Toniolo and Linder, Equation (13)

Average Error: 34.7 → 29.0

Time: 46.2s

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}\;U \leq -2.1113638748749556 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}}\right)}}\\ \mathbf{elif}\;U \leq 9.18875628408121 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt{\sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\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 (<= U -2.1113638748749556e-105)
   (*
    (sqrt
     (sqrt
      (*
       (* U (* 2.0 n))
       (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
    (sqrt
     (sqrt
      (*
       (*
        (* U (* 2.0 n))
        (*
         (cbrt (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))
         (cbrt (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
       (*
        (cbrt
         (*
          (cbrt (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))
          (cbrt
           (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
        (cbrt
         (cbrt
          (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))))))
   (if (<= U 9.18875628408121e+50)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
     (*
      (sqrt
       (sqrt
        (*
         (* U (* 2.0 n))
         (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
      (sqrt
       (*
        (cbrt
         (sqrt
          (*
           (* U (* 2.0 n))
           (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
        (*
         (cbrt
          (sqrt
           (*
            (* U (* 2.0 n))
            (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U)))))))))
         (cbrt
          (sqrt
           (*
            (* U (* 2.0 n))
            (+
             t
             (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U))))))))))))))))

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 (U <= -2.1113638748749556e-105) {
		tmp = sqrt(sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))))) * sqrt(sqrt(((U * (2.0 * n)) * (cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))) * cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))))) * (cbrt(cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))) * cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))))) * cbrt(cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))))))));
	} else if (U <= 9.18875628408121e+50) {
		tmp = sqrt((2.0 * n) * (U * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))))));
	} else {
		tmp = sqrt(sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))))) * sqrt(cbrt(sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))))) * (cbrt(sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U)))))))) * cbrt(sqrt((U * (2.0 * n)) * (t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))))))));
	}
	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 U < -2.11136387487495559e-105

   1. Initial program 30.2

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

   2. Simplified 26.4

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

   4. Applied add-sqr-sqrt_binary64 26.6

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

   6. Applied add-cube-cbrt_binary64 26.6

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

   7. Applied associate-*r*_binary64 26.6

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

   9. Applied add-cube-cbrt_binary64 26.7

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

   10. Applied cbrt-prod_binary64 26.7

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

   if -2.11136387487495559e-105 < U < 9.1887562840812092e50

   1. Initial program 38.2

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

   2. Simplified 34.3

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

   4. Applied associate-*l*_binary64 30.7

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

   if 9.1887562840812092e50 < U

   1. Initial program 29.8

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

   2. Simplified 26.2

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

   4. Applied add-sqr-sqrt_binary64 26.4

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

   6. Applied add-cube-cbrt_binary64 26.7

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

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.1113638748749556 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}}\right)}}\\ \mathbf{elif}\;U \leq 9.18875628408121 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt{\sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020274 
(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*))))))