Average Error: 13.3 → 9.7
Time: 52.2s
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \le 5.578770179921855 \cdot 10^{-16}:\\ \;\;\;\;\left(\sqrt{{\left((\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*\right)}^{\frac{1}{2}}} \cdot \sqrt{{\left((\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \left(\left(\sqrt[3]{\frac{M}{2}} \cdot \sqrt[3]{\frac{M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{M}{2}}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*\right)}^{\frac{1}{2}}}\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

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 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)) < 5.578770179921855e-16

    1. Initial program 8.9

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

    2. Initial simplification9.0

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

    4. Applied pow1/29.0

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

    6. Applied add-sqr-sqrt9.1

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

    8. Applied *-un-lft-identity9.1

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

    9. Applied add-cube-cbrt9.1

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

    10. Applied times-frac9.1

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

    11. Simplified9.1

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

    if 5.578770179921855e-16 < (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))

    1. Initial program 60.0

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

    2. Initial simplification57.9

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

    4. Applied pow1/257.9

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

    5. Taylor expanded around 0 16.7

      \[\leadsto \color{blue}{1} \cdot w0\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \le 5.578770179921855 \cdot 10^{-16}:\\ \;\;\;\;\left(\sqrt{{\left((\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*\right)}^{\frac{1}{2}}} \cdot \sqrt{{\left((\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \left(\left(\sqrt[3]{\frac{M}{2}} \cdot \sqrt[3]{\frac{M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{M}{2}}}{\frac{d}{D}}\right)\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*\right)}^{\frac{1}{2}}}\right) \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Runtime

Time bar (total: 52.2s)Debug logProfile

herbie shell --seed 2018257 +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))))))