Toniolo and Linder, Equation (13)

Average Error: 34.8 → 29.8

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 8.120471958061494211765127711129383515551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r205196 = 2.0;
        double r205197 = n;
        double r205198 = r205196 * r205197;
        double r205199 = U;
        double r205200 = r205198 * r205199;
        double r205201 = t;
        double r205202 = l;
        double r205203 = r205202 * r205202;
        double r205204 = Om;
        double r205205 = r205203 / r205204;
        double r205206 = r205196 * r205205;
        double r205207 = r205201 - r205206;
        double r205208 = r205202 / r205204;
        double r205209 = pow(r205208, r205196);
        double r205210 = r205197 * r205209;
        double r205211 = U_;
        double r205212 = r205199 - r205211;
        double r205213 = r205210 * r205212;
        double r205214 = r205207 - r205213;
        double r205215 = r205200 * r205214;
        double r205216 = sqrt(r205215);
        return r205216;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r205217 = t;
        double r205218 = 8.120471958061494e+92;
        bool r205219 = r205217 <= r205218;
        double r205220 = 2.0;
        double r205221 = n;
        double r205222 = r205220 * r205221;
        double r205223 = U;
        double r205224 = r205222 * r205223;
        double r205225 = l;
        double r205226 = Om;
        double r205227 = r205226 / r205225;
        double r205228 = r205225 / r205227;
        double r205229 = r205220 * r205228;
        double r205230 = r205217 - r205229;
        double r205231 = cbrt(r205225);
        double r205232 = r205231 * r205231;
        double r205233 = cbrt(r205226);
        double r205234 = r205233 * r205233;
        double r205235 = r205232 / r205234;
        double r205236 = 2.0;
        double r205237 = r205220 / r205236;
        double r205238 = pow(r205235, r205237);
        double r205239 = r205221 * r205238;
        double r205240 = r205231 / r205233;
        double r205241 = pow(r205240, r205237);
        double r205242 = r205239 * r205241;
        double r205243 = r205225 / r205226;
        double r205244 = pow(r205243, r205237);
        double r205245 = r205242 * r205244;
        double r205246 = U_;
        double r205247 = r205223 - r205246;
        double r205248 = r205245 * r205247;
        double r205249 = r205230 - r205248;
        double r205250 = r205224 * r205249;
        double r205251 = sqrt(r205250);
        double r205252 = sqrt(r205224);
        double r205253 = r205236 * r205237;
        double r205254 = pow(r205243, r205253);
        double r205255 = r205221 * r205254;
        double r205256 = r205247 * r205255;
        double r205257 = fma(r205220, r205228, r205256);
        double r205258 = r205217 - r205257;
        double r205259 = sqrt(r205258);
        double r205260 = r205252 * r205259;
        double r205261 = r205219 ? r205251 : r205260;
        return r205261;
}

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 t < 8.120471958061494e+92

   1. Initial program 34.2

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

   3. Applied associate-/l* 31.4

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

   5. Applied sqr-pow 31.4

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

   6. Applied associate-*r* 30.5

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

   8. Applied add-cube-cbrt 30.5

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

   9. Applied add-cube-cbrt 30.6

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

   10. Applied times-frac 30.6

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

   11. Applied unpow-prod-down 30.6

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

   12. Applied associate-*r* 30.6

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

   if 8.120471958061494e+92 < t

   1. Initial program 37.5

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

   3. Applied associate-/l* 35.0

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

   5. Applied sqr-pow 35.0

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

   6. Applied associate-*r* 34.4

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

   8. Applied sqrt-prod 25.9

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

   9. Simplified 26.4

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

4. Final simplification 29.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.120471958061494211765127711129383515551 \cdot 10^{92}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))