Toniolo and Linder, Equation (13)

Average Error: 34.1 → 29.1

Time: 34.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.30463153096067 \cdot 10^{+73}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.414462230876741 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9399217652603554 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \left(\frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.0898665093457604 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\\ \end{array}\]

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
 (if (<= l -9.30463153096067e+73)
   (-
    (*
     (* l (sqrt 2.0))
     (sqrt
      (*
       n
       (*
        U
        (-
         (/ (* n U*) (pow Om 2.0))
         (+ (/ (* n U) (pow Om 2.0)) (* 2.0 (/ 1.0 Om)))))))))
   (if (<= l -5.414462230876741e-230)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (/ l Om) (+ (* n (/ (* l (- U* U)) Om)) (* l -2.0)))))))
     (if (<= l -1.9399217652603554e-272)
       (sqrt
        (*
         2.0
         (*
          U
          (*
           n
           (-
            t
            (+
             (/ (* U (* n (pow l 2.0))) (pow Om 2.0))
             (* 2.0 (/ (pow l 2.0) Om))))))))
       (if (<= l 2.0898665093457604e+99)
         (sqrt
          (*
           (* U (* 2.0 n))
           (+ t (* (/ l Om) (+ (* l -2.0) (* (- U* U) (* n (/ l Om))))))))
         (*
          (* l (sqrt 2.0))
          (sqrt
           (*
            (* n U)
            (-
             (/ (* n U*) (pow Om 2.0))
             (+ (/ (* n U) (pow Om 2.0)) (* 2.0 (/ 1.0 Om))))))))))))

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 tmp;
	if (l <= -9.30463153096067e+73) {
		tmp = -((l * sqrt(2.0)) * sqrt(n * (U * (((n * U_42_) / pow(Om, 2.0)) - (((n * U) / pow(Om, 2.0)) + (2.0 * (1.0 / Om)))))));
	} else if (l <= -5.414462230876741e-230) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((n * ((l * (U_42_ - U)) / Om)) + (l * -2.0))))));
	} else if (l <= -1.9399217652603554e-272) {
		tmp = sqrt(2.0 * (U * (n * (t - (((U * (n * pow(l, 2.0))) / pow(Om, 2.0)) + (2.0 * (pow(l, 2.0) / Om)))))));
	} else if (l <= 2.0898665093457604e+99) {
		tmp = sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + ((U_42_ - U) * (n * (l / Om)))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * U) * (((n * U_42_) / pow(Om, 2.0)) - (((n * U) / pow(Om, 2.0)) + (2.0 * (1.0 / Om)))));
	}
	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 < -9.30463153096067058e73

   1. Initial program 52.6

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

   2. Simplified 42.7

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

   3. Taylor expanded around -inf 36.7

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

   if -9.30463153096067058e73 < l < -5.41446223087674113e-230

   1. Initial program 26.5

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

   2. Simplified 27.3

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

   4. Applied associate-*l*_binary64_360 28.3

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

   5. Simplified 26.1

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

   if -5.41446223087674113e-230 < l < -1.93992176526035541e-272

   1. Initial program 20.3

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

   2. Simplified 23.7

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

   3. Taylor expanded around 0 31.8

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

   if -1.93992176526035541e-272 < l < 2.08986650934576038e99

   1. Initial program 27.0

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

   2. Simplified 28.2

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

   4. Applied associate-*r*_binary64_359 25.6

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

   if 2.08986650934576038e99 < l

   1. Initial program 55.1

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

   2. Simplified 44.6

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

   3. Taylor expanded around inf 35.9

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

4. Final simplification 29.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.30463153096067 \cdot 10^{+73}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -5.414462230876741 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell \cdot \left(U* - U\right)}{Om} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.9399217652603554 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - \left(\frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.0898665093457604 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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*))))))