Toniolo and Linder, Equation (13)

Average Error: 34.1 → 27.5

Time: 32.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1970951713919756 \cdot 10^{+152}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1846863934485895 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7567949355391264 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2263965288855882 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\\ \end{array}\]

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 -3.1970951713919756e+152)
   (-
    (*
     (* 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 1.1846863934485895e-96)
     (sqrt
      (*
       (* U (* 2.0 n))
       (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
     (if (<= l 2.7567949355391264e-73)
       (*
        (sqrt (* U (* 2.0 n)))
        (sqrt (+ t (* (/ l Om) (+ (* l -2.0) (* (/ l Om) (* n (- U* U))))))))
       (if (<= l 3.2263965288855882e+156)
         (sqrt
          (*
           (* U (* 2.0 n))
           (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U)))))))
         (*
          (* 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 <= -3.1970951713919756e+152) {
		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 <= 1.1846863934485895e-96) {
		tmp = sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} else if (l <= 2.7567949355391264e-73) {
		tmp = sqrt(U * (2.0 * n)) * sqrt(t + ((l / Om) * ((l * -2.0) + ((l / Om) * (n * (U_42_ - U))))));
	} else if (l <= 3.2263965288855882e+156) {
		tmp = sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U))))));
	} 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 4 regimes

2. if l < -3.19709517139197558e152

   1. Initial program 63.8

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

   2. Simplified 49.0

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

   3. Taylor expanded around -inf 31.8

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

   if -3.19709517139197558e152 < l < 1.18468639344858954e-96 or 2.7567949355391264e-73 < l < 3.2263965288855882e156

   1. Initial program 27.4

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

   2. Simplified 28.5

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

   4. Applied associate-*r*_binary64_359 26.0

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

   if 1.18468639344858954e-96 < l < 2.7567949355391264e-73

   1. Initial program 28.6

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

   2. Simplified 30.4

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

   4. Applied sqrt-prod_binary64_435 42.0

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

   5. Simplified 42.0

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

   6. Simplified 42.0

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

   if 3.2263965288855882e156 < 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 48.6

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

   3. Taylor expanded around inf 32.9

      \[\leadsto \color{blue}{\sqrt{\left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{U \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right) \cdot \left(U \cdot n\right)} \cdot \left(\ell \cdot \sqrt{2}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1970951713919756 \cdot 10^{+152}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.1846863934485895 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.7567949355391264 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 3.2263965288855882 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021013 
(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*))))))