Toniolo and Linder, Equation (13)

Average Error: 33.3 → 25.3

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.107259989070679 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot 2} \cdot \sqrt{t - \left(2 \cdot \ell - \left(U - U*\right) \cdot \left(-n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r8616019 = 2.0;
        double r8616020 = n;
        double r8616021 = r8616019 * r8616020;
        double r8616022 = U;
        double r8616023 = r8616021 * r8616022;
        double r8616024 = t;
        double r8616025 = l;
        double r8616026 = r8616025 * r8616025;
        double r8616027 = Om;
        double r8616028 = r8616026 / r8616027;
        double r8616029 = r8616019 * r8616028;
        double r8616030 = r8616024 - r8616029;
        double r8616031 = r8616025 / r8616027;
        double r8616032 = pow(r8616031, r8616019);
        double r8616033 = r8616020 * r8616032;
        double r8616034 = U_;
        double r8616035 = r8616022 - r8616034;
        double r8616036 = r8616033 * r8616035;
        double r8616037 = r8616030 - r8616036;
        double r8616038 = r8616023 * r8616037;
        double r8616039 = sqrt(r8616038);
        return r8616039;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r8616040 = t;
        double r8616041 = 3.107259989070679e+84;
        bool r8616042 = r8616040 <= r8616041;
        double r8616043 = 2.0;
        double r8616044 = n;
        double r8616045 = l;
        double r8616046 = Om;
        double r8616047 = r8616045 / r8616046;
        double r8616048 = r8616044 * r8616047;
        double r8616049 = U;
        double r8616050 = r8616048 * r8616049;
        double r8616051 = -2.0;
        double r8616052 = r8616045 * r8616051;
        double r8616053 = U_;
        double r8616054 = r8616049 - r8616053;
        double r8616055 = r8616048 * r8616054;
        double r8616056 = r8616052 - r8616055;
        double r8616057 = r8616050 * r8616056;
        double r8616058 = r8616043 * r8616057;
        double r8616059 = r8616043 * r8616044;
        double r8616060 = r8616059 * r8616049;
        double r8616061 = r8616060 * r8616040;
        double r8616062 = r8616058 + r8616061;
        double r8616063 = sqrt(r8616062);
        double r8616064 = sqrt(r8616063);
        double r8616065 = r8616064 * r8616064;
        double r8616066 = r8616049 * r8616044;
        double r8616067 = r8616066 * r8616043;
        double r8616068 = sqrt(r8616067);
        double r8616069 = r8616043 * r8616045;
        double r8616070 = -r8616048;
        double r8616071 = r8616054 * r8616070;
        double r8616072 = r8616069 - r8616071;
        double r8616073 = r8616072 * r8616047;
        double r8616074 = r8616040 - r8616073;
        double r8616075 = sqrt(r8616074);
        double r8616076 = r8616068 * r8616075;
        double r8616077 = r8616042 ? r8616065 : r8616076;
        return r8616077;
}

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 < 3.107259989070679e+84

   1. Initial program 33.0

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

   3. Applied *-un-lft-identity 33.0

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

   4. Applied associate-*r* 33.0

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

   5. Simplified 29.3

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

   7. Applied sub-neg 29.3

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

   8. Applied distribute-rgt-in 29.3

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

   9. Simplified 25.3

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

   11. Applied add-sqr-sqrt 25.3

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

   12. Applied sqrt-prod 25.5

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

   if 3.107259989070679e+84 < t

   1. Initial program 34.5

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

   3. Applied *-un-lft-identity 34.5

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

   4. Applied associate-*r* 34.5

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

   5. Simplified 32.2

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

   7. Applied sqrt-prod 24.6

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

   8. Simplified 24.6

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

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.107259989070679 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \cdot \sqrt{\sqrt{2 \cdot \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot 2} \cdot \sqrt{t - \left(2 \cdot \ell - \left(U - U*\right) \cdot \left(-n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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*))))))