Toniolo and Linder, Equation (13)

Average Error: 34.9 → 28.1

Time: 31.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.704024741456845 \cdot 10^{+171}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.521405414691739 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)}\\ \end{array}\]

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 -6.704024741456845e+171)
   (-
    (*
     (* 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 7.521405414691739e+143)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (/ l Om) (+ (* l -2.0) (* (* n (/ l Om)) (- U* U))))))))
     (*
      (* l (sqrt 2.0))
      (sqrt
       (*
        (* n U)
        (- (/ (* n U*) (* Om Om)) (+ (/ (* n U) (* Om Om)) (/ 2.0 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 <= -6.704024741456845e+171) {
		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 <= 7.521405414691739e+143) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((l * -2.0) + ((n * (l / Om)) * (U_42_ - U)))))));
	} else {
		tmp = (l * sqrt(2.0)) * sqrt((n * U) * (((n * U_42_) / (Om * Om)) - (((n * U) / (Om * Om)) + (2.0 / 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 3 regimes

2. if l < -6.70402474145684512e171

   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 52.5

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

   4. Applied associate-*l*_binary64_360 51.8

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

   5. Simplified 55.8

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

   6. Taylor expanded around -inf 32.3

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

   if -6.70402474145684512e171 < l < 7.52140541469173887e143

   1. Initial program 29.0

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

   2. Simplified 29.3

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

   4. Applied associate-*r*_binary64_359 27.0

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

   5. Simplified 27.0

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

   7. Applied associate-*l*_binary64_360 27.0

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

   if 7.52140541469173887e143 < l

   1. Initial program 61.7

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

   2. Simplified 48.3

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

   3. Taylor expanded around inf 34.2

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

   4. Simplified 34.2

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

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.704024741456845 \cdot 10^{+171}:\\ \;\;\;\;-\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{n \cdot U}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.521405414691739 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{n \cdot U}{Om \cdot Om} + \frac{2}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021064 
(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*))))))