Toniolo and Linder, Equation (13)

Average Error: 34.3 → 26.0

Time: 1.2min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 1.992415001715832 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(2 \cdot n\right) \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(\left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \sqrt[3]{U}\right)\right)\right)}\\ \end{array}\]

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 (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (* (/ l Om) (+ (* n (* (/ l Om) (- U* U))) (* l -2.0)))))))
   (if (<=
        (sqrt
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        1.992415001715832e+147)
     (sqrt
      (+
       (* (* (* 2.0 n) U) t)
       (*
        (* (* 2.0 n) U)
        (* (/ l Om) (+ (* l -2.0) (* (- U* U) (* n (/ l Om))))))))
     (if (<=
          (sqrt
           (*
            (* (* 2.0 n) U)
            (-
             (- t (* 2.0 (/ (* l l) Om)))
             (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
          INFINITY)
       (*
        (sqrt (* (* 2.0 n) U))
        (sqrt (+ t (* (/ l Om) (+ (* n (* (/ l Om) (- U* U))) (* l -2.0))))))
       (sqrt
        (+
         (* 2.0 (* t (* n U)))
         (*
          (* 2.0 n)
          (*
           (* (cbrt U) (cbrt U))
           (*
            (+ (* n (* (/ l Om) (- U* U))) (* l -2.0))
            (* (/ l Om) (cbrt U)))))))))))

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 (sqrt(((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((n * ((l / Om) * (U_42_ - U))) + (l * -2.0))))));
	} else if (sqrt(((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 1.992415001715832e+147) {
		tmp = sqrt((((2.0 * n) * U) * t) + (((2.0 * n) * U) * ((l / Om) * ((l * -2.0) + ((U_42_ - U) * (n * (l / Om)))))));
	} else if (sqrt(((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * n) * U) * sqrt(t + ((l / Om) * ((n * ((l / Om) * (U_42_ - U))) + (l * -2.0))));
	} else {
		tmp = sqrt((2.0 * (t * (n * U))) + ((2.0 * n) * ((cbrt(U) * cbrt(U)) * (((n * ((l / Om) * (U_42_ - U))) + (l * -2.0)) * ((l / Om) * cbrt(U))))));
	}
	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 (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 56.2

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

   2. Simplified 56.2

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

   4. Applied associate-*r*_binary64_359 56.1

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

   5. Simplified 56.1

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

   7. Applied associate-*l*_binary64_360 38.1

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

   8. Simplified 38.9

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

   10. Applied *-commutative_binary64_350 38.9

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99241500171583206e147

   1. Initial program 1.9

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

   2. Simplified 5.5

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

   4. Applied associate-*r*_binary64_359 1.4

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

   5. Simplified 1.4

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

   7. Applied distribute-lft-in_binary64_368 1.4

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

   if 1.99241500171583206e147 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 62.7

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

   2. Simplified 52.2

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

   4. Applied associate-*r*_binary64_359 52.0

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

   5. Simplified 52.0

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

   7. Applied sqrt-prod_binary64_435 48.8

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

   8. Simplified 48.8

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

   9. Simplified 49.1

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   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 59.3

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

   4. Applied associate-*r*_binary64_359 58.0

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

   5. Simplified 58.0

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

   7. Applied associate-*l*_binary64_360 52.6

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

   8. Simplified 52.8

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

   10. Applied distribute-rgt-in_binary64_369 52.8

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

   11. Applied distribute-rgt-in_binary64_369 52.8

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

   12. Simplified 53.7

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

   13. Simplified 53.7

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

   15. Applied add-cube-cbrt_binary64_454 53.8

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

   16. Applied associate-*l*_binary64_360 53.8

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

   17. Simplified 49.9

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

4. Final simplification 26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 1.992415001715832 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right) + \left(2 \cdot n\right) \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(\left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right) \cdot \left(\frac{\ell}{Om} \cdot \sqrt[3]{U}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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*))))))