Average Error: 13.5 → 10.2
Time: 37.1s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -3.4098815546152694 \cdot 10^{-281}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\\ \end{array}\]

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 -3.4098815546152694e-281 < (/ h l)

    1. Initial program 12.8

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

    2. Initial simplification12.6

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

    3. Taylor expanded around 0 6.2

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

    if -inf.0 < (/ h l) < -3.4098815546152694e-281

    1. Initial program 14.2

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

    2. Initial simplification14.2

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

    4. Applied div-inv14.2

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

    5. Applied associate-/r*15.0

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

    6. Taylor expanded around -inf 15.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -3.4098815546152694 \cdot 10^{-281}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\frac{\frac{M}{2}}{d}}{\frac{1}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*}\\ \end{array}\]

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2018258 +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))))))