Toniolo and Linder, Equation (13)

Average Error: 34.5 → 27.8

Time: 22.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - t_1\right)\\ t_4 := t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, t_1\right)\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(t \cdot U\right) - 4 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}\\ \mathbf{elif}\;t_3 \leq 4.707788325681701 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{t_2 \cdot t_4}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2} \cdot \sqrt{t_4}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}\right)\right)}^{0.25}\right)}^{2}\\ \end{array} \]

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 (* (* n (pow (/ l Om) 2.0)) (- U U*)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (- (- t (* 2.0 (/ (* l l) Om))) t_1)))
        (t_4 (- t (fma 2.0 (* l (/ l Om)) t_1))))
   (if (<= t_3 0.0)
     (sqrt (* n (- (* 2.0 (* t U)) (* 4.0 (/ (* (pow l 2.0) U) Om)))))
     (if (<= t_3 4.707788325681701e+300)
       (sqrt (* t_2 t_4))
       (if (<= t_3 INFINITY)
         (* (sqrt t_2) (sqrt t_4))
         (pow
          (pow (* n (- (* 2.0 (* U t)) (* 4.0 (/ (* (* l l) U) Om)))) 0.25)
          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 = (n * pow((l / Om), 2.0)) * (U - U_42_);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) - t_1);
	double t_4 = t - fma(2.0, (l * (l / Om)), t_1);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * ((2.0 * (t * U)) - (4.0 * ((pow(l, 2.0) * U) / Om)))));
	} else if (t_3 <= 4.707788325681701e+300) {
		tmp = sqrt((t_2 * t_4));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_2) * sqrt(t_4);
	} else {
		tmp = pow(pow((n * ((2.0 * (U * t)) - (4.0 * (((l * l) * U) / Om)))), 0.25), 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 57.6

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

   2. Simplified 57.6

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

   3. Taylor expanded in n around 0 42.2

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(t \cdot U\right) - 4 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}} \]

   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*)))) < 4.7077883256817012e300

   1. Initial program 1.6

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

   2. Simplified 1.6

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

   3. Applied egg-rr 1.6

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

   if 4.7077883256817012e300 < (*.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 63.2

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

   2. Simplified 63.2

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

   3. Applied egg-rr 53.8

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

   4. Applied egg-rr 49.8

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

   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 64.0

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

   3. Applied egg-rr 64.0

      \[\leadsto \color{blue}{{\left({\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in n around 0 59.2

      \[\leadsto {\left({\color{blue}{\left(n \cdot \left(2 \cdot \left(t \cdot U\right) - 4 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)\right)}}^{0.25}\right)}^{2} \]

   5. Simplified 59.2

      \[\leadsto {\left({\color{blue}{\left(n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}\right)\right)}}^{0.25}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot \left(t \cdot U\right) - 4 \cdot \frac{{\ell}^{2} \cdot U}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4.707788325681701 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}\right)\right)}^{0.25}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(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*))))))