Average Error: 13.4 → 8.9
Time: 2.7m
Precision: 64
Internal Precision: 384
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{1}{\ell}}} \le +\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2}\right)}^{2} \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

  1. Split input into 2 regimes

  2. if (sqrt (- 1 (* (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (* (cbrt (/ 1 l)) (cbrt (/ 1 l)))) (cbrt (/ 1 l))))) < +inf.0

    1. Initial program 13.2

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv13.2

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*10.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied unpow210.0

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

    7. Applied associate-*l*8.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt8.8

      \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \sqrt[3]{\frac{1}{\ell}}\right)}}\]

    10. Applied associate-*r*8.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{1}{\ell}}}}\]

    if +inf.0 < (sqrt (- 1 (* (* (* (/ (* M D) (* 2 d)) (* (/ (* M D) (* 2 d)) h)) (* (cbrt (/ 1 l)) (cbrt (/ 1 l)))) (cbrt (/ 1 l)))))

    1. Initial program 62.9

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv62.9

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*62.9

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied times-frac13.7

      \[\leadsto w0 \cdot \sqrt{1 - \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\]

    7. Applied unpow-prod-down43.8

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left({\left(\frac{M}{2}\right)}^{2} \cdot {\left(\frac{D}{d}\right)}^{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

    8. Applied associate-*l*41.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M}{2}\right)}^{2} \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))