Toniolo and Linder, Equation (13)

Average Error: 34.5 → 28.8

Time: 22.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\right)\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2.5945646542098083 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + n \cdot \left(n \cdot \left(U \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array}\]

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 (<=
      (*
       (* (* 2.0 n) U)
       (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U* U))))
      0.0)
   (*
    (sqrt 2.0)
    (*
     (sqrt n)
     (sqrt
      (*
       U
       (+
        t
        (- (* n (* (pow (/ l Om) 2.0) (- U* U))) (* l (* 2.0 (/ l Om)))))))))
   (if (<=
        (*
         (* (* 2.0 n) U)
         (+
          (- t (* 2.0 (/ (* l l) Om)))
          (* (* n (pow (/ l Om) 2.0)) (- U* U))))
        2.5945646542098083e+276)
     (sqrt
      (*
       (* (* 2.0 n) U)
       (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U* U)))))
     (sqrt
      (*
       2.0
       (+
        (* n (* U t))
        (*
         n
         (-
          (* n (* U (* (pow (/ l Om) 2.0) (- U* U))))
          (* 2.0 (* U (* l (/ l Om))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * (((double) (l * l)) / Om))))) - ((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U - U_42_))))))))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * (((double) (l * l)) / Om))))) + ((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U_42_ - U)))))))) <= 0.0)) {
		tmp = ((double) (((double) sqrt(2.0)) * ((double) (((double) sqrt(n)) * ((double) sqrt(((double) (U * ((double) (t + ((double) (((double) (n * ((double) (((double) pow((l / Om), 2.0)) * ((double) (U_42_ - U)))))) - ((double) (l * ((double) (2.0 * (l / Om)))))))))))))))));
	} else {
		double tmp_1;
		if ((((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * (((double) (l * l)) / Om))))) + ((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U_42_ - U)))))))) <= 2.5945646542098083e+276)) {
			tmp_1 = ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * (((double) (l * l)) / Om))))) + ((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U_42_ - U))))))))));
		} else {
			tmp_1 = ((double) sqrt(((double) (2.0 * ((double) (((double) (n * ((double) (U * t)))) + ((double) (n * ((double) (((double) (n * ((double) (U * ((double) (((double) pow((l / Om), 2.0)) * ((double) (U_42_ - U)))))))) - ((double) (2.0 * ((double) (U * ((double) (l * (l / Om)))))))))))))))));
		}
		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 3 regimes

2. if (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 0.0

   1. Initial program Error: 57.6 bits

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

   2. Simplified Error: 41.6 bits

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

   4. Applied sqrt-prod Error: 41.7 bits

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

   6. Applied sqrt-prod Error: 41.3 bits

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

   7. Simplified Error: 41.3 bits

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

   if 0.0 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 2.5945646542098083e276

   1. Initial program Error: 1.8 bits

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

   if 2.5945646542098083e276 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))

   1. Initial program Error: 61.0 bits

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

   2. Simplified Error: 54.3 bits

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

   4. Applied distribute-lft-in Error: 54.3 bits

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

   5. Applied distribute-lft-in Error: 54.3 bits

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

   7. Applied sub-neg Error: 54.3 bits

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

   8. Applied distribute-lft-in Error: 54.3 bits

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

   9. Simplified Error: 53.6 bits

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

   10. Simplified Error: 53.6 bits

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

4. Final simplification Error: 28.8 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)\right)}\right)\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2.5945646542098083 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + n \cdot \left(n \cdot \left(U \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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*))))))