Toniolo and Linder, Equation (13)

Average Error: 34.7 → 28.4

Time: 23.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ t_2 := -t_1\\ \mathbf{if}\;\ell \leq -3.223783303575266 \cdot 10^{+210}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.3600349946707946 \cdot 10^{+176}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -1.171204455798095 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.6334565220071494 \cdot 10^{-122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 5.253788384023585 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.0262748173296868 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \end{array} \]

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
 (let* ((t_1
         (*
          (sqrt
           (*
            n
            (*
             U
             (- (/ (* n U*) (* Om Om)) (+ (/ 2.0 Om) (/ (* n U) (* Om Om)))))))
          (* l (sqrt 2.0))))
        (t_2 (- t_1)))
   (if (<= l -3.223783303575266e+210)
     t_2
     (let* ((t_3
             (sqrt
              (*
               (* n 2.0)
               (*
                U
                (+
                 t
                 (* (/ l Om) (+ (* n (* (/ l Om) (- U* U))) (* l -2.0)))))))))
       (if (<= l -2.3600349946707946e+176)
         t_3
         (if (<= l -1.171204455798095e+125)
           t_2
           (if (<= l -2.6334565220071494e-122)
             t_3
             (if (<= l 5.253788384023585e-221)
               (sqrt
                (*
                 (* U (* n 2.0))
                 (-
                  (- t (* 2.0 (/ (* l l) Om)))
                  (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
               (if (<= l 1.0262748173296868e+142) t_3 t_1)))))))))

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 t_1 = sqrt(n * (U * (((n * U_42_) / (Om * Om)) - ((2.0 / Om) + ((n * U) / (Om * Om)))))) * (l * sqrt(2.0));
	double t_2 = -t_1;
	double tmp;
	if (l <= -3.223783303575266e+210) {
		tmp = t_2;
	} else {
		double t_3 = sqrt((n * 2.0) * (U * (t + ((l / Om) * ((n * ((l / Om) * (U_42_ - U))) + (l * -2.0))))));
		double tmp_1;
		if (l <= -2.3600349946707946e+176) {
			tmp_1 = t_3;
		} else if (l <= -1.171204455798095e+125) {
			tmp_1 = t_2;
		} else if (l <= -2.6334565220071494e-122) {
			tmp_1 = t_3;
		} else if (l <= 5.253788384023585e-221) {
			tmp_1 = sqrt((U * (n * 2.0)) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))));
		} else if (l <= 1.0262748173296868e+142) {
			tmp_1 = t_3;
		} else {
			tmp_1 = t_1;
		}
		tmp = tmp_1;
	}
	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 < -3.22378330357526614e210 or -2.36003499467079459e176 < l < -1.1712044557980949e125

   1. Initial program 57.2

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

   2. Simplified 46.4

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

   3. Taylor expanded around -inf 36.1

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

   4. Simplified 36.1

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

   if -3.22378330357526614e210 < l < -2.36003499467079459e176 or -1.1712044557980949e125 < l < -2.63345652200714938e-122 or 5.2537883840235853e-221 < l < 1.02627481732969e142

   1. Initial program 31.1

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

   2. Simplified 30.6

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

   4. Applied associate-*l*_binary64 29.6

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

   5. Simplified 29.6

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

   7. Applied *-un-lft-identity_binary64 29.6

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

   8. Applied associate-*l*_binary64 29.6

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

   9. Simplified 27.6

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

   if -2.63345652200714938e-122 < l < 5.2537883840235853e-221

   1. Initial program 25.3

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

   if 1.02627481732969e142 < l

   1. Initial program 61.8

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

   2. Simplified 46.8

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

   3. Taylor expanded around inf 34.5

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

   4. Simplified 34.5

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

4. Final simplification 28.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.223783303575266 \cdot 10^{+210}:\\ \;\;\;\;-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq -2.3600349946707946 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq -1.171204455798095 \cdot 10^{+125}:\\ \;\;\;\;-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq -2.6334565220071494 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 5.253788384023585 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.0262748173296868 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right) + \ell \cdot -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021202 
(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*))))))