Average Error: 13.7 → 10.6
Time: 37.4s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -2.721589799344764 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{\sqrt{(\left(\left(\frac{\frac{1}{2}}{\sqrt[3]{\frac{d}{D}}} \cdot \frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}} \cdot \sqrt{\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}}\right) \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.457553485633167 \cdot 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array}\]

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 3 regimes

  2. if (/ (* M D) (* 2 d)) < -2.721589799344764e-57

    1. Initial program 26.6

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

    2. Initial simplification26.0

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

    4. Applied add-sqr-sqrt26.0

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

    5. Applied sqrt-prod26.1

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

    7. Applied add-cube-cbrt26.1

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

    8. Applied div-inv26.1

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

    9. Applied times-frac26.1

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

    10. Simplified26.1

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

    if -2.721589799344764e-57 < (/ (* M D) (* 2 d)) < 3.457553485633167e-125

    1. Initial program 7.0

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

    2. Initial simplification7.0

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

    3. Taylor expanded around 0 1.9

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

    if 3.457553485633167e-125 < (/ (* M D) (* 2 d))

    1. Initial program 20.8

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -2.721589799344764 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{\sqrt{(\left(\left(\frac{\frac{1}{2}}{\sqrt[3]{\frac{d}{D}}} \cdot \frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}\right) \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}} \cdot \sqrt{\sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}}\right) \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.457553485633167 \cdot 10^{-125}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed 2018346 +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))))))