Toniolo and Linder, Equation (13)

Average Error: 33.4 → 25.3

Time: 41.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0865908327746399 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3438079 = 2.0;
        double r3438080 = n;
        double r3438081 = r3438079 * r3438080;
        double r3438082 = U;
        double r3438083 = r3438081 * r3438082;
        double r3438084 = t;
        double r3438085 = l;
        double r3438086 = r3438085 * r3438085;
        double r3438087 = Om;
        double r3438088 = r3438086 / r3438087;
        double r3438089 = r3438079 * r3438088;
        double r3438090 = r3438084 - r3438089;
        double r3438091 = r3438085 / r3438087;
        double r3438092 = pow(r3438091, r3438079);
        double r3438093 = r3438080 * r3438092;
        double r3438094 = U_;
        double r3438095 = r3438082 - r3438094;
        double r3438096 = r3438093 * r3438095;
        double r3438097 = r3438090 - r3438096;
        double r3438098 = r3438083 * r3438097;
        double r3438099 = sqrt(r3438098);
        return r3438099;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3438100 = 2.0;
        double r3438101 = n;
        double r3438102 = r3438100 * r3438101;
        double r3438103 = U;
        double r3438104 = r3438102 * r3438103;
        double r3438105 = t;
        double r3438106 = l;
        double r3438107 = r3438106 * r3438106;
        double r3438108 = Om;
        double r3438109 = r3438107 / r3438108;
        double r3438110 = r3438109 * r3438100;
        double r3438111 = r3438105 - r3438110;
        double r3438112 = r3438106 / r3438108;
        double r3438113 = pow(r3438112, r3438100);
        double r3438114 = r3438101 * r3438113;
        double r3438115 = U_;
        double r3438116 = r3438103 - r3438115;
        double r3438117 = r3438114 * r3438116;
        double r3438118 = r3438111 - r3438117;
        double r3438119 = r3438104 * r3438118;
        double r3438120 = sqrt(r3438119);
        double r3438121 = 0.0;
        bool r3438122 = r3438120 <= r3438121;
        double r3438123 = r3438105 * r3438101;
        double r3438124 = r3438100 * r3438106;
        double r3438125 = r3438115 - r3438103;
        double r3438126 = r3438101 * r3438125;
        double r3438127 = r3438112 * r3438126;
        double r3438128 = r3438124 - r3438127;
        double r3438129 = -r3438128;
        double r3438130 = r3438101 * r3438112;
        double r3438131 = r3438129 * r3438130;
        double r3438132 = r3438123 + r3438131;
        double r3438133 = r3438103 * r3438132;
        double r3438134 = r3438100 * r3438133;
        double r3438135 = sqrt(r3438134);
        double r3438136 = 1.0865908327746399e+112;
        bool r3438137 = r3438120 <= r3438136;
        double r3438138 = r3438137 ? r3438120 : r3438135;
        double r3438139 = r3438122 ? r3438135 : r3438138;
        return r3438139;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 0.0 or 1.0865908327746399e+112 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 54.8

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

   2. Simplified 46.1

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

   4. Applied add-cube-cbrt 46.2

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

   5. Applied associate-*l* 46.2

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

   7. Applied sub-neg 46.2

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

   8. Applied distribute-lft-in 46.2

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

   9. Applied distribute-lft-in 46.2

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

   10. Simplified 46.1

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

   11. Simplified 41.3

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

   if 0.0 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0865908327746399e+112

   1. Initial program 1.7

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

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0865908327746399 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(t \cdot n + \left(-\left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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*))))))