Toniolo and Linder, Equation (13)

Average Error: 33.2 → 26.6

Time: 37.0s

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}\;\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{\sqrt{\left(n \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\right)\right)\right) \cdot \left(U \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(n \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\right)\right)\right) \cdot \left(U \cdot 2\right)}}\\ \mathbf{elif}\;\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 8.770828183655835 \cdot 10^{+295}:\\ \;\;\;\;\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{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right) \cdot n} \cdot \sqrt{U \cdot 2}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1058056 = 2.0;
        double r1058057 = n;
        double r1058058 = r1058056 * r1058057;
        double r1058059 = U;
        double r1058060 = r1058058 * r1058059;
        double r1058061 = t;
        double r1058062 = l;
        double r1058063 = r1058062 * r1058062;
        double r1058064 = Om;
        double r1058065 = r1058063 / r1058064;
        double r1058066 = r1058056 * r1058065;
        double r1058067 = r1058061 - r1058066;
        double r1058068 = r1058062 / r1058064;
        double r1058069 = pow(r1058068, r1058056);
        double r1058070 = r1058057 * r1058069;
        double r1058071 = U_;
        double r1058072 = r1058059 - r1058071;
        double r1058073 = r1058070 * r1058072;
        double r1058074 = r1058067 - r1058073;
        double r1058075 = r1058060 * r1058074;
        double r1058076 = sqrt(r1058075);
        return r1058076;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1058077 = 2.0;
        double r1058078 = n;
        double r1058079 = r1058077 * r1058078;
        double r1058080 = U;
        double r1058081 = r1058079 * r1058080;
        double r1058082 = t;
        double r1058083 = l;
        double r1058084 = r1058083 * r1058083;
        double r1058085 = Om;
        double r1058086 = r1058084 / r1058085;
        double r1058087 = r1058086 * r1058077;
        double r1058088 = r1058082 - r1058087;
        double r1058089 = r1058083 / r1058085;
        double r1058090 = pow(r1058089, r1058077);
        double r1058091 = r1058078 * r1058090;
        double r1058092 = U_;
        double r1058093 = r1058080 - r1058092;
        double r1058094 = r1058091 * r1058093;
        double r1058095 = r1058088 - r1058094;
        double r1058096 = r1058081 * r1058095;
        double r1058097 = 0.0;
        bool r1058098 = r1058096 <= r1058097;
        double r1058099 = r1058092 - r1058080;
        double r1058100 = r1058085 / r1058083;
        double r1058101 = r1058078 / r1058100;
        double r1058102 = r1058101 / r1058100;
        double r1058103 = -2.0;
        double r1058104 = r1058083 / r1058100;
        double r1058105 = fma(r1058103, r1058104, r1058082);
        double r1058106 = fma(r1058099, r1058102, r1058105);
        double r1058107 = cbrt(r1058106);
        double r1058108 = r1058107 * r1058107;
        double r1058109 = r1058107 * r1058108;
        double r1058110 = r1058078 * r1058109;
        double r1058111 = r1058080 * r1058077;
        double r1058112 = r1058110 * r1058111;
        double r1058113 = sqrt(r1058112);
        double r1058114 = sqrt(r1058113);
        double r1058115 = r1058114 * r1058114;
        double r1058116 = 8.770828183655835e+295;
        bool r1058117 = r1058096 <= r1058116;
        double r1058118 = sqrt(r1058096);
        double r1058119 = r1058106 * r1058078;
        double r1058120 = sqrt(r1058119);
        double r1058121 = sqrt(r1058111);
        double r1058122 = r1058120 * r1058121;
        double r1058123 = r1058117 ? r1058118 : r1058122;
        double r1058124 = r1058098 ? r1058115 : r1058123;
        return r1058124;
}

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 (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 0.0

   1. Initial program 57.5

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

   2. Simplified 37.2

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

   4. Applied add-cube-cbrt 37.4

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

   6. Applied add-sqr-sqrt 37.4

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

   if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 8.770828183655835e+295

   1. Initial program 1.9

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

   if 8.770828183655835e+295 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

   1. Initial program 59.5

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

   2. Simplified 50.5

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

   4. Applied add-cube-cbrt 50.6

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

   6. Applied sqrt-prod 51.4

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

   7. Simplified 51.4

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

4. Final simplification 26.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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{\sqrt{\left(n \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\right)\right)\right) \cdot \left(U \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(n \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\right)\right)\right) \cdot \left(U \cdot 2\right)}}\\ \mathbf{elif}\;\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 8.770828183655835 \cdot 10^{+295}:\\ \;\;\;\;\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{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right) \cdot n} \cdot \sqrt{U \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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*))))))