Toniolo and Linder, Equation (13)

Average Error: 32.9 → 26.7

Time: 39.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.6344908840558971 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.316465484859485 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2211186 = 2.0;
        double r2211187 = n;
        double r2211188 = r2211186 * r2211187;
        double r2211189 = U;
        double r2211190 = r2211188 * r2211189;
        double r2211191 = t;
        double r2211192 = l;
        double r2211193 = r2211192 * r2211192;
        double r2211194 = Om;
        double r2211195 = r2211193 / r2211194;
        double r2211196 = r2211186 * r2211195;
        double r2211197 = r2211191 - r2211196;
        double r2211198 = r2211192 / r2211194;
        double r2211199 = pow(r2211198, r2211186);
        double r2211200 = r2211187 * r2211199;
        double r2211201 = U_;
        double r2211202 = r2211189 - r2211201;
        double r2211203 = r2211200 * r2211202;
        double r2211204 = r2211197 - r2211203;
        double r2211205 = r2211190 * r2211204;
        double r2211206 = sqrt(r2211205);
        return r2211206;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2211207 = 2.0;
        double r2211208 = n;
        double r2211209 = r2211207 * r2211208;
        double r2211210 = U;
        double r2211211 = r2211209 * r2211210;
        double r2211212 = t;
        double r2211213 = l;
        double r2211214 = r2211213 * r2211213;
        double r2211215 = Om;
        double r2211216 = r2211214 / r2211215;
        double r2211217 = r2211216 * r2211207;
        double r2211218 = r2211212 - r2211217;
        double r2211219 = r2211213 / r2211215;
        double r2211220 = pow(r2211219, r2211207);
        double r2211221 = r2211208 * r2211220;
        double r2211222 = U_;
        double r2211223 = r2211210 - r2211222;
        double r2211224 = r2211221 * r2211223;
        double r2211225 = r2211218 - r2211224;
        double r2211226 = r2211211 * r2211225;
        double r2211227 = sqrt(r2211226);
        double r2211228 = 1.6344908840558971e-152;
        bool r2211229 = r2211227 <= r2211228;
        double r2211230 = r2211215 / r2211213;
        double r2211231 = r2211208 / r2211230;
        double r2211232 = r2211231 * r2211223;
        double r2211233 = r2211207 * r2211213;
        double r2211234 = r2211232 + r2211233;
        double r2211235 = r2211234 / r2211230;
        double r2211236 = r2211212 - r2211235;
        double r2211237 = r2211210 * r2211236;
        double r2211238 = sqrt(r2211237);
        double r2211239 = sqrt(r2211209);
        double r2211240 = r2211238 * r2211239;
        double r2211241 = 5.316465484859485e+126;
        bool r2211242 = r2211227 <= r2211241;
        double r2211243 = r2211209 * r2211237;
        double r2211244 = sqrt(r2211243);
        double r2211245 = r2211242 ? r2211227 : r2211244;
        double r2211246 = r2211229 ? r2211240 : r2211245;
        return r2211246;
}

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 3 regimes

2. if (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.6344908840558971e-152

   1. Initial program 53.7

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

   2. Simplified 37.6

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

   4. Applied associate-*r* 53.5

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

   6. Applied pow1 53.5

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

   7. Applied pow1 53.5

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

   8. Applied pow-prod-down 53.5

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

   9. Simplified 38.3

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

   11. Applied unpow-prod-down 38.3

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

   12. Applied sqrt-prod 38.0

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

   13. Simplified 37.5

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

   14. Simplified 37.5

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

   if 1.6344908840558971e-152 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 5.316465484859485e+126

   1. Initial program 1.6

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

   if 5.316465484859485e+126 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 56.3

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

   2. Simplified 48.8

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

   4. Applied associate-*r* 48.9

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

   6. Applied pow1 48.9

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

   7. Applied pow1 48.9

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

   8. Applied pow-prod-down 48.9

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

   9. Simplified 47.6

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

   11. Applied *-un-lft-identity 47.6

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

   12. Applied associate-*r* 47.6

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

   13. Simplified 47.5

       \[\leadsto \sqrt{{\left(\left(\color{blue}{\left(t - \frac{2 \cdot \ell + \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)} \cdot U\right) \cdot \left(n \cdot 2\right)\right)}^{1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.6344908840558971 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.316465484859485 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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*))))))