Average Error: 13.6 → 7.6
Time: 2.7m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\frac{M}{d} \cdot \left(\left(h \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \le 4.712989610511589 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{1 - \left(\frac{M}{d} \cdot \left(\left(h \cdot D\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\frac{D}{2} \cdot \frac{M}{d}}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (sqrt (- 1 (* (* (/ M d) (* (* h D) 1/2)) (/ (* (/ D 2) (/ M d)) l)))) < 4.712989610511589e+137

    1. Initial program 6.9

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

    3. Applied div-inv6.9

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

    4. Applied associate-*r*3.1

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

    6. Applied unpow23.1

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

    7. Applied associate-*l*1.8

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

    8. Taylor expanded around 0 2.7

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

    9. Applied simplify0.3

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

    if 4.712989610511589e+137 < (sqrt (- 1 (* (* (/ M d) (* (* h D) 1/2)) (/ (* (/ D 2) (/ M d)) l))))

    1. Initial program 44.8

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

    3. Applied div-inv44.8

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

    4. Applied associate-*r*43.3

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

    6. Applied add-cube-cbrt43.4

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

    7. Applied unpow-prod-down43.4

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

    8. Applied associate-*l*41.9

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018195 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))