Toniolo and Linder, Equation (13)

Average Error: 33.3 → 29.8

Time: 2.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2.435100367453427 \cdot 10^{-204}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 3.00502928168256 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot -2 - n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;t \le 4.3432694513312935 \cdot 10^{+85}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16756071 = 2.0;
        double r16756072 = n;
        double r16756073 = r16756071 * r16756072;
        double r16756074 = U;
        double r16756075 = r16756073 * r16756074;
        double r16756076 = t;
        double r16756077 = l;
        double r16756078 = r16756077 * r16756077;
        double r16756079 = Om;
        double r16756080 = r16756078 / r16756079;
        double r16756081 = r16756071 * r16756080;
        double r16756082 = r16756076 - r16756081;
        double r16756083 = r16756077 / r16756079;
        double r16756084 = pow(r16756083, r16756071);
        double r16756085 = r16756072 * r16756084;
        double r16756086 = U_;
        double r16756087 = r16756074 - r16756086;
        double r16756088 = r16756085 * r16756087;
        double r16756089 = r16756082 - r16756088;
        double r16756090 = r16756075 * r16756089;
        double r16756091 = sqrt(r16756090);
        return r16756091;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16756092 = t;
        double r16756093 = 2.435100367453427e-204;
        bool r16756094 = r16756092 <= r16756093;
        double r16756095 = n;
        double r16756096 = 2.0;
        double r16756097 = l;
        double r16756098 = U;
        double r16756099 = r16756097 * r16756098;
        double r16756100 = r16756096 * r16756099;
        double r16756101 = -2.0;
        double r16756102 = r16756097 * r16756101;
        double r16756103 = Om;
        double r16756104 = r16756097 / r16756103;
        double r16756105 = r16756104 * r16756095;
        double r16756106 = U_;
        double r16756107 = r16756098 - r16756106;
        double r16756108 = r16756105 * r16756107;
        double r16756109 = r16756102 - r16756108;
        double r16756110 = r16756100 * r16756109;
        double r16756111 = r16756110 / r16756103;
        double r16756112 = r16756095 * r16756092;
        double r16756113 = r16756112 * r16756096;
        double r16756114 = r16756098 * r16756113;
        double r16756115 = fma(r16756095, r16756111, r16756114);
        double r16756116 = 0.5;
        double r16756117 = pow(r16756115, r16756116);
        double r16756118 = 3.00502928168256e-47;
        bool r16756119 = r16756092 <= r16756118;
        double r16756120 = r16756095 * r16756096;
        double r16756121 = r16756098 * r16756120;
        double r16756122 = r16756092 * r16756121;
        double r16756123 = r16756096 * r16756098;
        double r16756124 = r16756095 * r16756123;
        double r16756125 = r16756107 * r16756104;
        double r16756126 = r16756095 * r16756125;
        double r16756127 = r16756102 - r16756126;
        double r16756128 = r16756127 * r16756104;
        double r16756129 = r16756124 * r16756128;
        double r16756130 = r16756122 + r16756129;
        double r16756131 = sqrt(r16756130);
        double r16756132 = 4.3432694513312935e+85;
        bool r16756133 = r16756092 <= r16756132;
        double r16756134 = sqrt(r16756121);
        double r16756135 = r16756097 * r16756097;
        double r16756136 = r16756135 / r16756103;
        double r16756137 = r16756136 * r16756096;
        double r16756138 = r16756092 - r16756137;
        double r16756139 = pow(r16756104, r16756096);
        double r16756140 = r16756139 * r16756095;
        double r16756141 = r16756140 * r16756107;
        double r16756142 = r16756138 - r16756141;
        double r16756143 = sqrt(r16756142);
        double r16756144 = r16756134 * r16756143;
        double r16756145 = r16756133 ? r16756117 : r16756144;
        double r16756146 = r16756119 ? r16756131 : r16756145;
        double r16756147 = r16756094 ? r16756117 : r16756146;
        return r16756147;
}

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 t < 2.435100367453427e-204 or 3.00502928168256e-47 < t < 4.3432694513312935e+85

   1. Initial program 33.2

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

   3. Applied sub-neg 33.2

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

   4. Applied associate--l+ 33.2

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

   5. Applied distribute-rgt-in 33.2

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

   6. Simplified 30.8

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

   8. Applied associate-*r* 30.8

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

   10. Applied associate-*r/ 31.2

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

   11. Applied associate-*l/ 30.8

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

   12. Applied associate-*l/ 31.1

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

   13. Applied associate-*r/ 31.1

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

   14. Applied associate-*l/ 32.8

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

   15. Applied sub-div 32.8

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

   16. Applied associate-*l/ 33.2

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

   17. Simplified 32.2

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

   19. Applied pow1 32.2

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

   20. Applied sqrt-pow1 32.2

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

   21. Simplified 30.9

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

   if 2.435100367453427e-204 < t < 3.00502928168256e-47

   1. Initial program 31.7

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

   3. Applied sub-neg 31.7

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

   4. Applied associate--l+ 31.7

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

   5. Applied distribute-rgt-in 31.7

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

   6. Simplified 29.1

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

   8. Applied associate-*r* 28.7

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

   10. Applied associate-*l* 29.1

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

   11. Simplified 28.4

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

   if 4.3432694513312935e+85 < t

   1. Initial program 34.6

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

   3. Applied sqrt-prod 27.0

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

4. Final simplification 29.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.435100367453427 \cdot 10^{-204}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 3.00502928168256 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot -2 - n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;t \le 4.3432694513312935 \cdot 10^{+85}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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*))))))