Toniolo and Linder, Equation (13)

Average Error: 34.1 → 28.2

Time: 16.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := \frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\\ \mathbf{if}\;\ell \leq -8.771456429987835 \cdot 10^{+227}:\\ \;\;\;\;\left(\frac{\sqrt{2} \cdot t}{\ell} \cdot \sqrt{\frac{n \cdot U}{t_2}}\right) \cdot -0.5 - \sqrt{n \cdot \left(U \cdot t_2\right)} \cdot t_1\\ \mathbf{elif}\;\ell \leq -8.032622395303857 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\ell \cdot \frac{U* - U}{Om}\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.135898270312147 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.0389359801185104 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U* - U}{\sqrt[3]{Om}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(n \cdot U\right) \cdot t_2}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

↓

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0)))
        (t_2
         (-
          (/ (* n U*) (pow Om 2.0))
          (+ (/ (* n U) (pow Om 2.0)) (* 2.0 (/ 1.0 Om))))))
   (if (<= l -8.771456429987835e+227)
     (-
      (* (* (/ (* (sqrt 2.0) t) l) (sqrt (/ (* n U) t_2))) -0.5)
      (* (sqrt (* n (* U t_2))) t_1))
     (if (<= l -8.032622395303857e-172)
       (sqrt
        (*
         (* n 2.0)
         (* U (+ t (* (/ l Om) (+ (* n (* l (/ (- U* U) Om))) (* l -2.0)))))))
       (if (<= l 2.135898270312147e-97)
         (sqrt
          (*
           (* n 2.0)
           (* U (+ t (* (/ l Om) (+ (* l -2.0) (/ (* n (* l U*)) Om)))))))
         (if (<= l 4.0389359801185104e+172)
           (sqrt
            (*
             (* n 2.0)
             (*
              U
              (+
               t
               (*
                (/ l Om)
                (+
                 (* l -2.0)
                 (*
                  n
                  (*
                   (/ l (* (cbrt Om) (cbrt Om)))
                   (/ (- U* U) (cbrt Om))))))))))
           (* t_1 (sqrt (* (* n U) t_2)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = ((n * U_42_) / pow(Om, 2.0)) - (((n * U) / pow(Om, 2.0)) + (2.0 * (1.0 / Om)));
	double tmp;
	if (l <= -8.771456429987835e+227) {
		tmp = ((((sqrt(2.0) * t) / l) * sqrt((n * U) / t_2)) * -0.5) - (sqrt(n * (U * t_2)) * t_1);
	} else if (l <= -8.032622395303857e-172) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((n * (l * ((U_42_ - U) / Om))) + (l * -2.0))))));
	} else if (l <= 2.135898270312147e-97) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + ((n * (l * U_42_)) / Om))))));
	} else if (l <= 4.0389359801185104e+172) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((l * -2.0) + (n * ((l / (cbrt(Om) * cbrt(Om))) * ((U_42_ - U) / cbrt(Om)))))))));
	} else {
		tmp = t_1 * sqrt((n * U) * t_2);
	}
	return tmp;
}

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 5 regimes

2. if l < -8.77145642998783535e227

   1. Initial program 64.0

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

   2. Simplified 54.9

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

   4. Applied associate-*l*_binary64 56.1

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

   5. Simplified 58.5

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

   6. Taylor expanded around -inf 37.2

      \[\leadsto \color{blue}{-\left(\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{U \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right) + 0.5 \cdot \left(\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{U \cdot n}{\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{U \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}}\right)\right)} \]

   if -8.77145642998783535e227 < l < -8.03262239530385652e-172

   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. Simplified 30.7

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

   4. Applied associate-*l*_binary64 30.3

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

   5. Simplified 30.8

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

   7. Applied *-un-lft-identity_binary64 30.8

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

   8. Applied times-frac_binary64 29.5

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

   if -8.03262239530385652e-172 < l < 2.13589827031214706e-97

   1. Initial program 24.6

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

   2. Simplified 26.9

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

   4. Applied associate-*l*_binary64 28.7

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

   5. Simplified 25.0

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

   6. Taylor expanded around inf 25.2

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

   if 2.13589827031214706e-97 < l < 4.03893598011851e172

   1. Initial program 32.5

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

   2. Simplified 30.7

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

   4. Applied associate-*l*_binary64 29.1

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

   5. Simplified 30.1

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

   7. Applied add-cube-cbrt_binary64 30.1

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

   8. Applied times-frac_binary64 28.0

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

   if 4.03893598011851e172 < l

   1. Initial program 64.0

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

   2. Simplified 51.6

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

   3. Taylor expanded around inf 32.2

      \[\leadsto \color{blue}{\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{U \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(U \cdot n\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.771456429987835 \cdot 10^{+227}:\\ \;\;\;\;\left(\frac{\sqrt{2} \cdot t}{\ell} \cdot \sqrt{\frac{n \cdot U}{\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}}\right) \cdot -0.5 - \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq -8.032622395303857 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\ell \cdot \frac{U* - U}{Om}\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.135898270312147 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.0389359801185104 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U* - U}{\sqrt[3]{Om}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021196 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))