Average Error: 33.5 → 25.7
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(U* - U\right) + \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right))_* \cdot \left(2 \cdot \left(n \cdot U\right)\right)} \le 5.733892914833802 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{\left((\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(n \cdot 2\right)\right) \cdot U + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{if}\;\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(U* - U\right) + \left(t - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right))_* \cdot \left(2 \cdot \left(n \cdot U\right)\right)} \le 3.7578672918789407 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)} + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left((\left(\frac{n}{Om} \cdot \frac{n}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U* \cdot U\right)\right) + \left(n \cdot \left(t \cdot U\right)\right))_* - \left(\frac{n}{Om} \cdot \frac{n}{Om}\right) \cdot \left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot U\right)\right)\right)}\\ \end{array}\]

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 (/ (/ n (/ Om l)) (/ Om l)) (- U* U) (- t (/ (* 2 l) (/ Om l)))) (* 2 (* n U)))) < 5.733892914833802e-143

    1. Initial program 52.8

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

    3. Applied add-cube-cbrt52.8

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

    4. Applied prod-diff52.8

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

    5. Applied distribute-rgt-in52.8

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

    6. Applied simplify52.7

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

    7. Applied simplify52.7

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

    9. Applied associate-*r*34.6

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

    if 5.733892914833802e-143 < (sqrt (* (fma (/ (/ n (/ Om l)) (/ Om l)) (- U* U) (- t (/ (* 2 l) (/ Om l)))) (* 2 (* n U)))) < 3.7578672918789407e+152

    1. Initial program 8.2

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

    3. Applied add-cube-cbrt8.7

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

    4. Applied prod-diff8.7

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

    5. Applied distribute-rgt-in8.7

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

    6. Applied simplify1.7

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

    7. Applied simplify0.9

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

    9. Applied add-cube-cbrt1.5

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)} \cdot \sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)}\right) \cdot \sqrt[3]{(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_* \cdot \left(\left(n \cdot 2\right) \cdot U\right)}} + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\]

    if 3.7578672918789407e+152 < (sqrt (* (fma (/ (/ n (/ Om l)) (/ Om l)) (- U* U) (- t (/ (* 2 l) (/ Om l)))) (* 2 (* n U))))

    1. Initial program 60.2

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

    2. Taylor expanded around inf 59.1

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

    3. Applied simplify56.1

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +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*))))))