Toniolo and Linder, Equation (13)

Average Error: 34.7 → 26.9

Time: 31.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4.670885049003775 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot U}{\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}}} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell} \cdot -0.5\right) - \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

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 (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      0.0)
   (*
    (sqrt (* 2.0 n))
    (sqrt (* U (+ t (* (/ l Om) (+ (* n (* (/ l Om) (- U* U))) (* l -2.0)))))))
   (if (<=
        (*
         (* (* 2.0 n) U)
         (-
          (- t (* 2.0 (/ (* l l) Om)))
          (* (* n (pow (/ l Om) 2.0)) (- U U*))))
        4.670885049003775e+267)
     (sqrt
      (*
       (* (* 2.0 n) U)
       (+ t (* (/ l Om) (+ (* l -2.0) (* (- U* U) (* n (/ l Om))))))))
     (if (<=
          (*
           (* (* 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) (+ (* l -2.0) (* (/ l Om) (* n (- U* U))))))))
       (-
        (*
         (sqrt (/ (* n U) (+ (* (/ n Om) (- (/ U* Om) (/ U Om))) (/ -2.0 Om))))
         (* (/ (* t (sqrt 2.0)) l) -0.5))
        (*
         (sqrt (* (* n U) (+ (* (/ n Om) (- (/ U* Om) (/ U Om))) (/ -2.0 Om))))
         (* l (sqrt 2.0))))))))

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 ((((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) * sqrt(U * (t + ((l / Om) * ((n * ((l / Om) * (U_42_ - U))) + (l * -2.0)))));
	} else if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 4.670885049003775e+267) {
		tmp = sqrt(((2.0 * n) * U) * (t + ((l / Om) * ((l * -2.0) + ((U_42_ - U) * (n * (l / Om)))))));
	} else if ((((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) * ((l * -2.0) + ((l / Om) * (n * (U_42_ - U))))));
	} else {
		tmp = (sqrt((n * U) / (((n / Om) * ((U_42_ / Om) - (U / Om))) + (-2.0 / Om))) * (((t * sqrt(2.0)) / l) * -0.5)) - (sqrt((n * U) * (((n / Om) * ((U_42_ / Om) - (U / Om))) + (-2.0 / Om))) * (l * sqrt(2.0)));
	}
	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 (*.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 57.5

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

   2. Simplified 52.7

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

   4. Applied associate-*l*_binary64_360 38.1

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

   5. Simplified 38.6

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

   7. Applied sqrt-prod_binary64_435 39.8

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

   8. Simplified 38.8

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

   if 0.0 < (*.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*)))) < 4.67088504900377462e267

   1. Initial program 1.7

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

   2. Simplified 5.2

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

   4. Applied associate-*r*_binary64_359 1.2

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

   if 4.67088504900377462e267 < (*.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 59.7

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

   2. Simplified 50.7

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

   4. Applied sqrt-prod_binary64_435 49.5

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

   if +inf.0 < (*.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 63.4

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

   4. Applied add-cube-cbrt_binary64_454 63.4

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

   5. Applied associate-*l*_binary64_360 63.4

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

   6. Simplified 63.5

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

   7. Taylor expanded around -inf 52.6

      \[\leadsto \color{blue}{-\left(\sqrt{\left(\left(\frac{n \cdot \left(U \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}{{Om}^{2}} + 2 \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{3}}{Om}\right) - \frac{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left(n \cdot U*\right)}{{Om}^{2}}\right) \cdot \left(U \cdot n\right)} \cdot \left(\ell \cdot \sqrt{2}\right) + 0.5 \cdot \left(\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{U \cdot n}{\left(\frac{n \cdot \left(U \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}{{Om}^{2}} + 2 \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{3}}{Om}\right) - \frac{{\left(\sqrt[3]{-1}\right)}^{3} \cdot \left(n \cdot U*\right)}{{Om}^{2}}}}\right)\right)}\]

   8. Simplified 50.2

      \[\leadsto \color{blue}{\sqrt{\frac{U \cdot n}{\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}}} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell} \cdot -0.5\right) - \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}\right)} \cdot \left(\sqrt{2} \cdot \ell\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 26.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4.670885049003775 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{n \cdot U}{\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}}} \cdot \left(\frac{t \cdot \sqrt{2}}{\ell} \cdot -0.5\right) - \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right) + \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021028 
(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*))))))