Average Error: 13.6 → 9.6
Time: 42.9s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -4.8227378484592 \cdot 10^{-311}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \frac{1}{\frac{\frac{d}{D}}{\frac{M}{2}}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\\ \end{array}\]

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 (/ h l) < -inf.0 or -4.8227378484592e-311 < (/ h l)

    1. Initial program 13.7

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

    2. Initial simplification13.6

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

    4. Applied clear-num13.6

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

    6. Applied *-un-lft-identity13.6

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

    7. Applied div-inv13.6

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

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

    10. Simplified13.7

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

    11. Taylor expanded around 0 6.5

      \[\leadsto \color{blue}{1} \cdot w0\]

    if -inf.0 < (/ h l) < -4.8227378484592e-311

    1. Initial program 13.4

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

    2. Initial simplification13.1

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

    4. Applied clear-num13.1

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

    6. Applied *-un-lft-identity13.1

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

    7. Applied div-inv13.1

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

    8. Applied times-frac13.2

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

    9. Simplified13.2

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

    10. Simplified13.2

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -4.8227378484592 \cdot 10^{-311}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right) \cdot \frac{1}{\frac{\frac{d}{D}}{\frac{M}{2}}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\\ \end{array}\]

Runtime

Time bar (total: 42.9s)Debug logProfile

herbie shell --seed 2018248 +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))))))