Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.3

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -7122680.873158953:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le -1.8663328166415404 \cdot 10^{-262}:\\ \;\;\;\;{\left(\left(U \cdot \left(t - (\left(\ell \cdot 2\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)\right) \cdot \left(2 \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 2.2093515262227777 \cdot 10^{-301}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot {\left(\left(2 \cdot n\right) \cdot \left(t - (\left(\ell \cdot 2\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r15011039 = 2.0;
        double r15011040 = n;
        double r15011041 = r15011039 * r15011040;
        double r15011042 = U;
        double r15011043 = r15011041 * r15011042;
        double r15011044 = t;
        double r15011045 = l;
        double r15011046 = r15011045 * r15011045;
        double r15011047 = Om;
        double r15011048 = r15011046 / r15011047;
        double r15011049 = r15011039 * r15011048;
        double r15011050 = r15011044 - r15011049;
        double r15011051 = r15011045 / r15011047;
        double r15011052 = pow(r15011051, r15011039);
        double r15011053 = r15011040 * r15011052;
        double r15011054 = U_;
        double r15011055 = r15011042 - r15011054;
        double r15011056 = r15011053 * r15011055;
        double r15011057 = r15011050 - r15011056;
        double r15011058 = r15011043 * r15011057;
        double r15011059 = sqrt(r15011058);
        return r15011059;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r15011060 = U;
        double r15011061 = -7122680.873158953;
        bool r15011062 = r15011060 <= r15011061;
        double r15011063 = t;
        double r15011064 = l;
        double r15011065 = 2.0;
        double r15011066 = r15011064 * r15011065;
        double r15011067 = Om;
        double r15011068 = r15011064 / r15011067;
        double r15011069 = U_;
        double r15011070 = r15011060 - r15011069;
        double r15011071 = r15011068 * r15011070;
        double r15011072 = n;
        double r15011073 = r15011071 * r15011072;
        double r15011074 = r15011066 + r15011073;
        double r15011075 = r15011074 * r15011068;
        double r15011076 = r15011063 - r15011075;
        double r15011077 = r15011065 * r15011072;
        double r15011078 = r15011060 * r15011077;
        double r15011079 = r15011076 * r15011078;
        double r15011080 = 0.5;
        double r15011081 = pow(r15011079, r15011080);
        double r15011082 = -1.8663328166415404e-262;
        bool r15011083 = r15011060 <= r15011082;
        double r15011084 = r15011072 * r15011068;
        double r15011085 = r15011068 * r15011084;
        double r15011086 = r15011070 * r15011085;
        double r15011087 = fma(r15011066, r15011068, r15011086);
        double r15011088 = r15011063 - r15011087;
        double r15011089 = r15011060 * r15011088;
        double r15011090 = r15011089 * r15011077;
        double r15011091 = pow(r15011090, r15011080);
        double r15011092 = 2.2093515262227777e-301;
        bool r15011093 = r15011060 <= r15011092;
        double r15011094 = sqrt(r15011060);
        double r15011095 = r15011077 * r15011088;
        double r15011096 = pow(r15011095, r15011080);
        double r15011097 = r15011094 * r15011096;
        double r15011098 = r15011093 ? r15011081 : r15011097;
        double r15011099 = r15011083 ? r15011091 : r15011098;
        double r15011100 = r15011062 ? r15011081 : r15011099;
        return r15011100;
}

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 U < -7122680.873158953 or -1.8663328166415404e-262 < U < 2.2093515262227777e-301

   1. Initial program 32.3

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

   3. Applied *-un-lft-identity 32.3

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

   4. Applied times-frac 29.3

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

   5. Simplified 29.3

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

   7. Applied pow1 29.3

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

   8. Applied sqrt-pow1 29.3

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

   9. Simplified 30.1

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

   11. Applied pow1 30.1

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

   12. Applied pow1 30.1

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

   13. Applied pow-prod-down 30.1

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

   14. Applied pow1 30.1

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

   15. Applied pow-prod-down 30.1

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

   16. Applied pow-pow 30.1

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

   17. Simplified 29.2

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

   if -7122680.873158953 < U < -1.8663328166415404e-262

   1. Initial program 34.7

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

   3. Applied *-un-lft-identity 34.7

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

   4. Applied times-frac 31.8

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

   5. Simplified 31.8

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

   7. Applied pow1 31.8

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

   8. Applied sqrt-pow1 31.8

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

   9. Simplified 30.4

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

   11. Applied associate-*r* 27.1

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

   if 2.2093515262227777e-301 < 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. Using strategy rm

   3. Applied *-un-lft-identity 32.8

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

   4. Applied times-frac 30.1

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

   5. Simplified 30.1

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

   7. Applied pow1 30.1

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

   8. Applied sqrt-pow1 30.1

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

   9. Simplified 29.1

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

   11. Applied unpow-prod-down 22.4

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

   12. Simplified 22.4

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

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7122680.873158953:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le -1.8663328166415404 \cdot 10^{-262}:\\ \;\;\;\;{\left(\left(U \cdot \left(t - (\left(\ell \cdot 2\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)\right) \cdot \left(2 \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 2.2093515262227777 \cdot 10^{-301}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot {\left(\left(2 \cdot n\right) \cdot \left(t - (\left(\ell \cdot 2\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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*))))))