
        double r2317529 = r2317519 - r2317528;
        double r2317530 = cbrt(r2317529);
        double r2317531 = r2317530 * r2317530;
        double r2317532 = n;
        double r2317533 = pow(r2317525, r2317526);
        double r2317534 = r2317532 * r2317533;
        double r2317535 = r2317531 * r2317534;
        double r2317536 = r2317530 * r2317535;
        double r2317537 = fma(r2317525, r2317527, r2317536);
        double r2317538 = r2317522 - r2317537;
        double r2317539 = r2317526 * r2317532;
        double r2317540 = r2317539 * r2317519;
        double r2317541 = r2317538 * r2317540;
        double r2317542 = sqrt(r2317541);
        double r2317543 = 2.3872108003406555e-169;
        bool r2317544 = r2317519 <= r2317543;
        double r2317545 = r2317529 * r2317534;
        double r2317546 = fma(r2317527, r2317525, r2317545);
        double r2317547 = r2317522 - r2317546;
        double r2317548 = r2317547 * r2317519;
        double r2317549 = r2317548 * r2317539;
        double r2317550 = sqrt(r2317549);
        double r2317551 = log1p(r2317550);
        double r2317552 = expm1(r2317551);
        double r2317553 = r2317544 ? r2317552 : r2317542;
        double r2317554 = r2317521 ? r2317542 : r2317553;
        return r2317554;
}

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*))))))