Average Error: 32.8 → 25.8
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(\frac{\ell}{\frac{Om}{n}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{if}\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(\frac{\ell}{\frac{Om}{n}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \le 4.443906106860638 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(\frac{\ell}{\frac{Om}{n}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \end{array}\]

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 2 regimes

  2. if (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* (* (/ l (/ Om n)) (cbrt (/ l Om))) (pow (cbrt (/ l Om)) 2)) (- U U*)))))) < 0.0 or 4.443906106860638e+124 < (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* (* (/ l (/ Om n)) (cbrt (/ l Om))) (pow (cbrt (/ l Om)) 2)) (- U U*))))))

    1. Initial program 50.1

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

    3. Applied add-sqr-sqrt55.7

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

    4. Applied prod-diff55.7

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

    5. Applied distribute-rgt-in55.7

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

    6. Applied simplify45.4

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

    7. Applied simplify44.7

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \color{blue}{\left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}}\]

    if 0.0 < (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* (* (/ l (/ Om n)) (cbrt (/ l Om))) (pow (cbrt (/ l Om)) 2)) (- U U*)))))) < 4.443906106860638e+124

    1. Initial program 10.4

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

    3. Applied associate-*l*3.5

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

    5. Applied add-cube-cbrt3.6

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

    6. Applied unpow-prod-down3.5

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

    7. Applied associate-*r*1.7

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

    8. Applied simplify1.4

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))