Toniolo and Linder, Equation (13)

Average Error: 34.5 → 25.6

Time: 46.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t_3 \leq 7.5785213595 \cdot 10^{-314}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, t_1, t\right)\right)}\\ \sqrt{t_4 \cdot t_4} \end{array}\\ \mathbf{elif}\;t_3 \leq 3.85959885474332 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t_3}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2} \cdot \sqrt{t + \frac{\ell}{Om} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

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 (fma l -2.0 (* (- U* U) (* n (/ l Om)))))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 7.5785213595e-314)
     (let* ((t_4 (sqrt (* (* 2.0 n) (* U (fma (/ l Om) t_1 t))))))
       (sqrt (* t_4 t_4)))
     (if (<= t_3 3.85959885474332e+306)
       (sqrt t_3)
       (if (<= t_3 INFINITY)
         (* (sqrt t_2) (sqrt (+ t (* (/ l Om) t_1))))
         (*
          (sqrt
           (*
            n
            (*
             U
             (- (/ (* n U*) (* Om Om)) (+ (/ 2.0 Om) (/ (* n U) (* 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 t_1 = fma(l, -2.0, ((U_42_ - U) * (n * (l / Om))));
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 7.5785213595e-314) {
		double t_4_1 = sqrt((2.0 * n) * (U * fma((l / Om), t_1, t)));
		tmp = sqrt(t_4_1 * t_4_1);
	} else if (t_3 <= 3.85959885474332e+306) {
		tmp = sqrt(t_3);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_2) * sqrt(t + ((l / Om) * t_1));
	} else {
		tmp = sqrt(n * (U * (((n * U_42_) / (Om * Om)) - ((2.0 / Om) + ((n * U) / (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 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*)))) < 7.5785213595e-314

   1. Initial program 55.9

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

   2. Simplified 50.1

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

   3. Applied associate-*l*_binary64 35.4

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

   4. Applied add-sqr-sqrt_binary64 35.4

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

   5. Simplified 35.4

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

   6. Simplified 35.4

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

   if 7.5785213595e-314 < (*.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*)))) < 3.85959885474331978e306

   1. Initial program 1.4

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

   if 3.85959885474331978e306 < (*.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 63.8

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

   2. Simplified 52.4

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

   3. Applied sqrt-prod_binary64 49.1

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

   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 63.6

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

   3. Taylor expanded in l around inf 51.5

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

   4. Simplified 51.5

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

4. Final simplification 25.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 7.5785213595 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right), t\right)\right)}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 3.85959885474332 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021339 
(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*))))))