Toniolo and Linder, Equation (13)

Average Error: 34.5 → 23.9

Time: 55.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.9128334740045034 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{U \cdot \left(n \cdot \left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right) - \frac{n}{Om} \cdot \frac{U}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.452413834329494 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\sqrt[3]{U} \cdot \sqrt[3]{U}} \cdot \sqrt{\sqrt[3]{U} \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right) - \ell \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7140618391835295 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5185536863072047 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{\sqrt[3]{U} \cdot \sqrt[3]{U}} \cdot \sqrt{\sqrt[3]{U} \cdot \left(2 \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) - \ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{2}{Om} + \frac{U \cdot n}{Om \cdot Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

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 (<= l -2.9128334740045034e+150)
   (*
    (sqrt 2.0)
    (*
     l
     (-
      (sqrt
       (*
        U
        (*
         n
         (- (- (* (/ n Om) (/ U* Om)) (/ 2.0 Om)) (* (/ n Om) (/ U Om)))))))))
   (if (<= l -1.452413834329494e-170)
     (*
      (sqrt (* (cbrt U) (cbrt U)))
      (sqrt
       (*
        (cbrt U)
        (*
         (* 2.0 n)
         (+ t (* (/ l Om) (- (* (* n (/ l Om)) (- U* U)) (* l 2.0))))))))
     (if (<= l 1.7140618391835295e-236)
       (sqrt
        (*
         (* 2.0 (* U n))
         (+
          (- t (* 2.0 (/ (* l l) Om)))
          (* n (* (- U* U) (pow (/ l Om) 2.0))))))
       (if (<= l 4.5185536863072047e+182)
         (*
          (sqrt (* (cbrt U) (cbrt U)))
          (sqrt
           (*
            (cbrt U)
            (*
             2.0
             (*
              n
              (+ t (* (/ l Om) (- (* n (* (/ l Om) (- U* U))) (* l 2.0)))))))))
         (*
          (sqrt
           (*
            (* U n)
            (- (* (/ n Om) (/ U* Om)) (+ (/ 2.0 Om) (/ (* U n) (* Om 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 (l <= -2.9128334740045034e+150) {
		tmp = sqrt(2.0) * (l * -sqrt(U * (n * ((((n / Om) * (U_42_ / Om)) - (2.0 / Om)) - ((n / Om) * (U / Om))))));
	} else if (l <= -1.452413834329494e-170) {
		tmp = sqrt(cbrt(U) * cbrt(U)) * sqrt(cbrt(U) * ((2.0 * n) * (t + ((l / Om) * (((n * (l / Om)) * (U_42_ - U)) - (l * 2.0))))));
	} else if (l <= 1.7140618391835295e-236) {
		tmp = sqrt((2.0 * (U * n)) * ((t - (2.0 * ((l * l) / Om))) + (n * ((U_42_ - U) * pow((l / Om), 2.0)))));
	} else if (l <= 4.5185536863072047e+182) {
		tmp = sqrt(cbrt(U) * cbrt(U)) * sqrt(cbrt(U) * (2.0 * (n * (t + ((l / Om) * ((n * ((l / Om) * (U_42_ - U))) - (l * 2.0)))))));
	} else {
		tmp = sqrt((U * n) * (((n / Om) * (U_42_ / Om)) - ((2.0 / Om) + ((U * n) / (Om * 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 5 regimes

2. if l < -2.9128334740045034e150

   1. Initial program 63.2

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

   2. Simplified 46.6

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

   3. Taylor expanded around -inf 35.7

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

   4. Simplified 32.6

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

   if -2.9128334740045034e150 < l < -1.45241383432949406e-170

   1. Initial program 29.5

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

   2. Simplified 26.3

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

   4. Applied add-cube-cbrt_binary64_454 26.6

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

   5. Applied associate-*l*_binary64_360 26.6

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

   6. Simplified 26.6

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

   8. Applied sqrt-prod_binary64_435 21.6

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

   if -1.45241383432949406e-170 < l < 1.71406183918352948e-236

   1. Initial program 24.2

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

   2. Simplified 24.4

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

   if 1.71406183918352948e-236 < l < 4.5185536863072047e182

   1. Initial program 30.3

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

   2. Simplified 26.2

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

   4. Applied add-cube-cbrt_binary64_454 26.5

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

   5. Applied associate-*l*_binary64_360 26.5

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

   6. Simplified 26.5

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

   8. Applied sqrt-prod_binary64_435 20.6

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

   9. Simplified 21.5

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

   if 4.5185536863072047e182 < 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 51.3

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

   4. Applied add-cube-cbrt_binary64_454 51.5

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

   5. Applied associate-*l*_binary64_360 51.5

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

   6. Simplified 51.5

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

   7. Taylor expanded around inf 33.3

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

   8. Simplified 30.8

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{U \cdot n}{Om \cdot Om} + \frac{2}{Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \ell\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9128334740045034 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \left(-\sqrt{U \cdot \left(n \cdot \left(\left(\frac{n}{Om} \cdot \frac{U*}{Om} - \frac{2}{Om}\right) - \frac{n}{Om} \cdot \frac{U}{Om}\right)\right)}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.452413834329494 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\sqrt[3]{U} \cdot \sqrt[3]{U}} \cdot \sqrt{\sqrt[3]{U} \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right) - \ell \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.7140618391835295 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + n \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.5185536863072047 \cdot 10^{+182}:\\ \;\;\;\;\sqrt{\sqrt[3]{U} \cdot \sqrt[3]{U}} \cdot \sqrt{\sqrt[3]{U} \cdot \left(2 \cdot \left(n \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) - \ell \cdot 2\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{2}{Om} + \frac{U \cdot n}{Om \cdot Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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*))))))