Toniolo and Linder, Equation (13)

Average Error: 32.9 → 28.6

Time: 45.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -4.4451804611309516 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U - U*}{\sqrt[3]{Om}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le -3.1769776407334526 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 3.336337607476201 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 3.2878796761193827 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) - \left(\frac{\frac{\ell \cdot \ell}{Om}}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1277256 = 2.0;
        double r1277257 = n;
        double r1277258 = r1277256 * r1277257;
        double r1277259 = U;
        double r1277260 = r1277258 * r1277259;
        double r1277261 = t;
        double r1277262 = l;
        double r1277263 = r1277262 * r1277262;
        double r1277264 = Om;
        double r1277265 = r1277263 / r1277264;
        double r1277266 = r1277256 * r1277265;
        double r1277267 = r1277261 - r1277266;
        double r1277268 = r1277262 / r1277264;
        double r1277269 = pow(r1277268, r1277256);
        double r1277270 = r1277257 * r1277269;
        double r1277271 = U_;
        double r1277272 = r1277259 - r1277271;
        double r1277273 = r1277270 * r1277272;
        double r1277274 = r1277267 - r1277273;
        double r1277275 = r1277260 * r1277274;
        double r1277276 = sqrt(r1277275);
        return r1277276;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1277277 = U;
        double r1277278 = -4.4451804611309516e+70;
        bool r1277279 = r1277277 <= r1277278;
        double r1277280 = t;
        double r1277281 = 2.0;
        double r1277282 = l;
        double r1277283 = Om;
        double r1277284 = r1277282 / r1277283;
        double r1277285 = r1277282 * r1277284;
        double r1277286 = r1277281 * r1277285;
        double r1277287 = r1277280 - r1277286;
        double r1277288 = n;
        double r1277289 = r1277288 * r1277282;
        double r1277290 = r1277289 * r1277284;
        double r1277291 = cbrt(r1277283);
        double r1277292 = r1277291 * r1277291;
        double r1277293 = r1277290 / r1277292;
        double r1277294 = U_;
        double r1277295 = r1277277 - r1277294;
        double r1277296 = r1277295 / r1277291;
        double r1277297 = r1277293 * r1277296;
        double r1277298 = r1277287 - r1277297;
        double r1277299 = r1277281 * r1277288;
        double r1277300 = r1277299 * r1277277;
        double r1277301 = r1277298 * r1277300;
        double r1277302 = sqrt(r1277301);
        double r1277303 = -3.1769776407334526e-289;
        bool r1277304 = r1277277 <= r1277303;
        double r1277305 = r1277282 * r1277281;
        double r1277306 = r1277288 * r1277284;
        double r1277307 = r1277284 * r1277306;
        double r1277308 = r1277307 * r1277295;
        double r1277309 = fma(r1277305, r1277284, r1277308);
        double r1277310 = r1277280 - r1277309;
        double r1277311 = r1277277 * r1277310;
        double r1277312 = r1277311 * r1277299;
        double r1277313 = sqrt(r1277312);
        double r1277314 = 3.336337607476201e-207;
        bool r1277315 = r1277277 <= r1277314;
        double r1277316 = r1277295 * r1277284;
        double r1277317 = r1277316 * r1277289;
        double r1277318 = r1277317 / r1277283;
        double r1277319 = r1277287 - r1277318;
        double r1277320 = r1277319 * r1277300;
        double r1277321 = sqrt(r1277320);
        double r1277322 = 3.2878796761193827e+96;
        bool r1277323 = r1277277 <= r1277322;
        double r1277324 = r1277282 * r1277282;
        double r1277325 = r1277324 / r1277283;
        double r1277326 = -2.0;
        double r1277327 = fma(r1277325, r1277326, r1277280);
        double r1277328 = r1277325 / r1277283;
        double r1277329 = r1277328 * r1277288;
        double r1277330 = r1277329 * r1277295;
        double r1277331 = r1277327 - r1277330;
        double r1277332 = r1277300 * r1277331;
        double r1277333 = sqrt(r1277332);
        double r1277334 = r1277323 ? r1277313 : r1277333;
        double r1277335 = r1277315 ? r1277321 : r1277334;
        double r1277336 = r1277304 ? r1277313 : r1277335;
        double r1277337 = r1277279 ? r1277302 : r1277336;
        return r1277337;
}

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 4 regimes

2. if U < -4.4451804611309516e+70

   1. Initial program 27.5

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

   3. Applied *-un-lft-identity 27.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 24.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 24.8

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

   7. Applied unpow2 24.8

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

   8. Applied associate-*r* 24.4

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

   10. Applied associate-*r/ 24.7

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

   11. Applied associate-*l/ 24.8

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

   12. Applied associate-*l/ 24.9

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

   14. Applied add-cube-cbrt 24.9

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

   15. Applied times-frac 25.7

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

   if -4.4451804611309516e+70 < U < -3.1769776407334526e-289 or 3.336337607476201e-207 < U < 3.2878796761193827e+96

   1. Initial program 33.1

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

   3. Applied *-un-lft-identity 33.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 30.3

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 30.3

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

   7. Applied unpow2 30.3

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

   8. Applied associate-*r* 29.2

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

   10. Applied associate-*l* 26.9

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

   11. Simplified 26.9

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

   if -3.1769776407334526e-289 < U < 3.336337607476201e-207

   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. Using strategy rm

   3. Applied *-un-lft-identity 42.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 41.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 41.6

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

   7. Applied unpow2 41.6

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

   8. Applied associate-*r* 40.7

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

   10. Applied associate-*r/ 41.7

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

   11. Applied associate-*l/ 41.9

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

   12. Applied associate-*l/ 41.6

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

   14. Applied associate-*l* 41.1

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

   if 3.2878796761193827e+96 < U

   1. Initial program 26.1

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

   2. Simplified 26.6

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

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -4.4451804611309516 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U - U*}{\sqrt[3]{Om}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le -3.1769776407334526 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 3.336337607476201 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 3.2878796761193827 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) - \left(\frac{\frac{\ell \cdot \ell}{Om}}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))