Average Error: 13.5 → 9.5
Time: 1.1m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right) \le -1.1694263660327407 \cdot 10^{+297}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D \cdot e^{\log \frac{1}{2}}\right) \cdot \left(\frac{1}{d} \cdot M\right)\right) \cdot \frac{M \cdot \frac{1}{2}}{\frac{d}{h} \cdot \frac{\ell}{D}}}\\ \mathbf{elif}\;\left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right) \le 5.196697411112677 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(\sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}} \cdot \left(\sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*}\\ \end{array}\]

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 3 regimes

  2. if (* (* 1/2 (/ (* M (* D h)) (* l d))) (* (/ -1/2 (/ -1 D)) (/ (/ -1 d) (/ -1 M)))) < -1.1694263660327407e+297

    1. Initial program 51.7

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

    2. Initial simplification52.1

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

    4. Applied add-cube-cbrt52.1

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

    5. Taylor expanded around 0 61.5

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

    6. Simplified54.4

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

    if -1.1694263660327407e+297 < (* (* 1/2 (/ (* M (* D h)) (* l d))) (* (/ -1/2 (/ -1 D)) (/ (/ -1 d) (/ -1 M)))) < 5.196697411112677e-24

    1. Initial program 6.1

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

    2. Initial simplification6.3

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

    4. Applied add-cube-cbrt6.3

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

    5. Taylor expanded around -inf 62.6

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

    6. Simplified1.4

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

    7. Taylor expanded around 0 0.4

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

    if 5.196697411112677e-24 < (* (* 1/2 (/ (* M (* D h)) (* l d))) (* (/ -1/2 (/ -1 D)) (/ (/ -1 d) (/ -1 M))))

    1. Initial program 17.6

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

    2. Initial simplification16.4

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

    4. Applied add-cube-cbrt16.4

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right) \le -1.1694263660327407 \cdot 10^{+297}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(D \cdot e^{\log \frac{1}{2}}\right) \cdot \left(\frac{1}{d} \cdot M\right)\right) \cdot \frac{M \cdot \frac{1}{2}}{\frac{d}{h} \cdot \frac{\ell}{D}}}\\ \mathbf{elif}\;\left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right) \le 5.196697411112677 \cdot 10^{-24}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{-1}{2}}{\frac{-1}{D}} \cdot \frac{\frac{-1}{d}}{\frac{-1}{M}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{(\left(\left(\sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}} \cdot \left(\sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt[3]{\frac{M}{2} \cdot \frac{D}{d}}\right)\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \left(\frac{-h}{\ell}\right) + 1)_*}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018215 +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))))))