Average Error: 13.2 → 9.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.459217635143209 \cdot 10^{+66}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}}\\ \mathbf{elif}\;\ell \le 1.395011616219124 \cdot 10^{-81}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2}}{\ell} \cdot \left(\frac{D}{\frac{2}{h}} \cdot \frac{M}{d}\right)}\\ \mathbf{elif}\;\ell \le 2.1563372481906454 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}}{\frac{1}{h}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}}\\ \end{array}\]

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 3 regimes

  2. if l < -3.459217635143209e+66 or 2.1563372481906454e-66 < l

    1. Initial program 9.1

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

    2. Initial simplification9.0

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv9.0

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

    5. Applied associate-/r*8.9

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}} \cdot w0\]
    6. Using strategy rm

    7. Applied div-inv8.9

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

    8. Applied *-un-lft-identity8.9

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

    9. Applied times-frac8.9

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

    10. Simplified8.9

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

    11. Simplified7.0

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

    if -3.459217635143209e+66 < l < 1.395011616219124e-81

    1. Initial program 18.3

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

    2. Initial simplification17.6

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv17.6

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

    5. Applied times-frac9.2

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

    6. Simplified11.6

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

    if 1.395011616219124e-81 < l < 2.1563372481906454e-66

    1. Initial program 19.1

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

    2. Initial simplification17.1

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
    3. Using strategy rm

    4. Applied div-inv17.1

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

    5. Applied associate-/r*12.4

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}} \cdot w0\]
    6. Using strategy rm

    7. Applied frac-times20.7

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

    8. Applied associate-/l/21.3

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.459217635143209 \cdot 10^{+66}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}}\\ \mathbf{elif}\;\ell \le 1.395011616219124 \cdot 10^{-81}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2}}{\ell} \cdot \left(\frac{D}{\frac{2}{h}} \cdot \frac{M}{d}\right)}\\ \mathbf{elif}\;\ell \le 2.1563372481906454 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\left(d \cdot 2\right) \cdot \left(d \cdot 2\right)\right) \cdot \ell}}{\frac{1}{h}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D}{d} \cdot \frac{M}{2}}{\frac{\frac{\ell}{h}}{\frac{D}{d} \cdot \frac{M}{2}}}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018285 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))