Average Error: 32.8 → 24.2
Time: 3.5m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)} \le 9.004677962347198 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left((\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(n \cdot 2\right)\right) \cdot U + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{if}\;\sqrt{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)} \le 4.481056891997075 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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 3 regimes

  2. if (sqrt (+ (* (fma (- U* U) (* (/ l Om) (* n (/ l Om))) (- t (* (/ l Om) (* l 2)))) (* (* n 2) U)) (* (* 0 n) (* U 2)))) < 9.004677962347198e-137

    1. Initial program 51.9

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

    3. Applied add-cube-cbrt52.0

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

    4. Applied prod-diff52.0

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

    5. Applied distribute-rgt-in52.0

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

    6. Applied simplify51.8

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

    7. Applied simplify51.8

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

    9. Applied associate-*r*34.5

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

    if 9.004677962347198e-137 < (sqrt (+ (* (fma (- U* U) (* (/ l Om) (* n (/ l Om))) (- t (* (/ l Om) (* l 2)))) (* (* n 2) U)) (* (* 0 n) (* U 2)))) < 4.481056891997075e+152

    1. Initial program 7.9

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

    3. Applied add-cube-cbrt8.5

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

    4. Applied prod-diff8.5

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

    5. Applied distribute-rgt-in8.5

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

    6. Applied simplify1.5

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

    7. Applied simplify0.8

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

    if 4.481056891997075e+152 < (sqrt (+ (* (fma (- U* U) (* (/ l Om) (* n (/ l Om))) (- t (* (/ l Om) (* l 2)))) (* (* n 2) U)) (* (* 0 n) (* U 2))))

    1. Initial program 60.3

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

    3. Applied sqrt-prod54.3

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed 2018170 +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*))))))