Average Error: 34.8 → 15.5
Time: 25.7s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U}\right) \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)}^{1.5}\]
\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)}
↓
{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U}\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)}^{1.5}
(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
 (pow
  (*
   (* (cbrt (* 2.0 n)) (cbrt U))
   (cbrt (+ t (* (/ l Om) (+ (* l -2.0) (* n (* (/ l Om) (- U* U))))))))
  1.5))
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_) {
	return pow(((cbrt(2.0 * n) * cbrt(U)) * cbrt(t + ((l / Om) * ((l * -2.0) + (n * ((l / Om) * (U_42_ - U))))))), 1.5);
}

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 34.8

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

  2. Simplified31.2

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

  4. Applied add-cube-cbrt_binary64_44631.5

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

  6. Applied pow3_binary64_49431.5

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

  7. Applied sqrt-pow1_binary64_43131.5

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

  8. Simplified31.5

    \[\leadsto {\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)\right)}\right)}^{\color{blue}{1.5}}\]
  9. Using strategy rm

  10. Applied cbrt-prod_binary64_44224.4

    \[\leadsto {\color{blue}{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)}}^{1.5}\]
  11. Using strategy rm

  12. Applied cbrt-prod_binary64_44215.5

    \[\leadsto {\left(\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U}\right)} \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)}^{1.5}\]

  13. Final simplification15.5

    \[\leadsto {\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{U}\right) \cdot \sqrt[3]{t + \frac{\ell}{Om} \cdot \left(\ell \cdot -2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U* - U\right)\right)\right)}\right)}^{1.5}\]

Reproduce

herbie shell --seed 2020281 
(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*))))))