Toniolo and Linder, Equation (13)

Average Error: 34.5 → 27.9

Time: 44.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.704885011833772 \cdot 10^{+180}:\\ \;\;\;\;-\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{U \cdot n}{Om \cdot Om} + \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq -6.804098699339296 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \frac{\frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \left(U* - U\right)}{\sqrt[3]{Om}} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.821733402645745 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{U \cdot n}{Om \cdot Om} + \frac{2}{Om}\right)\right)\right)\right)}\\ \end{array}\]

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 -7.704885011833772e+180)
   (-
    (*
     (sqrt
      (*
       (* U n)
       (- (/ (* n U*) (* Om Om)) (+ (/ (* U n) (* Om Om)) (/ 2.0 Om)))))
     (* l (sqrt 2.0))))
   (if (<= l -6.804098699339296e-204)
     (sqrt
      (*
       (* n 2.0)
       (*
        U
        (+
         t
         (*
          (/ l Om)
          (+
           (* n (/ (* (/ l (* (cbrt Om) (cbrt Om))) (- U* U)) (cbrt Om)))
           (* l -2.0)))))))
     (if (<= l 6.821733402645745e+150)
       (sqrt
        (*
         (* U (* n 2.0))
         (+ t (* (/ l Om) (+ (* l -2.0) (* (- U* U) (* n (/ l Om))))))))
       (*
        l
        (sqrt
         (*
          n
          (*
           U
           (*
            2.0
            (-
             (/ (* n U*) (* Om Om))
             (+ (/ (* U n) (* 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 <= -7.704885011833772e+180) {
		tmp = -(sqrt((U * n) * (((n * U_42_) / (Om * Om)) - (((U * n) / (Om * Om)) + (2.0 / Om)))) * (l * sqrt(2.0)));
	} else if (l <= -6.804098699339296e-204) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((n * (((l / (cbrt(Om) * cbrt(Om))) * (U_42_ - U)) / cbrt(Om))) + (l * -2.0))))));
	} else if (l <= 6.821733402645745e+150) {
		tmp = sqrt((U * (n * 2.0)) * (t + ((l / Om) * ((l * -2.0) + ((U_42_ - U) * (n * (l / Om)))))));
	} else {
		tmp = l * sqrt(n * (U * (2.0 * (((n * U_42_) / (Om * Om)) - (((U * n) / (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 4 regimes

2. if l < -7.704885011833772e180

   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 52.1

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

   3. Taylor expanded around -inf 34.6

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

   4. Simplified 33.0

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

   if -7.704885011833772e180 < l < -6.8040986993392963e-204

   1. Initial program 31.5

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

   2. Simplified 31.0

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

   4. Applied associate-*l*_binary64_360 31.0

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

   5. Simplified 30.4

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

   7. Applied add-cube-cbrt_binary64_454 30.5

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

   8. Applied associate-/r*_binary64_363 30.5

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

   9. Simplified 28.9

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

   if -6.8040986993392963e-204 < l < 6.8217334026457453e150

   1. Initial program 26.9

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

   2. Simplified 28.3

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

   4. Applied associate-*r*_binary64_359 25.5

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

   if 6.8217334026457453e150 < l

   1. Initial program 63.4

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

   2. Simplified 46.7

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

   4. Applied add-cube-cbrt_binary64_454 46.9

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

   5. Simplified 54.3

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

   6. Simplified 54.3

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

   7. Taylor expanded around inf 34.9

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(U \cdot \left(\left(\frac{n \cdot U*}{{Om}^{2}} - \left(\frac{U \cdot n}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right) \cdot {\left(\sqrt[3]{2}\right)}^{3}\right)\right)} \cdot \ell}\]

   8. Simplified 34.3

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

4. Final simplification 27.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.704885011833772 \cdot 10^{+180}:\\ \;\;\;\;-\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{U \cdot n}{Om \cdot Om} + \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq -6.804098699339296 \cdot 10^{-204}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \frac{\frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \left(U* - U\right)}{\sqrt[3]{Om}} + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.821733402645745 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \left(U \cdot \left(2 \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{U \cdot n}{Om \cdot Om} + \frac{2}{Om}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021097 
(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*))))))